Wikipedia - Benutzerbeiträge [de]

Nişancı Ahmed Pascha

2016-09-11T23:59:12Z

Paul August: /* Later years */ Fix name, link

{{Distinguish2|the governor of Egypt (1615–18) [[Nişancı Ahmed Pasha (17th century)]], or the Grand Vizier (1703) [[Kavanoz Ahmed Pasha|Sührablı Kavanoz Nişancı Ahmed Pasha]]}}
{{Ottoman Turkish name|Ahmet|Pasha}}
{{Use dmy dates|date=December 2013}}
{{Infobox Officeholder
| honorific-prefix = [[Nişancı]] · Şehla · [[wikt:kör#Turkish|Kör]] · [[Hajji|Hacı]]
| name = Ahmed
| honorific-suffix = [[Pasha]]
| image =
| imagesize = 250px
| smallimage =
| caption =
| order1 =
| office1 = [[Grand Vizier of the Ottoman Empire]]
| monarch1 = [[Mahmud I]]
| term_start1 = 23 June 1740
| term_end1 = 21 April 1742
| predecessor1 = [[Ivaz Mehmed Pasha]]
| successor1 = [[Hekimoğlu Ali Pasha]]
| order2 =
| office2 = [[Ottoman Governor of Egypt]]
| term_start2 = 1748
| term_end2 = 1751
| predecessor2 = [[Yeğen Ali Pasha]]
| successor2 = [[Seyyid Abdullah Pasha]]
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| governor_general =
| constituency =
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| birth_date =
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| death_date = February 1753
| death_place = [[Aleppo]], [[Aleppo Eyalet]], [[Ottoman Empire]]
| nationality = [[Ottoman Empire|Ottoman]]
| party =
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'''[[Nişancı]] Ahmed [[Pasha]]''' (died February 1753), also called '''Şehla Ahmed Pasha''' or '''Hacı Şehla Ahmed Pasha''' or '''Kör Vezir Ahmed Pasha''' ("Ahmed Pasha the Blind Vizier"), was an [[Ottoman Empire|Ottoman]] [[Grand Vizier of the Ottoman Empire|Grand Vizier]] during the reign of [[Mahmud I]].<ref name = turkbook>İsmail Hâmi Danişmend, Osmanlı Devlet Erkânı, Türkiye Yayınevi, İstanbul, 1971 (Turkish)</ref> He was also the [[Ottoman governor of Egypt]] from 1748 to 1751.<ref name="sicilli">{{citation|author=Mehmet Süreyya|editor1=Nuri Akbayar|editor2=Seyit A. Kahraman|title=Sicill-i Osmanî|url=https://books.google.com/books?id=btElAQAAMAAJ|year=1996|publisher=Türkiye Kültür Bakanlığı and Türkiye Ekonomik ve Toplumsal Tarih Vakfı|location=Beşiktaş, Istanbul|language=Turkish|origyear=1890}}</ref><ref name="Öztuna1994">{{cite book|author=Yılmaz Öztuna|title=Büyük Osmanlı Tarihi: Osmanlı Devleti'nin siyasî, medenî, kültür, teşkilât ve san'at tarihi|url=https://books.google.com/books?id=pBGQMgAACAAJ|volume=10|year=1994|publisher=Ötüken Neşriyat A.S.|language=Turkish|isbn=975-437-141-5|pages=412–416}}</ref><ref name="Crecelius1990">{{cite book|last=Crecelius|first=Daniel|title=Eighteenth Century Egypt: The Arabic Manuscript Sources|url=https://books.google.com/books?id=sVpyAAAAMAAJ|year=1990|publisher=Regina Books|location=Claremont, California|isbn=978-0-941690-42-3}}</ref>

== Early life ==
His family was from Alaiye (now [[Alanya]] in [[Antalya Province]], [[Turkey]]), but Ahmed was born in [[Söke]] (in [[Aydın Eyalet|Aydın Province]], Turkey) to his father Cafer. One of his uncles was a [[vizier]]. He was appointed as the chief stableman ({{lang-tr|imrahor}}). In 1738, he was promoted to be the governor of Aydın Province. In 1742, he returned to [[Constantinople]], the capital. He was appointed as the [[nişancı]] (one of the highest bureaucratic posts). Soon afterwards, he was promoted to be the [[Grand Vizier of the Ottoman Empire|grand vizier]] on 23 June 1740.

He was sometimes called ''Kör Vezir'' ("blind [[vizier]]") because he was somewhat cross-eyed.<ref name="JabartiPhilipp1994-303">{{cite book|author1='Abd al-Rahman Jabarti|author2=Thomas Philipp|author3=Moshe Perlmann|title=Abd Al-Rahmann Al-Jabarti's History of Egypt|url=https://books.google.com/books?id=Nw9hcgAACAAJ|volume=1|year=1994|publisher=Franz Steiner Verlag Stuttgart|page=303}}</ref>

== As Grand Vizier ==
His term in the office was one of the few periods of peace in the history of the [[Ottoman Empire]], as the war against the [[Habsburg Monarchy]] and the [[Russian Empire]] had just ended and [[Nader Shah]] of [[Afsharid Dynasty|Persia]] was occupied in [[Transoxiana]] and [[Daghestan]]. Despite the favorable conditions, Ahmed Pasha was unable to take advantage of the political state of peace and failed to follow his intended program of recovery and reform. Meanwhile, he was accused of dishonesty and indifference to state affairs. He was dismissed from the post on 21 April 1742, and was replaced by the more experienced [[Hekimoğlu Ali Pasha]], who had already once served a term as grand vizier 10 years ago.

== Later years ==
He was exiled to the island of [[Rhodes]]. Soon thereafter, however, he returned to government services. In 1743, he became the governor of the Sanjak of İçel (modern [[Mersin Province]], Turkey) and then the governor of [[Sidon Eyalet]]. After the beginning of the new phase of the war against Persia, he was tasked with commanding ({{lang-tr|serdar}}) the northern portion of the front, where he successfully defended [[Kars]] (in modern Turkey). He then worked as the governor of [[Aleppo Eyalet]] (in modern [[Syria]]) and [[Diyarbekir Eyalet]].

After the [[Treaty of Kerden]], he was appointed the governor of [[Baghdad Eyalet]] in 1747, the governor of [[Egypt Eyalet]] in 1748,<ref name="sicilli" /><ref name="Öztuna1994" /><ref name="Crecelius1990" /> and the governor of [[Adana Eyalet]] in 1751. However, Ahmed Pasha refused this last position in Adana, and in 1752, he returned to his former governorship in Aleppo, where he died in February 1753.<ref>Ayhan Buz: ''Osmanlı Sadrazamları'', Neden Kitap, İstanbul, 2009, ISBN 978-975-254-278-5 pp 227-231</ref>

Contemporaries in Ottoman Egypt described him as a man interested in the sciences and philosophy, but reported that he was disappointed when he discovered that Egypt's famed [[Al-Azhar University]] had ceased to teach about sciences and focused on only religious education.<ref name="Crecelius1990" /><ref name="JabartiPhilipp1994-305">{{cite book|author1='Abd al-Rahman Jabarti|author2=Thomas Philipp|author3=Moshe Perlmann|title=Abd Al-Rahmann Al-Jabarti's History of Egypt|url=https://books.google.com/books?id=Nw9hcgAACAAJ|volume=1|year=1994|publisher=Franz Steiner Verlag Stuttgart|page=305}}</ref> Reportedly, he found even the most educated Egyptians and [[ulema]] to be illiterate in basic mathematics, spending most of his time with the few he found that shared his interest in the sciences.<ref name="Crecelius1990" /><ref name="JabartiPhilipp1994-305" />

== See also ==
* [[List of Ottoman grand viziers]]
* [[List of Ottoman governors of Egypt]]

== References ==
{{reflist}}

{{s-start}}
{{s-off}}
{{succession box |
title = [[Grand Vizier of the Ottoman Empire]] |
before = [[Ivaz Mehmed Pasha]] |
after = [[Hekimoğlu Ali Pasha]] |
years = 22 June 1740 – 23 April 1742
}}
{{succession box |
title = [[Ottoman Governor of Egypt]] |
before = [[Yeğen Ali Pasha]] |
after = [[Seyyid Abdullah Pasha]] |
years = 1748–1751
}}
{{s-end}}

{{Grand Viziers of Ottoman Empire}}

{{DEFAULTSORT:Ahmed Pasha, Nisanci}}
[[Category:18th-century Ottoman Grand Viziers]]
[[Category:Pashas]]
[[Category:Nişancı]]
[[Category:1753 deaths]]
[[Category:Year of birth unknown]]
[[Category:Ottoman governors of Egypt]]

Benutzer:JonskiC/Kleinste-Quadrate-Schätzung

2012-11-16T01:32:11Z

Paul August: ce

{{Regression bar}}
The method of '''least squares''' is a standard approach to the approximate solution of [[overdetermined system]]s, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.

The most important application is in [[curve fitting|data fitting]]. The best fit in the least-squares sense minimizes the sum of squared [[errors and residuals in statistics|residuals]], a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the [[independent variable]] (the 'x' variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting [[errors-in-variables models]] may be considered instead of that for least squares.

Least squares problems fall into two categories: linear or [[ordinary least squares]] and [[non-linear least squares]], depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical [[regression analysis]]; it has a closed-form solution. A closed-form solution (or [[closed-form expression]]) is any formula that can be evaluated in a finite number of standard operations. The non-linear problem has no closed-form solution and is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, thus the core calculation is similar in both cases.

Least squares corresponds to the [[maximum likelihood]] criterion if the experimental errors have a [[normal distribution]] and can also be derived as a [[method of moments (statistics)|method of moments]] estimator.

The following discussion is mostly presented in terms of [[linear]] functions but the use of least-squares is valid and practical for more general families of functions. Also, by iteratively applying local [[quadratic approximation]] to the likelihood (through the [[Fisher information]]), the least-squares method may be used to fit a [[generalized linear model]].

For the topic of approximating a function by a sum of others using an objective function based on squared distances, see [[least squares (function approximation)]].

[[File:Linear least squares2.png|right|thumb|The result of fitting a set of data points with a quadratic function.]]

The least-squares method is usually credited to [[Carl Friedrich Gauss]] (1794)<ref name=brertscher>{{cite book|author = Bretscher, Otto|title = Linear Algebra With Applications, 3rd ed.|publisher = Prentice Hall|year = 1995|location = Upper Saddle River NJ}}</ref>, but it was first published by [[Adrien-Marie Legendre]].

==History==
===Context===

The method of least squares grew out of the fields of [[astronomy]] and [[geodesy]] as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the [[Age of Exploration]]. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas where before sailors had relied on land sightings to determine the positions of their ships.

The method was the culmination of several advances that took place during the course of the eighteenth century<ref name=stigler>{{cite book
| author = Stigler, Stephen M.
| title = The History of Statistics: The Measurement of Uncertainty Before 1900
| publisher = Belknap Press of Harvard University Press
| year = 1986
| location = Cambridge, MA
| isbn = 0-674-40340-1
}}</ref>:

*The combination of different observations taken under the ''same'' conditions contrary to simply trying one's best to observe and record a single observation accurately. This approach was notably used by [[Tobias Mayer]] while studying the [[libration]]s of the moon.
*The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, perhaps first expressed by [[Roger Cotes]].
*The combination of different observations taken under ''different'' conditions as notably performed by [[Roger Joseph Boscovich]] in his work on the shape of the earth and [[Pierre-Simon Laplace]] in his work in explaining the differences in motion of [[Jupiter]] and [[Saturn]].
*The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved, developed by Laplace in his ''Method of Least Squares''.

===The method===

[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|[[Carl Friedrich Gauss]]]]

[[Carl Friedrich Gauss]] is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen.<ref name=brertscher/> [[Adrien-Marie Legendre|Legendre]] was the first to publish the method, however.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid [[Ceres (asteroid)|Ceres]]. On January 1, 1801, the Italian astronomer [[Giuseppe Piazzi]] discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, astronomers desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated [[Kepler's laws of planetary motion|Kepler's nonlinear equations]] of planetary motion. The only predictions that successfully allowed Hungarian astronomer [[Franz Xaver von Zach]] to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium''.
In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. This result is known as the [[Gauss–Markov theorem]].

The idea of least-squares analysis was also independently formulated by the Frenchman [[Adrien-Marie Legendre]] in 1805 and the American [[Robert Adrain]] in 1808. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares.<ref>{{cite journal|doi=10.1111/j.1751-5823.1998.tb00406.x|author=J. Aldrich|year=1998|title=Doing Least Squares: Perspectives from Gauss and Yule|journal=International Statistical Review|volume=66|issue=1|pages= 61–81}}</ref>

==Problem statement==
{{Unreferenced section|date=February 2012}}
The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of ''n'' points (data pairs) <math>(x_i,y_i)\!</math>, ''i'' = 1, ..., ''n'', where <math>x_i\!</math> is an [[independent variable]] and <math>y_i\!</math> is a [[dependent variable]] whose value is found by observation. The model function has the form <math>f(x,\beta)</math>, where the ''m'' adjustable parameters are held in the vector <math>\boldsymbol \beta</math>. The goal is to find the parameter values for the model which "best" fits the data. The least squares method finds its optimum when the sum, ''S'', of squared residuals
:<math>S=\sum_{i=1}^{n}{r_i}^2</math>
is a minimum. A [[errors and residuals in statistics|residual]] is defined as the difference between the actual value of the dependent variable and the value predicted by the model.

:<math>r_i=y_i-f(x_i,\boldsymbol \beta)</math>.

An example of a model is that of the straight line. Denoting the intercept as <math>\beta_0</math> and the slope as <math>\beta_1</math>, the model function is given by <math>f(x,\boldsymbol \beta)=\beta_0+\beta_1 x</math>. See [[Linear_least_squares_(mathematics)#Motivational_example|linear least squares]] for a fully worked out example of this model.

A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, ''x'' and ''z'', say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point. 

== Limitations ==
This regression formulation considers only residuals in the dependent variable. There are two rather different contexts in which different implications apply:
*Regression for prediction. Here a model is fitted to provide a prediction rule for application in a similar situation to which the data used for fitting apply. Here the dependent variables corresponding to such future application would be subject to the same types of observation error as those in the data used for fitting. It is therefore logically consistent to use the least-squares prediction rule for such data.
*Regression for fitting a "true relationship". In standard [[regression analysis]], that leads to fitting by least squares, there is an implicit assumption that errors in the [[independent variable]] are zero or strictly controlled so as to be negligible. When errors in the [[independent variable]] are non-negligible, [[Errors-in-variables models|models of measurement error]] can be used; such methods can lead to [[parameter estimation|parameter estimates]], [[hypothesis testing]] and [[confidence interval]]s that take into account the presence of observation errors in the independent variables.{{Citation needed|date=February 2012}} An alternative approach is to fit a model by [[total least squares]]; this can be viewed as taking a pragmatic approach to balancing the effects of the different sources of error in formulating an objective function for use in model-fitting.

==Solving the least squares problem==
{{Unreferenced section|date=February 2012}}
The [[Maxima and minima|minimum]] of the sum of squares is found by setting the [[gradient]] to zero. Since the model contains ''m'' parameters there are ''m'' gradient equations.

:<math>\frac{\partial S}{\partial \beta_j}=2\sum_i r_i\frac{\partial r_i}{\partial \beta_j}=0,\ j=1,\ldots,m</math>

and since <math>r_i=y_i-f(x_i,\boldsymbol \beta)\,</math> the gradient equations become

:<math>-2\sum_i r_i\frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j}=0,\ j=1,\ldots,m</math>.

The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.

=== Linear least squares ===

{{main|Linear_least_squares_(mathematics)|l1=Linear least squares}}

A regression model is a linear one when the model comprises a [[linear combination]] of the parameters, i.e.,

:<math> f(x_i, \beta) = \sum_{j = 1}^{m} \beta_j \phi_j(x_{i})</math>

where the coefficients, <math>\phi_{j}</math>, are functions of <math> x_{i} </math>.

Letting

:<math> X_{ij}= \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j}= \phi_j(x_{i}) . \, </math>

we can then see that in that case the least square estimate (or estimator, in the context of a random sample), <math> \boldsymbol \beta</math> is given by

:<math> \boldsymbol{\hat\beta} =( X ^TX)^{-1}X^{T}\boldsymbol y.</math>

For a derivation of this estimate see [[Linear least squares (mathematics)]].

=== Non-linear least squares ===

{{main|Non-linear least squares}}

There is no closed-form solution to a non-linear least squares problem. Instead, numerical algorithms are used to find the value of the parameters <math>\beta</math> which minimize the objective. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation.
:<math>{\beta_j}^{k+1}={\beta_j}^k+\Delta \beta_j</math>
''k'' is an iteration number and the vector of increments, <math>\Delta \beta_j\,</math> is known as the shift vector. In some commonly used algorithms, at each iteration the model may be linearized by approximation to a first-order [[Taylor series]] expansion about <math> \boldsymbol \beta^k\!</math>

:<math>
\begin{align}
f(x_i,\boldsymbol \beta) & = f^k(x_i,\boldsymbol \beta) +\sum_j \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} \left(\beta_j-{\beta_j}^k \right) \\
& = f^k(x_i,\boldsymbol \beta) +\sum_j J_{ij} \Delta\beta_j.
\end{align}
</math>

The [[Jacobian matrix and determinant|Jacobian]], '''J''', is a function of constants, the independent variable ''and'' the parameters, so it changes from one iteration to the next. The residuals are given by

:<math>r_i=y_i- f^k(x_i,\boldsymbol \beta)- \sum_{j=1}^{m} J_{ij}\Delta\beta_j=\Delta y_i- \sum_{j=1}^{m} J_{ij}\Delta\beta_j</math>.

To minimize the sum of squares of <math>r_i</math>, the gradient equation is set to zero and solved for <math> \Delta \beta_j\!</math>

:<math>-2\sum_{i=1}^{n}J_{ij} \left( \Delta y_i-\sum_{j=1}^{m} J_{ij}\Delta \beta_j \right)=0</math>

which, on rearrangement, become ''m'' simultaneous linear equations, the '''normal equations'''.

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} J_{ij}J_{ik}\Delta \beta_k=\sum_{i=1}^{n} J_{ij}\Delta y_i \qquad (j=1,\ldots,m)\,</math>

The normal equations are written in matrix notation as

:<math>\mathbf{\left(J^TJ\right)\Delta \boldsymbol \beta=J^T\Delta y}.\,</math>


These are the defining equations of the [[Gauss–Newton algorithm]].

=== Differences between linear and non-linear least squares ===

* The model function, ''f'', in LLSQ (linear least squares) is a linear combination of parameters of the form <math>f = X_{i1}\beta_1 + X_{i2}\beta_2 +\cdots</math> The model may represent a straight line, a parabola or any other linear combination of functions. In NLLSQ (non-linear least squares) the parameters appear as functions, such as <math>\beta^2, e^{\beta x}</math> and so forth. If the derivatives <math>\partial f /\partial \beta_j</math> are either constant or depend only on the values of the independent variable, the model is linear in the parameters. Otherwise the model is non-linear.
*Algorithms for finding the solution to a NLLSQ problem require initial values for the parameters, LLSQ does not.
*Like LLSQ, solution algorithms for NLLSQ often require that the Jacobian be calculated. Analytical expressions for the partial derivatives can be complicated. If analytical expressions are impossible to obtain either the partial derivatives must be calculated by numerical approximation or an estimate must be made of the Jacobian.
*In NLLSQ non-convergence (failure of the algorithm to find a minimum) is a common phenomenon whereas the LLSQ is globally concave so non-convergence is not an issue.
*NLLSQ is usually an iterative process. The iterative process has to be terminated when a convergence criterion is satisfied. LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the [[Gauss–Seidel]] method.
*In LLSQ the solution is unique, but in NLLSQ there may be multiple minima in the sum of squares.
*Under the condition that the errors are uncorrelated with the predictor variables, LLSQ yields unbiased estimates, but even under that condition NLLSQ estimates are generally biased.
These differences must be considered whenever the solution to a non-linear least squares problem is being sought.

==Least squares, regression analysis and statistics==
{{Unreferenced section|date=February 2012}}

The methods of least squares and [[regression analysis]] are conceptually different. However, the method of least squares is often used to generate estimators and other statistics in regression analysis.

Consider a simple example drawn from physics. A spring should obey [[Hooke's law]] which states that the extension of a spring is proportional to the force, ''F'', applied to it.
:<math>f(F_i,k)=kF_i\!</math>
constitutes the model, where ''F'' is the independent variable. To estimate the [[force constant]], ''k'', a series of ''n'' measurements with different forces will produce a set of data, <math>(F_i, y_i), i=1,n\!</math>, where ''yi'' is a measured spring extension. Each experimental observation will contain some error. If we denote this error <math>\varepsilon</math>, we may specify an empirical model for our observations,

: <math> y_i = kF_i + \varepsilon_i. \, </math>

There are many methods we might use to estimate the unknown parameter ''k''. Noting that the ''n'' equations in the ''m'' variables in our data comprise an [[overdetermined system]] with one unknown and ''n'' equations, we may choose to estimate ''k'' using least squares. The sum of squares to be minimized is

:<math> S = \sum_{i=1}^{n} \left(y_i - kF_i\right)^2. </math>

The least squares estimate of the force constant, ''k'', is given by

:<math>\hat k=\frac{\sum_i F_i y_i}{\sum_i {F_i}^2}.</math>

Here it is assumed that application of the force '''''causes''''' the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.

In regression analysis the researcher specifies an empirical model. For example, a very common model is the straight line model which is used to test if there is a linear relationship between dependent and independent variable. If a linear relationship is found to exist, the variables are said to be [[correlated]]. However, [[Correlation_does_not_imply_causation|correlation does not prove causation]], as both variables may be correlated with other, hidden, variables, or the dependent variable may "reverse" cause the independent variables, or the variables may be otherwise spuriously correlated. For example, suppose there is a correlation between deaths by drowning and the volume of ice cream sales at a particular beach. Yet, both the number of people going swimming and the volume of ice cream sales increase as the weather gets hotter, and presumably the number of deaths by drowning is correlated with the number of people going swimming. Perhaps an increase in swimmers causes both the other variables to increase.

In order to make statistical tests on the results it is necessary to make assumptions about the nature of the experimental errors. A common (but not necessary) assumption is that the errors belong to a [[Normal distribution]]. The [[central limit theorem]] supports the idea that this is a good approximation in many cases.
* The [[Gauss–Markov theorem]]. In a linear model in which the errors have [[expected value|expectation]] zero conditional on the independent variables, are [[uncorrelated]] and have equal [[variance]]s, the best linear [[unbiased]] estimator of any linear combination of the observations, is its least-squares estimator. "Best" means that the least squares estimators of the parameters have minimum variance. The assumption of equal variance is valid when the errors all belong to the same distribution.
*In a linear model, if the errors belong to a [[Normal distribution]] the least squares estimators are also the [[maximum likelihood estimator]]s.

However, if the errors are not normally distributed, a [[central limit theorem]] often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large. For this reason, given the important property that the error mean is independent of the independent variables, the distribution of the error term is not an important issue in regression analysis. Specifically, it is not typically important whether the error term follows a normal distribution.

In a least squares calculation with unit weights, or in linear regression, the variance on the ''j''th parameter,
denoted <math>\text{var}(\hat{\beta}_j)</math>, is usually estimated with

:<math>\text{var}(\hat{\beta}_j)= \sigma^2\left( \left[X^TX\right]^{-1}\right)_{jj} \approx \frac{S}{n-m}\left( \left[X^TX\right]^{-1}\right)_{jj},</math>
where the true residual variance σ2 is replaced by an estimate based on the minimised value of the sum of squares objective function ''S''. The denominator, ''n-m'', is the [[Degrees of freedom (statistics)|statistical degrees of freedom]]; see [[Degrees of freedom (statistics)#Effective degrees of freedom|effective degrees of freedom]] for generalizations.

[[Confidence limits]] can be found if the [[probability distribution]] of the parameters is known, or an asymptotic approximation is made, or assumed. Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed. The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed. Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.

==Weighted least squares==
{{see also|Weighted mean|Linear least squares (mathematics)#Weighted linear least squares}}

A special case of [[Generalized least squares]] called '''weighted least squares''' occurs when all the off-diagonal entries of ''Ω'' (the correlation matrix of the residuals) are 0.

The expressions given above are based on the implicit assumption that the errors are uncorrelated with each other and with the independent variables and have equal variance. The [[Gauss–Markov theorem]] shows that, when this is so, <math>\hat{\boldsymbol{\beta}}</math> is a [[best linear unbiased estimator]] (BLUE). If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted. [[Alexander Aitken|Aitken]] showed that when a weighted sum of squared residuals is minimized, <math>\hat{\boldsymbol{\beta}}</math> is BLUE if each weight is equal to the reciprocal of the variance of the measurement.
:<math> S = \sum_{i=1}^{n} W_{ii}{r_i}^2,\qquad W_{ii}=\frac{1}{{\sigma_i}^2} </math>
The gradient equations for this sum of squares are

:<math>-2\sum_i W_{ii}\frac{\partial f(x_i,\boldsymbol {\beta})}{\partial \beta_j} r_i=0,\qquad j=1,\ldots,n</math>

which, in a linear least squares system give the modified normal equations,

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} X_{ij}W_{ii}X_{ik}\hat{ \beta}_k=\sum_{i=1}^{n} X_{ij}W_{ii}y_i, \qquad j=1,\ldots,m\,.</math>

When the observational errors are uncorrelated and the weight matrix, '''W''', is diagonal, these may be written as

:<math>\mathbf{\left(X^TWX\right)\hat {\boldsymbol {\beta}}=X^TWy}.</math>

If the errors are correlated, the resulting estimator is BLUE if the weight matrix is equal to the inverse of the [[variance-covariance matrix]] of the observations.

When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as <math>\mathbf{w_{ii}}=\sqrt \mathbf{W_{ii}}</math>. The normal equations can then be written as

:<math>\mathbf{\left(X'^TX'\right)\hat{\boldsymbol{\beta}}=X'^Ty'}\,</math>

where

: <math>\mathbf{X'}=\mathbf{wX}, \mathbf{y'}=\mathbf{wy}.\,</math>

For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.

:<math>\mathbf{\left(J^TWJ\right)\boldsymbol \Delta \beta=J^TW \boldsymbol\Delta y}.\,</math>

Note that for empirical tests, the appropriate '''W''' is not known for sure and must be
estimated. For this [[Feasible Generalized Least Squares]] (FGLS) techniques may be used.

==Relationship to principal components==

The first [[Principal component analysis|principal component]] about the mean of a set of points can be represented by that line which most closely approaches the data points (as measured by squared distance of closest approach, i.e. perpendicular to the line). In contrast, linear least squares tries to minimize the distance in the <math>y</math> direction only. Thus, although the two use a similar error metric, linear least squares is a method that treats one dimension of the data preferentially, while PCA treats all dimensions equally.

== Regularized versions ==
===Tikhonov regularization===
{{Main|Tikhonov regularization}}
In some contexts a [[Regularization (machine learning)|regularized]] version of the least squares solution may be preferable. [[Tikhonov regularization]] (or [[ridge regression]]) adds a constraint that <math>\|\beta\|^2</math>, the [[L2-norm|L2-norm]] of the parameter vector, is no greater than a given value. Equivalently, it may solve an unconstrained minimization of the least-squares penalty with <math>\alpha\|\beta\|^2</math> added, where <math>\alpha</math> is a constant (this is the [[Lagrange multipliers|Lagrangian]] form of the constrained problem). In a [[Bayesian statistics|Bayesian]] context, this is equivalent to placing a zero-mean [[normal distribution|normally distributed]] [[prior distribution|prior]] on the parameter vector.

===LASSO method===
An alternative [[Regularization (machine learning)|regularized]] version of least squares is ''LASSO'' (least absolute shrinkage and selection operator), which uses the constraint that <math>\|\beta\|^1</math>, the [[L1-norm|L1-norm]] of the parameter vector, is no greater than a given value. (As above, this is equivalent to an unconstrained minimization of the least-squares penalty with <math>\alpha\|\beta\|^1</math> added.) In a [[Bayesian statistics|Bayesian]] context, this is equivalent to placing a zero-mean [[Laplace distribution|Laplace]] [[prior distribution]] on the parameter vector.

One of the prime differences between LASSO and ridge regression is that in ridge regression, as the penalty is increased, all parameters are reduced while still remaining non-zero, while in LASSO, increasing the penalty will cause more and more of the parameters to be driven to zero.

This problem may be solved using [[quadratic programming]] or more general [[convex optimization]] methods, as well as by specific algorithms such as the [[least angle regression]] algorithm. The L1-regularized formulation is useful in some contexts due to its tendency to prefer solutions with fewer nonzero parameter values, effectively reducing the number of variables upon which the given solution is dependent.<ref>{{cite journal|author=Tibshirani, R. |year=1996|title=Regression shrinkage and selection via the lasso |journal=[[Journal of the Royal Statistical Society]], Series B|volume= 58|issue= 1| pages =267–288}}</ref> For this reason, the LASSO and its variants are fundamental to the field of [[compressed sensing]]. An extension of this approach is [[elastic net regularization]].

==See also==
* [[Gauss–Markov theorem|Best linear unbiased estimator]] (BLUE)
* [[Best linear unbiased prediction]] (BLUP)
* [[Gauss-Markov theorem]]
* [[L2 norm|''L''2 norm]]
* [[Least absolute deviation]]
* [[Measurement uncertainty]]
* [[Quadratic loss function]]
* [[Root mean square]]
* [[Squared deviations]]

==Notes==

<references />

==References==

*{{cite book|author=Å. Björck|isbn=978-0-89871-360-2|title=Numerical Methods for Least Squares Problems|publisher=SIAM|year=1996|url=http://www.ec-securehost.com/SIAM/ot51.html}}
*{{cite book| author=C.R. Rao, H. Toutenburg, A. Fieger, C. Heumann, T. Nittner and S. Scheid| title=Linear Models: Least Squares and Alternatives| series=Springer Series in Statistics|year=1999}}
*{{cite book|author=T. Kariya and H. Kurata |title=Generalized Least Squares|publisher= Wiley|year= 2004}}
*{{cite book|author=J. Wolberg|title=Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments|publisher= Springer|year=2005|isbn=3-540-25674-1}}
*{{cite book|author=T. Strutz| title=Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond) |publisher=Vieweg+Teubner | year=2010 | isbn= 978-3-8348-1022-9}}

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Benutzer:JonskiC/Kleinste-Quadrate-Schätzung

2012-04-18T20:58:08Z

Paul August: Undid revision 488062376 by 98.253.228.150 (talk) no caps here

{{Regression bar}}
The method of '''least squares''' is a standard approach to the approximate solution of [[overdetermined system]]s, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.

The most important application is in [[curve fitting|data fitting]]. The best fit in the least-squares sense minimizes the sum of squared [[errors and residuals in statistics|residuals]], a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the [[independent variable]] (the 'x' variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting [[errors-in-variables models]] may be considered instead of that for least squares.

Least squares problems fall into two categories: linear or [[ordinary least squares]] and [[non-linear least squares]], depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical [[regression analysis]]; it has a closed-form solution. A closed-form solution (or [[closed-form expression]]) is any formula that can be evaluated in a finite number of standard operations. The non-linear problem has no closed-form solution and is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, thus the core calculation is similar in both cases.

The least-squares method was first described by [[Carl Friedrich Gauss]] around 1794.<ref name=brertscher>{{cite book|author = Bretscher, Otto|title = Linear Algebra With Applications, 3rd ed.|publisher = Prentice Hall|year = 1995|location = Upper Saddle River NJ}}</ref> Least squares corresponds to the [[maximum likelihood]] criterion if the experimental errors have a [[normal distribution]] and can also be derived as a [[method of moments (statistics)|method of moments]] estimator.

The following discussion is mostly presented in terms of [[linear]] functions but the use of least-squares is valid and practical for more general families of functions. Also, by iteratively applying local [[quadratic approximation]] to the likelihood (through the [[Fisher information]]), the least-squares method may be used to fit a [[generalized linear model]].

For the topic of approximating a function by a sum of others using an objective function based on squared distances, see [[least squares (function approximation)]].

[[Image:Linear least squares2.png|right|thumb|The result of fitting a set of data points with a quadratic function.]]

==History==
===Context===

The method of least squares grew out of the fields of [[astronomy]] and [[geodesy]] as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the [[Age of Exploration]]. The accurate description of the behavior of celestial bodies was key to enabling ships to sail in open seas where before sailors had relied on land sightings to determine the positions of their ships.

The method was the culmination of several advances that took place during the course of the eighteenth century<ref name=stigler>{{cite book
| author = Stigler, Stephen M.
| title = The History of Statistics: The Measurement of Uncertainty Before 1900
| publisher = Belknap Press of Harvard University Press
| year = 1986
| location = Cambridge, MA
| isbn = 0674403401
}}</ref>:

*The combination of different observations taken under the ''same'' conditions contrary to simply trying one's best to observe and record a single observation accurately. This approach was notably used by [[Tobias Mayer]] while studying the [[libration]]s of the moon.
*The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, perhaps first expressed by [[Roger Cotes]].
*The combination of different observations taken under ''different'' conditions as notably performed by [[Roger Joseph Boscovich]] in his work on the shape of the earth and [[Pierre-Simon Laplace]] in his work in explaining the differences in motion of [[Jupiter]] and [[Saturn]].
*The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved, developed by Laplace in his ''Method of Least Squares''.

===The method===

[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|[[Carl Friedrich Gauss]]]]

[[Carl Friedrich Gauss]] is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen.<ref name=brertscher/> [[Adrien-Marie Legendre|Legendre]] was the first to publish the method, however.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid [[Ceres (asteroid)|Ceres]]. On January 1, 1801, the Italian astronomer [[Giuseppe Piazzi]] discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, astronomers desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated [[Kepler's laws of planetary motion|Kepler's nonlinear equations]] of planetary motion. The only predictions that successfully allowed Hungarian astronomer [[Franz Xaver von Zach]] to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium''.
In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. This result is known as the [[Gauss–Markov theorem]].

The idea of least-squares analysis was also independently formulated by the Frenchman [[Adrien-Marie Legendre]] in 1805 and the American [[Robert Adrain]] in 1808. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares.<ref>{{cite journal|doi=10.1111/j.1751-5823.1998.tb00406.x|author=J. Aldrich|year=1998|title=Doing Least Squares: Perspectives from Gauss and Yule|journal=International Statistical Review|volume=66|issue=1|pages= 61–81}}</ref>

==Problem statement==
{{Unreferenced section|date=February 2012}}
The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of ''n'' points (data pairs) <math>(x_i,y_i)\!</math>, ''i'' = 1, ..., ''n'', where <math>x_i\!</math> is an [[independent variable]] and <math>y_i\!</math> is a [[dependent variable]] whose value is found by observation. The model function has the form <math>f(x,\beta)</math>, where the ''m'' adjustable parameters are held in the vector <math>\boldsymbol \beta</math>. The goal is to find the parameter values for the model which "best" fits the data. The least squares method finds its optimum when the sum, ''S'', of squared residuals
:<math>S=\sum_{i=1}^{n}{r_i}^2</math>
is a minimum. A [[errors and residuals in statistics|residual]] is defined as the difference between the actual value of the dependent variable and the value predicted by the model.

:<math>r_i=y_i-f(x_i,\boldsymbol \beta)</math>.

An example of a model is that of the straight line. Denoting the intercept as <math>\beta_0</math> and the slope as <math>\beta_1</math>, the model function is given by <math>f(x,\boldsymbol \beta)=\beta_0+\beta_1 x</math>. See [[Linear_least_squares_(mathematics)#Motivational_example|linear least squares]] for a fully worked out example of this model.

A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, ''x'' and ''z'', say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point. 

== Limitations ==
This regression formulation considers only residuals in the dependent variable. There are two rather different contexts in which different implications apply:
*Regression for prediction. Here a model is fitted to provide a prediction rule for application in a similar situation to which the data used for fitting apply. Here the dependent variables corresponding to such future application would be subject to the same types of observation error as those in the data used for fitting. It is therefore logically consistent to use the least-squares prediction rule for such data.
*Regression for fitting a "true relationship". In standard [[regression analysis]], that leads to fitting by least squares, there is an implicit assumption that errors in the [[independent variable]] are zero or strictly controlled so as to be negligible. When errors in the [[independent variable]] are non-negligible, [[Errors-in-variables models|models of measurement error]] can be used; such methods can lead to [[parameter estimation|parameter estimates]], [[hypothesis testing]] and [[confidence interval]]s that take into account the presence of observation errors in the independent variables.{{Citation needed|date=February 2012}} An alternative approach is to fit a model by [[total least squares]]; this can be viewed as taking a pragmatic approach to balancing the effects of the different sources of error in formulating an objective function for use in model-fitting.

==Solving the least squares problem==
{{Unreferenced section|date=February 2012}}
The [[Maxima and minima|minimum]] of the sum of squares is found by setting the [[gradient]] to zero. Since the model contains ''m'' parameters there are ''m'' gradient equations.

:<math>\frac{\partial S}{\partial \beta_j}=2\sum_i r_i\frac{\partial r_i}{\partial \beta_j}=0,\ j=1,\ldots,m</math>

and since <math>r_i=y_i-f(x_i,\boldsymbol \beta)\,</math> the gradient equations become

:<math>-2\sum_i \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} r_i=0,\ j=1,\ldots,m</math>.

The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.

=== Linear least squares ===

{{main|Linear_least_squares_(mathematics)|l1=Linear least squares}}

A regression model is a linear one when the model comprises a [[linear combination]] of the parameters, i.e.,

:<math> f(x_i, \beta) = \sum_{j = 1}^{m} \beta_j \phi_j(x_{i})</math>

where the coefficients, <math>\phi_{j}</math>, are functions of <math> x_{i} </math>.

Letting

:<math> X_{ij}= \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j}= \phi_j(x_{i}) . \, </math>

we can then see that in that case the least square estimate (or estimator, in the context of a random sample), <math> \boldsymbol \beta</math> is given by

:<math> \boldsymbol{\hat\beta} =( X ^TX)^{-1}X^{T}\boldsymbol y.</math>

For a derivation of this estimate see [[Linear least squares (mathematics)]].

=== Non-linear least squares ===

{{main|Non-linear least squares}}

There is no closed-form solution to a non-linear least squares problem. Instead, numerical algorithms are used to find the value of the parameters <math>\beta</math> which minimize the objective. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation.
:<math>{\beta_j}^{k+1}={\beta_j}^k+\Delta \beta_j</math>
''k'' is an iteration number and the vector of increments, <math>\Delta \beta_j\,</math> is known as the shift vector. In some commonly used algorithms, at each iteration the model may be linearized by approximation to a first-order [[Taylor series]] expansion about <math> \boldsymbol \beta^k\!</math>

:<math>
\begin{align}
f(x_i,\boldsymbol \beta) & = f^k(x_i,\boldsymbol \beta) +\sum_j \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} \left(\beta_j-{\beta_j}^k \right) \\
& = f^k(x_i,\boldsymbol \beta) +\sum_j J_{ij} \Delta\beta_j.
\end{align}
</math>

The [[Jacobian matrix and determinant|Jacobian]], '''J''', is a function of constants, the independent variable ''and'' the parameters, so it changes from one iteration to the next. The residuals are given by

:<math>r_i=y_i- f^k(x_i,\boldsymbol \beta)- \sum_{j=1}^{m} J_{ij}\Delta\beta_j=\Delta y_i- \sum_{j=1}^{m} J_{ij}\Delta\beta_j</math>.

To minimize the sum of squares of <math>r_i</math>, the gradient equation is set to zero and solved for <math> \Delta \beta_j\!</math>

:<math>-2\sum_{i=1}^{n}J_{ij} \left( \Delta y_i-\sum_{j=1}^{m} J_{ij}\Delta \beta_j \right)=0</math>

which, on rearrangement, become ''m'' simultaneous linear equations, the '''normal equations'''.

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} J_{ij}J_{ik}\Delta \beta_k=\sum_{i=1}^{n} J_{ij}\Delta y_i \qquad (j=1,\ldots,m)\,</math>

The normal equations are written in matrix notation as

:<math>\mathbf{\left(J^TJ\right)\Delta \boldsymbol \beta=J^T\Delta y}.\,</math>


These are the defining equations of the [[Gauss–Newton algorithm]].

=== Differences between linear and non-linear least squares ===

* The model function, ''f'', in LLSQ (linear least squares) is a linear combination of parameters of the form <math>f = X_{i1}\beta_1 + X_{i2}\beta_2 +\cdots</math> The model may represent a straight line, a parabola or any other linear combination of functions. In NLLSQ (non-linear least squares) the parameters appear as functions, such as <math>\beta^2, e^{\beta x}</math> and so forth. If the derivatives <math>\partial f /\partial \beta_j</math> are either constant or depend only on the values of the independent variable, the model is linear in the parameters. Otherwise the model is non-linear.
*Algorithms for finding the solution to a NLLSQ problem require initial values for the parameters, LLSQ does not.
*Like LLSQ, solution algorithms for NLLSQ often require that the Jacobian be calculated. Analytical expressions for the partial derivatives can be complicated. If analytical expressions are impossible to obtain either the partial derivatives must be calculated by numerical approximation or an estimate must be made of the Jacobian.
*In NLLSQ non-convergence (failure of the algorithm to find a minimum) is a common phenomenon whereas the LLSQ is globally concave so non-convergence is not an issue.
*NLLSQ is usually an iterative process. The iterative process has to be terminated when a convergence criterion is satisfied. LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the [[Gauss–Seidel]] method.
*In LLSQ the solution is unique, but in NLLSQ there may be multiple minima in the sum of squares.
*Under the condition that the errors are uncorrelated with the predictor variables, LLSQ yields unbiased estimates, but even under that condition NLLSQ estimates are generally biased.
These differences must be considered whenever the solution to a non-linear least squares problem is being sought.

==Least squares, regression analysis and statistics==
{{Unreferenced section|date=February 2012}}

The methods of least squares and [[regression analysis]] are conceptually different. However, the method of least squares is often used to generate estimators and other statistics in regression analysis.

Consider a simple example drawn from physics. A spring should obey [[Hooke's law]] which states that the extension of a spring is proportional to the force, ''F'', applied to it.
:<math>f(F_i,k)=kF_i\!</math>
constitutes the model, where ''F'' is the independent variable. To estimate the [[force constant]], ''k'', a series of ''n'' measurements with different forces will produce a set of data, <math>(F_i, y_i), i=1,n\!</math>, where ''yi'' is a measured spring extension. Each experimental observation will contain some error. If we denote this error <math>\varepsilon</math>, we may specify an empirical model for our observations,

: <math> y_i = kF_i + \varepsilon_i. \, </math>

There are many methods we might use to estimate the unknown parameter ''k''. Noting that the ''n'' equations in the ''m'' variables in our data comprise an [[overdetermined system]] with one unknown and ''n'' equations, we may choose to estimate ''k'' using least squares. The sum of squares to be minimized is

:<math> S = \sum_{i=1}^{n} \left(y_i - kF_i\right)^2. </math>

The least squares estimate of the force constant, ''k'', is given by

:<math>\hat k=\frac{\sum_i F_i y_i}{\sum_i {F_i}^2}.</math>

Here it is assumed that application of the force '''''causes''''' the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.

In regression analysis the researcher specifies an empirical model. For example, a very common model is the straight line model which is used to test if there is a linear relationship between dependent and independent variable. If a linear relationship is found to exist, the variables are said to be [[correlated]]. However, [[Correlation_does_not_imply_causation|correlation does not prove causation]], as both variables may be correlated with other, hidden, variables, or the dependent variable may "reverse" cause the independent variables, or the variables may be otherwise spuriously correlated. For example, suppose there is a correlation between deaths by drowning and the volume of ice cream sales at a particular beach. Yet, both the number of people going swimming and the volume of ice cream sales increase as the weather gets hotter, and presumably the number of deaths by drowning is correlated with the number of people going swimming. Perhaps an increase in swimmers causes both the other variables to increase.

In order to make statistical tests on the results it is necessary to make assumptions about the nature of the experimental errors. A common (but not necessary) assumption is that the errors belong to a [[Normal distribution]]. The [[central limit theorem]] supports the idea that this is a good approximation in many cases.
* The [[Gauss–Markov theorem]]. In a linear model in which the errors have [[expected value|expectation]] zero conditional on the independent variables, are [[uncorrelated]] and have equal [[variance]]s, the best linear [[unbiased]] estimator of any linear combination of the observations, is its least-squares estimator. "Best" means that the least squares estimators of the parameters have minimum variance. The assumption of equal variance is valid when the errors all belong to the same distribution.
*In a linear model, if the errors belong to a [[Normal distribution]] the least squares estimators are also the [[maximum likelihood estimator]]s.

However, if the errors are not normally distributed, a [[central limit theorem]] often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large. For this reason, given the important property that the error mean is independent of the independent variables, the distribution of the error term is not an important issue in regression analysis. Specifically, it is not typically important whether the error term follows a normal distribution.

In a least squares calculation with unit weights, or in linear regression, the variance on the ''j''th parameter,
denoted <math>\text{var}(\hat{\beta}_j)</math>, is usually estimated with

:<math>\text{var}(\hat{\beta}_j)= \sigma^2\left( \left[X^TX\right]^{-1}\right)_{jj} \approx \frac{S}{n-m}\left( \left[X^TX\right]^{-1}\right)_{jj},</math>
where the true residual variance σ2 is replaced by an estimate based on the minimised value of the sum of squares objective function ''S''. The denominator, ''n-m'', is the [[Degrees of freedom (statistics)|statistical degrees of freedom]]; see [[Degrees of freedom (statistics)#Effective degrees of freedom|effective degrees of freedom]] for generalizations.

[[Confidence limits]] can be found if the [[probability distribution]] of the parameters is known, or an asymptotic approximation is made, or assumed. Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed. The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed. Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.

==Weighted least squares==
{{see also|Weighted mean|Linear least squares (mathematics)#Weighted linear least squares}}

The expressions given above are based on the implicit assumption that the errors are uncorrelated with each other and with the independent variables and have equal variance. The [[Gauss–Markov theorem]] shows that, when this is so, <math>\hat{\boldsymbol{\beta}}</math> is a [[best linear unbiased estimator]] (BLUE). If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted. [[Alexander Aitken|Aitken]] showed that when a weighted sum of squared residuals is minimized, <math>\hat{\boldsymbol{\beta}}</math> is BLUE if each weight is equal to the reciprocal of the variance of the measurement.
:<math> S = \sum_{i=1}^{n} W_{ii}{r_i}^2,\qquad W_{ii}=\frac{1}{{\sigma_i}^2} </math>
The gradient equations for this sum of squares are

:<math>-2\sum_i W_{ii}\frac{\partial f(x_i,\boldsymbol {\beta})}{\partial \beta_j} r_i=0,\qquad j=1,\ldots,n</math>

which, in a linear least squares system give the modified normal equations,

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} X_{ij}W_{ii}X_{ik}\hat{ \beta}_k=\sum_{i=1}^{n} X_{ij}W_{ii}y_i, \qquad j=1,\ldots,m\,.</math>

When the observational errors are uncorrelated and the weight matrix, '''W''', is diagonal, these may be written as

:<math>\mathbf{\left(X^TWX\right)\hat {\boldsymbol {\beta}}=X^TWy}.</math>

If the errors are correlated, the resulting estimator is BLUE if the weight matrix is equal to the inverse of the [[variance-covariance matrix]] of the observations.

When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as <math>\mathbf{w_{ii}}=\sqrt \mathbf{W_{ii}}</math>. The normal equations can then be written as

:<math>\mathbf{\left(X'^TX'\right)\hat{\boldsymbol{\beta}}=X'^Ty'}\,</math>

where

: <math>\mathbf{X'}=\mathbf{wX}, \mathbf{y'}=\mathbf{wy}.\,</math>

For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.

:<math>\mathbf{\left(J^TWJ\right)\boldsymbol \Delta \beta=J^TW \boldsymbol\Delta y}.\,</math>

Note that for empirical tests, the appropriate '''W''' is not known for sure and must be
estimated. For this [[Feasible Generalized Least Squares]] (FGLS) techniques may be used.

==Relationship to principal components==

The first [[Principal component analysis|principal component]] about the mean of a set of points can be represented by that line which most closely approaches the data points (as measured by squared distance of closest approach, i.e. perpendicular to the line). In contrast, linear least squares tries to minimize the distance in the <math>y</math> direction only. Thus, although the two use a similar error metric, linear least squares is a method that treats one dimension of the data preferentially, while PCA treats all dimensions equally.

==Tikhonov regularization==
In some contexts a [[Regularization (machine learning)|regularized]] version of the least squares solution may be preferable. [[Tikhonov regularization]] (or [[ridge regression]]) adds a constraint that <math>\|\beta\|^2</math>, the [[L2-norm|L2-norm]] of the parameter vector, is no greater than a given value. Equivalently, it may solve an unconstrained minimization of the least-squares penalty with <math>\alpha\|\beta\|^2</math> added, where <math>\alpha</math> is a constant (this is the [[Lagrange multipliers|Lagrangian]] form of the constrained problem). In a [[Bayesian statistics|Bayesian]] context, this is equivalent to placing a zero-mean [[normal distribution|normally-distributed]] [[prior distribution|prior]] on the parameter vector.

==LASSO method==
An alternative [[Regularization (machine learning)|regularized]] version of least squares is ''LASSO'' (least absolute shrinkage and selection operator), which uses the constraint that <math>\|\beta\|^1</math>, the [[L1-norm|L1-norm]] of the parameter vector, is no greater than a given value. (As above, this is equivalent to an unconstrained minimization of the least-squares penalty with <math>\alpha\|\beta\|^1</math> added.) In a [[Bayesian statistics|Bayesian]] context, this is equivalent to placing a zero-mean [[Laplace distribution|Laplace]] [[prior distribution]] on the parameter vector.

One of the prime differences between LASSO and ridge regression is that in ridge regression, as the penalty is increased, all parameters are reduced while still remaining non-zero, while in LASSO, increasing the penalty will cause more and more of the parameters to be driven to zero.

This problem may be solved using [[quadratic programming]] or more general [[convex optimization]] methods, as well as by specific algorithms such as the [[least angle regression]] algorithm. The L1-regularized formulation is useful in some contexts due to its tendency to prefer solutions with fewer nonzero parameter values, effectively reducing the number of variables upon which the given solution is dependent.<ref>{{cite journal|author=Tibshirani, R. |year=1996|title=Regression shrinkage and selection via the lasso |journal=[[Journal of the Royal Statistical Society]], Series B|volume= 58|issue= 1| pages =267–288}}</ref> For this reason, the LASSO and its variants are fundamental to the field of [[compressed sensing]]. An extension of this approach is [[elastic net regularization]].

==See also==
* [[Gauss–Markov theorem|Best linear unbiased estimator]] (BLUE)
* [[Best linear unbiased prediction]] (BLUP)
* [[Gauss-Markov theorem]]
* [[L2 norm|''L''2 norm]]
* [[Least absolute deviation]]
* [[Measurement uncertainty]]
* [[Quadratic loss function]]
* [[Root mean square]]
* [[Squared deviations]]

==Notes==

<references />

==References==

*{{cite book|author=Å. Björck|isbn=978-0-898713-60-2|title=Numerical Methods for Least Squares Problems|publisher=SIAM|year=1996|url=http://www.ec-securehost.com/SIAM/ot51.html}}
*{{cite book| author=C.R. Rao, H. Toutenburg, A. Fieger, C. Heumann, T. Nittner and S. Scheid| title=Linear Models: Least Squares and Alternatives| series=Springer Series in Statistics|year=1999}}
*{{cite book|author=T. Kariya and H. Kurata |title=Generalized Least Squares|publisher= Wiley|year= 2004}}
*{{cite book|author=J. Wolberg|title=Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments|publisher= Springer|year=2005|isbn=3540256741}}
*{{cite book|author=T. Strutz| title=Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond) |publisher=Vieweg+Teubner | year=2010 | isbn= 978-3-8348-1022-9}}

{{Least Squares and Regression Analysis|state=expanded}}
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[[Category:Regression analysis]]
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[[Category:Mathematical and quantitative methods (economics)]]
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Fjodor Michailowitsch Dostojewski

2011-12-05T00:07:18Z

Paul August: Reverted edits by 66.195.118.243 (talk) to last version by Goustien

{{Redirect2|Dostoyevsky|Dostoevsky|other uses|Dostoyevsky (disambiguation)}}
{{Eastern Slavic name|Mikhaylovich|Dostoyevsky}}
{{Infobox writer 
|name = Fyodor Dostoyevsky
|image = Dostoevsky.jpg
|caption = 1879
|birth_name = Fyodor Mikhaylovich Dostoyevsky
|birth_date = {{Birth date|1821|11|11}}
|birth_place = [[Moscow]], [[Russian Empire]]
|death_date = {{Death date and age|1881|2|9|1821|11|11}}
|death_place = [[Saint Petersburg]], Russian Empire
|occupation = [[Novelist]], [[short story]] writer, [[essay]]ist
|language = [[Russian language|Russian]]
|nationality = [[Russians|Russian]]
|period = 1846–1881
|genre =
|subject =
|movement =
|religion = [[Russian Orthodox]]
|notableworks = ''[[Notes from Underground]]'' ''[[Crime and Punishment]]'' ''[[The Idiot]]'' ''[[The Brothers Karamazov]]''
|spouse = Mariya Dmitriyevna Isayeva (1857–64) [her death]
[[Anna Grigoryevna Snitkina]] (1867–1881) [his death]
|children = Sofiya (1868), [[Lyubov Dostoyevskaya|Lyubov]] (1869—1926), Fyodor (1871–1922), Alexei (1875—1878)
|relatives =
|signature = Fyodor Dostoyevsky Signature.svg
}}
'''Fyodor Mikhaylovich Dostoyevsky''' ({{lang-rus|Фёдор Михайлович Достоевский|p=ˈfʲodər mʲɪˈxajləvʲɪtɕ dəstɐˈjefskʲɪj|a=ru-Dostoevsky.ogg}}; November 11, 1821 – February 9, 1881<ref>[[Old Style and New Style dates|Old Style date]] October 30, 1821 – January 29, 1881.</ref>) was a [[Russia]]n writer of [[novel]]s, [[short story|short stories]] and [[essay]]s.<ref name="pravoslavye.org.ua">[http://www.pravoslavye.org.ua/index.php?r_type=article&action=fullinfo&id=13375 Ukrainian origin of Dostoyevsky (Українське коріння Достоєвського)]</ref> He is best known for his novels ''[[Crime and Punishment]]'', ''[[The Idiot]]'' and ''[[The Brothers Karamazov]]''.

His name has been variously transcribed in English, his first name sometimes being rendered as Theodore. This is because, before the post-revolutionary [[Reforms_of_Russian_orthography#The_post-revolution_reform|orthographic reform]] which, amongst other things, replaced the cyrillic letter Ѳ ('th') with the cyrillic letter Ф ('f'), Dostoyevsky's name was written Ѳеодоръ (Theodor) Михайловичъ Достоевскій.

Dostoyevsky's literary works explore human psychology in the troubled political, social and spiritual context of 19th-century Russian society. With the embittered voice of the anonymous "underground man", Dostoyevsky wrote '' [[Notes from Underground]]'' (1864), which has been called the "best overture for [[existentialism]] ever written" by [[Walter Kaufmann (philosopher)|Walter Kaufmann]].<ref>Existentialism: from Dostoyevsky to Sartre, ed. Walter Kaufmann, Penguin Books, 1989 ISBN 0452009308 p. 12</ref> He is often acknowledged by critics as one of the greatest and most prominent psychologists in [[world literature]].<ref name="BritannicaRussianLit">{{Cite web|url=http://www.britannica.com/EBchecked/topic/513793/Russian-literature|publisher=Encyclopedia Britannica|accessdate=2008-04-11|title=Russian literature|quote=Dostoyevsky, who is generally regarded as one of the supreme psychologists in world literature, sought to demonstrate the compatibility of Christianity with the deepest truths of the psyche.}}</ref>

==Biography==
===Early life and studies===
[[File:Wki Dostoyevsky Street 2 Moscow Mariinsky Hospital.jpg|thumb|240px|Mariinsky Hospital in Moscow, Dostoyevsky's birthplace]]

Dostoyevsky's father Mikhail and grandfather, Andrey, were born in modern central [[Ukraine]].<ref>[http://www.scribd.com/doc/56070816/1/ORIGIN-OF-THE-DOSTOYEVSKY-FAMILY ORIGIN OF THE DOSTOYEVSKY FAMILY ... become priests in Ukraine.]</ref><ref>[http://books.google.hr/books?id=mDKphT8_XLsC&pg=PA3&lpg=PA3&dq=dostoevsky+family+comes+from+ukraine&source=bl&ots=E6D5zSXZm3&sig=qPHtXSuAujHQu9l9vg4M-2huDps&hl=hr&ei=VJK2TriUO-L64QT43JD7Aw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBgQ6AEwAA#v=onepage&q&f=false ''Dostoevsky: his life and work'', Princeton University.]</ref> Mikhail was a doctor and a devout Christian, who practised at the Mariinsky Hospital for the Poor in Moscow.

Dostoyevsky was born in Moscow to Mikhail and Maria Dostoyevsky, the second of seven children.<ref>''The Best Short Stories of Dostoevsky'' Translated with an Introduction by David Magarshack. New York: The Modern Library, Random House; 1971.</ref> The family lived in a small apartment in the Mariinsky Hospital grounds. The hospital was located in one of the city's worst areas near a cemetery for criminals, a lunatic asylum, and an orphanage for abandoned infants. This urban landscape made a lasting impression on the young Dostoyevsky, whose interest in and compassion for the poor, oppressed and tormented was apparent in his life and works. Although it was forbidden by his parents, Dostoyevsky liked to wander out to the hospital garden, where the patients sat to catch a glimpse of the sun. The young Dostoyevsky appreciated spending time with these patients and listening to their stories.{{cn|date=November 2011}}

Stories of Dostoyevsky's father's despotic treatment of his children may be tempered by records of his care for his children and their upbringing. After returning home from work, the father would take a nap while his children, ordered to keep absolutely silent, stood by in shifts and swatted the flies that came near his head. But he was also careful to send his children to private schools where they would not be beaten. In the opinion of the biographer Joseph Frank, the father figure in ''[[The Brothers Karamazov]]'' is not based on Dostoyevsky's own father. Letters and personal accounts demonstrate that they had a fairly loving relationship.

[[File:Dostoevskij 1847.jpg|thumb|upright|The young Dostoyevsky, in an 1847 portrait by [[Konstiantyn Trutovsky|Trutovsky]]]]

In 1837 shortly after his mother died of [[tuberculosis]] Dostoyevsky and his brother were sent to St Petersburg to attend the [[Military engineering-technical university|Nikolayev Military Engineering Institute]], now called the [[Military engineering-technical university|Military Engineering-Technical University]].<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]]</ref>

Fyodor Dostoyevsky's father died in 1839. Though it has never been proven, it is believed by some that he was murdered by his own [[serf]]s.<ref>[http://worldebooklibrary.com/eBooks/Coradella_Collegiate_Bookshelf_Collection/Dostoevsky-notesfromtheunderground.pdf Notes from the Underground] Coradella Collegita Bookshelf edition, ''About the Author''.</ref> According to one account, the serfs became enraged during one of his drunken fits of violence, and after restraining him, poured [[vodka]] into his mouth until he drowned. A similar account appears in ''Notes from Underground''. Another story holds that Mikhail died of natural causes, and a neighboring landowner invented the story of his murder so that he might buy the estate at a cheaper price. Some, like Sigmund Freud in his 1928 article, "[[Dostoevsky and Parricide]]", have argued that his father's personality had influenced the character of Fyodor Pavlovich Karamazov, the "wicked and sentimental buffoon", father of the main characters in his 1880 novel ''[[The Brothers Karamazov]]'', but such claims fail to withstand scrutiny.

From the age of nine Dostoyevsky suffered sporadically from [[epilepsy]] throughout his life <ref>[http://www.epilepsy.com/epilepsy/famous_writers.html Epilepsy.com] Famous authors with epilepsy.</ref> and his experiences are thought<ref>Dostoyevsky, Fyodor, Richard Pevear, and Larissa Volokhonsky. The Idiot. New York: Vintage, 2001. Print. Introduction pp. xix</ref> to have formed the basis for his description of Prince Myshkin's epilepsy in his novel ''[[The Idiot]]'' and that of Smerdyakov in ''The Brothers Karamazov'', among others.

At the [[Military engineering-technical university|Saint Petersburg Institute of Military Engineering]]<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]],</ref> Dostoyevsky was taught mathematics, a subject he despised. However, he also studied literature by [[Shakespeare]], [[Blaise Pascal|Pascal]], [[Victor Hugo]] and [[E.T.A. Hoffmann]]. Though he focused on areas different from mathematics, he did well in the exams and received a commission in 1841.

===Early publications===

In 1841, influenced by the German poet/playwright [[Friedrich Schiller]], Dostoyevsky wrote two romantic plays: ''[[Maria Stuart (play)|Mary Stuart]]'' and ''[[Boris Godunov]]''. The plays have not been preserved. Dostoyevsky described himself as a "dreamer" when young. In the years when he wrote his great masterpieces he sometimes made fun of Schiller.

In 1842 Dostoyevsky was made a lieutenant.

In 1843 he left the Engineering Academy. In the same year he completed a translation into Russian of [[Balzac]]'s novel ''[[Eugénie Grandet]]'', but it brought him little attention.

Dostoyevsky started to write his own fiction in late 1844 after leaving the army. In 1846 his first work, the epistolary short novel, ''[[Poor Folk]]'', printed in the almanac ''A Petersburg Collection'', met with great acclaim. As legend has it, the editor of the magazine, poet [[Nikolai Nekrasov]], walked into the office of liberal critic [[Vissarion Belinsky]] and announced, "A new [[Nikolai Gogol|Gogol]] has arisen!" Belinsky, his followers, and many others agreed. After the novel was published in book form at the beginning of the next year, Dostoyevsky became a literary celebrity at the age of 24.

In 1846 Belinsky and others reacted negatively to his novella, ''[[The Double: A Petersburg Poem|The Double]]'', a psychological study of a bureaucrat whose alter ego overtakes his life. Dostoyevsky's fame began to fade. Much of his work after ''[[Poor Folk]]'' received ambivalent reviews, and it seemed that Belinsky's prediction that Dostoyevsky would be one of the greatest writers of Russia was mistaken.

===Exile in Siberia===
[[File:Omsk Dostoyevskiy Monument.jpg|thumb|upright|Statue of Dostoyevsky in [[Omsk]]]]
Dostoyevsky was incarcerated on 23 April 1849 for being part of the [[Liberalism|liberal]] intellectual group the [[Petrashevsky Circle]]. Emperor [[Nicholas I of Russia|Nicolas I]], after seeing the [[Revolutions of 1848]] in Europe, was harsh on any type of underground organization which might put [[autocracy]] in jeopardy. On November 16 of that year Dostoyevsky, with other members of the Petrashevsky Circle, was [[death sentence|sentenced to death]]. After a [[mock execution]], in which he and other members of the group stood outside in freezing weather waiting to be shot by a firing squad, Dostoyevsky's sentence was commuted to four years of [[exile]] with hard labour at a [[katorga]] prison camp in [[Omsk]], [[Siberia]]. Later Dostoyevsky described his years of suffering to his brother, as being, "shut up in a coffin." In describing the dilapidated barracks which "should have been torn down years ago", he wrote:
''{{quote|In summer, intolerable closeness; in winter, unendurable cold. All the floors were rotten. Filth on the floors an inch thick; one could slip and fall... We were packed like herrings in a barrel... There was no room to turn around. From dusk to dawn it was impossible not to behave like pigs... Fleas, lice, and black beetles by the bushel...''<ref>Frank 76. Quoted from Pisma, I: 135–37.</ref>}} This experience inspired him to write ''[[The House of the Dead (novel)|The House of the Dead]]''.

Dostoyevsky was released from prison in 1854, and was required to serve in the Siberian Regiment. He spent the following five years as a private (and later lieutenant) in the Regiment's Seventh Line Battalion, stationed at the fortress of [[Semey|Semipalatinsk]], now in [[Kazakhstan]].

While there, he began a relationship with Maria Dmitrievna Isayeva, the wife of an acquaintance in Siberia. After her husband's death, they married in February 1857.

===Post-prison maturation as a writer===
[[File:Valikhanov.jpg|thumb|upright|Dostoyevsky (right) and the [[Kazakhs|Kazakh]] scholar [[Shokan Walikhanuli]] in 1859]]

Dostoyevsky's experiences in prison and the army changed his political and religious convictions. First, his ordeal caused him to repudiate contemporary Western European philosophical movements and to pay greater tribute in his writings to traditional, rustic Russian values, exemplified in the [[Slavophile]] concept of ''[[sobornost]]''. Even more significantly he had what his biographer Joseph Frank describes as a [[Religious conversion|conversion]] experience in prison, which greatly strengthened his Christian, and specifically [[Russian Orthodox|Orthodox]], faith.{{sfn|Frank|1987|pp=124–27}} Dostoyevsky would later depict his conversion experience in the short story, ''[[The Peasant Marey]]'' (1876).

In his writings Dostoyevsky started to extol the virtues of humility, submission, and suffering.<ref name="Nab81Censors">[[Vladimir Nabokov]] (1981) ''[[Lectures on Russian Literature]]'', lecture on ''Russian Writers, Censors, and Readers'', p.14</ref> He now displayed a more critical stance on contemporary European philosophy and turned with intellectual rigour against the [[Nihilist movement|Nihilist]] and Socialist movements; and much of his post-prison work—particularly the novel, ''[[The Possessed (novel)|The Possessed]]'', and the essays, ''[[A Writer's Diary|The Diary of a Writer]]''—contains both criticism of socialist and nihilist ideas, as well as thinly veiled parodies of contemporary Western-influenced Russian intellectuals ([[Timofey Granovsky|Timofey Granovskiy]]), revolutionaries ([[Sergey Nechayev|Sergey Nyechayev]]), and even fellow novelists ([[Ivan Turgenev|Ivan Turgyenyev]]).<ref>Dostoevsky the Thinker James P. Scanlan. Dostoevsky the Thinker. Ithaca: Cornell University Press, 2002. xiii, p. 251</ref><ref>[http://ourworld.compuserve.com/homepages/jim_forest/pevear.htm Dostoevsky's View of Evil] Reprinted from ''In Communion'', April 1998.</ref> In social circles Dostoyevsky allied himself with known conservatives, such as the statesman [[Konstantin Pobedonostsev|Konstantin Pobyedonostsyev]]. His post-prison essays praised the tenets of the [[Pochvennichestvo|Pochvyennichyestvo]] movement, a late-19th century Russian nativist ideology closely aligned with [[Slavophile|Slavophilism]].

Dostoyevsky's post-prison fiction abandoned the Western European-style domestic melodramas and quaint character studies of his youthful work in favor of dark, more complex story-lines and situations, played-out by brooding, tortured characters—often styled partly on Dostoyevsky himself—who agonized over [[existentialism|existential]] themes of spiritual torment, religious awakening, and the psychological confusion caused by the conflict between traditional Russian culture and the influx of modern, Western philosophy. Nonetheless, this does not take from the debt which Dostoyevsky owed to earlier Western-influenced writers such as [[Gogol]], whose work grew from the irrational and anti-authoritarian spiritualist ideas contained within the [[Romantic movement]] which had immediately preceded Dostoyevsky in West Europe. However, Dostoyevsky's major novels focused on the idea that [[utopia]] and [[positivist]] ideas were unrealistic and unobtainable.<ref>{{Cite book|last = Sirotkina|first = Irina|title = Diagnosing Literary Genius: A Cultural History of Psychiatry in Russia, 1880|year = 1996|publisher = [[Johns Hopkins University Press]]|page = 55|isbn = 0801867827}}</ref>

===Later literary career===
[[File:Dostoevskij 1863.jpg|thumb|upright|Dostoyevsky in 1863]][[File:Fyodor Dostoevsky house.jpg|thumb|right|170px|Dostoyevsky's last address where he died, now a memorial and literary museum, St Petersburg.]]

In December 1859 Dostoyevsky returned to [[Saint Petersburg]], where he ran a series of unsuccessful literary journals, ''[[Vremya (magazine)|Vremya]]'' (Time) and ''[[Epoch (Russian magazine)|Epokha]]'' (Epoch), with his older brother [[Mikhail Dostoyevsky|Mikhail]].<ref>{{Cite book|title=F. M. Dostoyevsky. Collection of works in 15 volumes |volume=11|year=1993 |publisher=Nauka |location=Leningrad|pages=361–365 |chapter=A few words about Mikhail Mikhailovich Dostoyevsky}}</ref> The former was shut down as a consequence of its coverage of the [[January Uprising|Polish Uprising of 1863]].

In 1863 Dostoyevsky traveled to western Europe, and frequented gambling casinos. There he met [[Polina Suslova|Apollinaria Suslova]], the model for his "proud women", such as the two characters named Katerina Ivanovna, in ''[[Crime and Punishment]]'' and in ''[[The Brothers Karamazov]]''.

In 1864 Dostoyevsky was devastated by his wife's death; which was followed shortly thereafter by his brother's death. He was financially crippled by business debts; furthermore, he decided to assume the responsibility of his deceased brother's outstanding debts, as well providing for his wife's son from her earlier marriage and his brother's widow and children. He sank into a deep depression, frequenting gambling parlors and accumulating massive losses at the tables. He became dominated by his [[Problem gambling|gambling compulsion]]. He completed ''Crime and Punishment'' in a hurry because he was in urgent need of an advance from his publisher, having been left practically penniless after a gambling spree. He wrote ''[[The Gambler (novel)|The Gambler]]'' simultaneously in order to satisfy an agreement with his publisher, Stellovsky who, if he did not receive a new work, would claim the copyrights to all Dostoyevsky's writings.<ref>{{cite web|title=Fyodor Dostoevsky||publisher=Russia Today (RT)|url=http://russiapedia.rt.com/prominent-russians/literature/fyodor-dostoevsky/|accessdate=12 July 2011}}</ref>

Wishing to escape creditors at home and to visit casinos abroad, Dostoyevsky traveled to western Europe. There he attempted to rekindle a love affair with Suslova, but she refused his marriage proposal. Dostoyevsky was heartbroken, but soon met [[Anna Dostoyevskaya|Anna Grigorevna Snitkina]], a twenty-year-old [[stenographer]]. Shortly before marrying her in 1867, he dictated ''The Gambler'' to her.<ref>{{cite book |last=Dostoevsky |first=Fyodor |others=Notes and Introduction by Maire Jaanus. Translated by [[Constance Garnett]] |title=The Brothers Karamazov |series=Barnes & Noble Classics |year=2004 |origyear=First published 1879–1880 |publisher=Barnes & Noble Books |location=New York, NY |isbn=978-1-59308-045-7 |oclc=34325193 |page=703 |chapter=Endnotes |quote=Anna Grigorievna Snitkina, Dostoyevsky's second wife, was a stenographer to whom Dostoyevsky dictated his novel ''The Gambler'' in 1866; they married the following year.}}</ref>

From 1873 to 1881 he published the ''Writer's Diary'', a monthly journal of short stories, sketches, and articles on current events. The journal was an enormous success.

Dostoyevsky influenced, and was himself influenced by, the philosopher [[Vladimir Sergeyevich Solovyov]]. Solovyov was a source for the characters [[Ivan Karamazov]] and [[Alyosha Karamazov]].<ref>Zouboff, Peter, Solovyov on Godmanhood: Solovyov’s Lectures on Godmanhood Harmon Printing House: Poughkeepsie, New York, 1944; see Czeslaw Milosz’s introduction to Solovyov’s War, Progress and the End of History. Lindisfarne Press: Hudson, New York 1990.</ref>

[[File:Fyodor Mikahailovich Dostoyevsky's Study in St Petersburg.jpg|right|thumb|180px|Dostoyevsky's study in [[Saint Petersburg]].]]
In 1877 Dostoyevsky gave a [[eulogy]] at the funeral of his friend, the poet [[Nikolai Alekseevich Nekrasov|Nekrasov]], to much controversy{{Who|date=June 2009}}.

On 8 June 1880, shortly before he died, he gave his famous [[Alexander Pushkin|Pushkin]] speech at the unveiling of the [[Pushkin Square|Pushkin monument in Moscow]].<ref>Dostoyevsky [http://az.lib.ru/d/dostoewskij_f_m/text_0340.shtml Az.lib.ru Пушкинская речь (Pushkin's style)] (in Russian)</ref> In his later years Dostoyevsky lived for an extended period at the resort of [[Staraya Russa]] in northwestern Russia, which was closer to [[Saint Petersburg]] and less expensive than German resorts.

===Death===
Dostoyevsky died in St. Petersburg on {{OldStyleDate|9 February|1881|28 January}} of a lung hemorrhage associated with [[emphysema]] and an [[epileptic seizure]]. The copy of the New Testament given to him in Siberia sat on his lap. He was interred in [[Tikhvin Cemetery]] at the [[Alexander Nevsky Monastery]] in [[Saint Petersburg]]. Forty thousand mourners attended his funeral.<ref>Dostoevsky, Fyodor; Introduction to The Idiot, Wordsworth Ed. Ltd, 1996.</ref> His tombstone is inscribed with the words of Christ, ''Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.'' (from [[Gospel of John|the Gospel According to John]] 12:24) - which are also the [[Epigraph (literature)|epigraph]] of his final novel, ''The Brothers Karamazov''.)

The rented apartment where Dostoevsky spent the last few years of his life and wrote his last novel, ''The Brothers Karamazov'', and where he died is situated at 5 Kuznechnyi pereulok. It has been restored, by reference to old photographs, as it looked when he lived there, and opened in 1971 as the Dostoyevsky House Museum. It is a popular tourist attraction in Saint Petersburg.<ref>{{cite book |title=St Petersburg |last= Woodworth|first=Bradley |authorlink= |coauthors= Harold Bloom, Constance Richards|year=2005 |editor=Harold Bloom|publisher=Infobase Publishing |location= |isbn=0791083845, 9780791083840 |page=69 |pages= |url=http://books.google.com/books?id=tMn6qHyTIywC&dq=dostoevsky+house+museum,+St+Petersburg&source=gbs_navlinks_s |accessdate=19 November 2010}}</ref>

==Influence==
[[File:Vasily Perov - Портрет Ф.М.Достоевского - Google Art Project.jpg|thumb|upright|Portrait of Dostoyevsky in 1872 painted by [[Vasily Perov]].]]

Some, like journalist [[Otto Friedrich]],<ref>{{Cite news|publisher=Time Magazine|url=http://www.time.com/time/magazine/article/0,9171,943893,00.html?promoid=googlep|accessdate=2008-04-10|title=Freaking-Out with Fyodor|author=Otto Friedrich|date=6 September 1971}}</ref> consider Dostoyevsky to be one of Europe's major novelists, while others like [[Vladimir Nabokov]] maintain that from a point of view of enduring art and individual genius, he is a rather mediocre writer who produced wastelands of literary [[platitude]]s.<ref>Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 98</ref>

Dostoyevsky investigated in his novels religious concerns, particularly those of [[Eastern Orthodox Christianity]].<ref name="BritannicaRussianLit"/> "Dostoyevsky and the Religion of Suffering," the essay devoted to Dostoyevsky in [[Eugène-Melchior de Vogüé]]'s ''Le roman russe'' (1886), was an influential early analysis of the novelist's work, introducing Dostoyevsky and other Russian novelists to the West. Nabokov argued in his University courses at [[Cornell University|Cornell]], that such religious propaganda, rather than artistic qualities, was the main reason Dostoyevsky was praised and regarded as a 'Prophet' in Soviet Russia.<ref>Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 104</ref>{{Clarify|Why would the atheistic Soviets praise him for religious propaganda? Was the "propaganda" unconvincing?|date=July 2011}}

[[James Joyce]] and [[Virginia Woolf]] praised his prose. [[Ernest Hemingway]] cited Dostoyevsky as an influence on his work, in his posthumous collection of sketches ''[[A Moveable Feast]]''. [[Kurt Vonnegut]] in his novel [[Slaughterhouse-Five]] mentions Dostoevsky in such way:

''{{quote|[Eliot] Rosewater said an interesting thing to Billy [Pilgrim] one time ... He said that everything there was to know about life is in "The Brothers Karamazov," by Fyodor Dostoevsky. "But that isn't enough anymore," said Rosewater.}}''

According to Arthur Power's ''Conversations with James Joyce'', Joyce praised Dostoyevsky's prose:

''{{quote|...he is the man more than any other who has created modern prose, and intensified it to its present-day pitch. It was his explosive power which shattered the Victorian novel with its simpering maidens and ordered commonplaces; books which were without imagination or violence.}}''
In her essay ''The Russian Point of View'', Virginia Woolf said:

''{{quote|The novels of Dostoevsky are seething whirlpools, gyrating sandstorms, waterspouts which hiss and boil and suck us in. They are composed purely and wholly of the stuff of the soul. Against our wills we are drawn in, whirled round, blinded, suffocated, and at the same time filled with a giddy rapture. Out of [[Shakespeare]] there is no more exciting reading.''<ref>[http://etext.library.adelaide.edu.au/w/woolf/virginia/w91c/chapter16.html The Russian Point of View] Virginia Woolf.</ref>}}

[[File:Dostoevsky-Library Moscow Russia.jpg|thumb|upright|Dostoyevsky monument at the [[Russian State Library]] in Moscow.]]
Dostoyevsky displayed a nuanced understanding of human psychology in his major works. He created an opus of vitality and almost hypnotic power, characterized by feverishly dramatized scenes where his characters are frequently in scandalous and explosive atmospheres, engaged in passionate dialogue. The quest for God, the [[problem of evil]] and the suffering of the innocent are the themes which haunt the majority of his novels.

His characters fall into a few distinct categories: humble and self-effacing Christians ([[Prince Myshkin]], [[Sonya Marmeladova]], [[Alyosha Karamazov]], [[Saint Ambrose of Optina]]), self-destructive [[nihilism|nihilists]] ([[Svidrigailov]], [[Smerdyakov]], [[Stavrogin]], [[Notes from Underground|the underground man]]){{Citation needed|date=May 2009}}, cynical debauchees ([[Fyodor Karamazov]], [[Dmitri Karamazov]]), and rebellious intellectuals ([[Raskolnikov]], [[Ivan Karamazov]], [[Ippolit]]); also, his characters are driven by ideas rather than by biological or social imperatives. In comparison with the [[Literary realism|realistic]] characters of [[Leo Tolstoy|Tolstoy]] those of Dostoyevsky are more symbolic of the ideas they represent; thus Dostoyevsky is often cited as a forerunner of [[Symbolism (arts)|Literary Symbolism]], especially [[Russian Symbolism]] (see [[Alexander Blok]]).<ref>Dostoievsky by A. Steinberg p. 112</ref>
[[File:Dostoevsky MR280908.jpg|thumb|upright|Dostoyevsky statue, erected 1918, in front of [[Mariinsky Hospital]], the writer's birthplace in Moscow.]]
Dostoyevsky's novels are compressed in time (many cover only a few days); and this enables him to get rid of one of the dominant presentations of [[realism (arts)|realist]] prose, that of the corrosion of human life in the process of the time flux; his characters embody spiritual values that are timeless. Other themes include suicide, wounded pride, collapsed family values, spiritual regeneration through suffering, rejection of the West and affirmation of the [[Russian Orthodox Church]] and of [[tsarism]]. Literary scholars such as [[Mikhail Bakhtin]] have characterized his work as "[[Polyphony (literature)|polyphonic]]": Dostoyevsky does not appear to aim for a "single vision", and beyond simply describing situations from various angles, Dostoyevsky engendered fully dramatic novels of ideas, where conflicting views and characters are left to develop unevenly into unbearable crescendo.

Dostoyevsky and the other giant of late 19th century [[Russian literature]], [[Lev Nikolayevich Tolstoy]], never met in person, though each praised, criticized, and influenced the other (Dostoyevsky remarked of Tolstoy's ''[[Anna Karenina]]'' that it was a "flawless work of art"; [[Henri Troyat]] reports that Tolstoy once remarked of ''[[Crime and Punishment]]'' that, "Once you read the first few chapters you know pretty much how the novel will end up").{{Citation needed|date=August 2007}} A meeting was arranged but there was a confusion about where the meeting was to take place; and the two never rescheduled. Tolstoy wept when he learned of Dostoyevsky's death.<ref>Letter from Leo Tolstoy to Nikolai Strakhov, from [http://www.archive.org/stream/lettersoffyodorm00dostuoft#page/n389/mode/2up Letters of Fyodor Michailovitch Dostoevsky to his Family and Friends, page 337], Chatto and Windus, London, 1914.</ref> A copy of ''[[The Brothers Karamazov]]'' was found on the nightstand next to Tolstoy's deathbed at the [[Lev Tolstoy (settlement)|Astapovo]] railway station.

[[File:Grab-dostojewsky.JPG|thumb|upright|Dostoyevsky's tomb at the [[Alexander Nevsky Monastery]]]]

===Dostoyevsky on Jews in Russia===
{{Cleanup|section|date=May 2011}}
Several writers and critics (including Joseph Frank, [[Maxim D. Shrayer]]<ref>Shrayer, Maxim D. “The Jewish Question and The Brothers Karamazov.” In: A New Word on “The Brothers Karamazov.” Ed. Robert Louis Jackson. Evanston: Northwestern University Press, 2004. 210-233</ref>, Stephen Cassedy, David I. Goldstein, [[Gary Saul Morson]], and Felix Dreizin) have offered insights and suppositions regarding Dostoyevsky’s views on [[Jews]] and organized [[Jewry in Russia]]. One view is that Dostoyevsky perceived Jewish [[ethnocentrism]] and influence to be threatening the Russian peasantry in border regions.{{Citation needed|date=May 2011}} In ''[[A Writer's Diary]]'', Dostoyevsky wrote:

<blockquote>Thus, Jewry is thriving precisely there where the people are still ignorant, or not free, or economically backward. It is there that Jewry has a champ libre. And instead of raising, by its influence, the level of education, instead of increasing knowledge, generating economic fitness in the native population—instead of this the Jew, wherever he has settled, has still more humiliated and debauched the people; there humaneness was still more debased and the educational level fell still lower; there inescapable, inhuman misery, and with it despair, spread still more disgustingly. Ask the native population in our border regions: What is propelling the Jew—and has been propelling him for centuries? You will receive a unanimous answer: mercilessness. He has been prompted so many centuries only by pitilessness to us, only by the thirst for our sweat and blood.

And, in truth, the whole activity of the Jews in these border regions of ours consisted of rendering the native population as much as possible inescapably dependent on them, taking advantage of the local laws. They have always managed to be on friendly terms with those upon whom the people were dependent. Point to any other tribe from among Russian aliens which could rival the Jew by his dreadful influence in this connection! You will find no such tribe. In this respect the Jew preserves all his originality as compared with other Russian aliens, and of course, the reason therefore is that status of status of his, that spirit of which specifically breathes pitilessness for everything that is not Jew, with disrespect for any people and tribe, for every human creature who is not a Jew...<ref name="M. Dostoevsky 1949">Dostoevsky, F. M. ''The Diary of a Writer'', trans. Boris Brasol (New York: Charles Scribner's Sons), 1949.</ref></blockquote>

Dostoyevsky has been noted as both having expressed [[antisemitic]] sentiments as well as standing up for the rights of the Jewish people. In a review of Joseph Frank's book, ''The Mantle of the Prophet'', [[Orlando Figes]] notes that ''A Writer's Diary'' is "filled with politics, literary criticism, and pan-[[Slav]] diatribes about the virtues of the Russian Empire, [and] represents a major challenge to the Dostoyevsky fan, not least on account of its frequent expressions of anti-semitism."<ref>Figes, Orlando. "Dostoevsky's leap of faith This volume concludes a magnificent biography which is also a cultural history", ''Sunday Telegraph'' (London), p.13. September 29, 2002.</ref> Frank, in his foreword for David I. Goldstein's book ''Dostoevsky and the Jews'', attempts to place Dostoyevsky as a product of his time. Frank notes that Dostoyevsky made antisemitic remarks, but that Dostoyevsky's writing and stance, by and large, was one where Dostoyevsky held a great deal of guilt for his comments and positions that were antisemitic.<ref>Frank, Joseph. "Foreword" p. xiv. in Goldstein, David I. ''Dostoevsky and the Jews'', University of Texas Press, 1981. ISBN 0292715285</ref>

Steven Cassedy alleges in his book, ''Dostoevsky's Religion'', that much of the depiction of Dostoyevsky's views as antisemitic omits that Dostoyevsky expressed support for the equal rights of the Russian Jewish population, an unpopular position in Russia at the time.<ref name="Cassedy1">{{Cite book|title= Dostoevsky's Religion |last= Cassedy |first= Steven |year= 2005 |publisher= [[Stanford University Press]] |isbn= 0804751374 |pages= 67–80}}</ref> Cassedy also notes that this criticism of Dostoyevsky also appears to deny his sincerity when he said that he was for equal rights for the Russian Jewish populace and the [[Russian serfdom|serf]]s of his own country (since neither group at that point in history had equal rights).<ref name=Cassedy1/> Cassedy again notes when Dostoyevsky stated that he did not hate Jewish people and was not antisemitic.<ref name=Cassedy1/> Even though Dostoyevsky spoke of the potential negative influence of Jewish people, Dostoyevsky advised emperor [[Alexander II of Russia]] to give them rights to positions of influence in Russian society, such as allowing them access to Professorships at Universities. According to Cassedy, labeling Dostoyevsky anti-Semitic does not take into consideration Dostoyevsky's expressed desire to reconcile Jews and Christians peacefully in a single universal brotherhood of mankind.<ref name=Cassedy1 />

==Dostoyevsky and existentialism==
[[File:Fyodor Mikahailovich Dostoyevsky's Handwriting 1838.jpg|right|thumb|180px|Dostoyevsky's handwriting.]]
With the publication of ''[[Crime and Punishment]]'', in 1866, Dostoyevsky became one of Russia's most prominent authors. [[Will Durant]], in ''[[The Pleasures of Philosophy]]'' (1953), called Dostoyevsky one of the founding fathers of the philosophical movement known as [[existentialism]], and cited ''[[Notes from Underground]]'' in particular as a founding work of existentialism. For Dostoyevsky, war is the people's rebellion against the idea that [[reason]] guides everything, and reason is not the ultimate guiding principle for history or [[human|mankind]]. After his 1849 exile to the city of [[Omsk]], Siberia, Dostoyevsky focused on questions of [[suffering]] and [[wiktionary:despair|despair]] in many of his works.

[[Friedrich Nietzsche]] referred to Dostoyevsky as "the only psychologist from whom I have something to learn: he belongs to the happiest windfalls of my life, happier even than the discovery of [[Stendhal]]." He said that ''Notes from Underground'' "cried truth from the blood." According to [[Kontinent|Mihajlo Mihajlov]]'s "The Great Catalyzer: Nietzsche and Russian Neo-Idealism", Nietzsche constantly refers to Dostoyevsky in his notes and drafts throughout the winter of 1886–1887. Nietzsche also wrote abstracts of several of Dostoyevsky's works.

[[Freud]] wrote an article titled ''[[Dostoevsky and Parricide]]'', asserting that the greatest works in world literature are all about [[parricide]]. Though critical of Dostoyevsky's work overall, he regarded ''[[The Brothers Karamazov]]'' as among the three greatest works of literature.

==Works==
===Fiction===
Dostoyevsky's works of fiction includes 2 translations, 15 novels and novellas, and 17 short stories. Many of his longer novels were first published in [[Serial (literature)|serialized form]] in [[literary magazine]]s and [[journal]]s (see the individual articles). The years given below indicate the year in which the novel's final part or first complete book edition was published. in English many of his novels and stories are known by several titles.

===Translated books===
*''[[Eugénie Grandet]]', ([[Honore de Balzac]]) (1843)
* ''La dernière Aldini'' ([[George Sand]]) (1843)

====Novels and novellas====
*''[[Poor Folk]]'' (Бедные люди [''Bednye lyudi''], 1846)
*''[[The Double: A Petersburg Poem]]'' (Двойник: Петербургская поэма [''Dvoynik: Peterburgskaya poema''], 1846)
*''[[Netochka Nezvanova (novel)|Netochka Nezvanova]]'' (Неточка Незванова [''Netochka Nezvanova''], 1849)
*''[[Uncle's Dream]]'' (Дядюшкин сон [''Dyadyushkin son''], 1859)
*''[[The Village of Stepanchikovo]]'' (Село Степанчиково и его обитатели [''Selo Stepanchikovo i ego obitateli''], 1859)
*''[[Humiliated and Insulted]]'' (Униженные и оскорбленные [''Unizhennye i oskorblennye''], 1861)
*''[[The House of the Dead (novel)|The House of the Dead]]'' (Записки из мертвого дома [''Zapiski iz mertvogo doma''], 1862)
*''[[Notes from Underground]]'' (Записки из подполья [''Zapiski iz podpolya''], 1864)
*''[[Crime and Punishment]]'' (Преступление и наказание [''Prestuplenie i nakazanie''], 1866)
*''[[The Gambler (novel)|The Gambler]]'' (Игрок [''Igrok''], 1867)
*''[[The Idiot]]'' (Идиот [''Idiot''], 1869). Translated into English by [[Henry Carlisle]] and [[Olga Carlisle]].
*''[[The Eternal Husband]]'' (Вечный муж [''Vechnyj muzh''], 1870)
*''[[The Possessed (novel)|Demons]]'' (Бесы [''Besy''], 1872)
*''[[The Adolescent]]'' (Подросток [''Podrostok''], 1875)
*''[[The Brothers Karamazov]]'' (Братья Карамазовы [''Brat'ya Karamazovy''], 1880)

====Short stories====
*"[[Mr. Prokharchin]]" ("Господин Прохарчин" ["Gospodin Prokharchin"], 1846)
*"Novel in Nine Letters" ("Роман в девяти письмах" ["Roman v devyati pis'mah"], 1847)
*"The Landlady" ("Хозяйка" ["Hozyajka"], 1847)
*"The Jealous Husband" ("Чужая жена и муж под кроватью" ["Chuzhaya zhena i muzh pod krovat'yu"], 1848)
*"A Weak Heart" ("Слабое сердце" ["Slaboe serdze"], 1848)
*"Polzunkov" ("Ползунков" ["Polzunkov"], 1848)
*"[[An Honest Thief|The Honest Thief]]" ("Честный вор" ["Chestnyj vor"], 1848)
*"[[A Christmas Tree and a Wedding|The Christmas Tree and a Wedding]]" ("Елка и свадьба" ["Elka i svad'ba"], 1848)
*"[[White Nights (short story)|White Nights]]" ("Белые ночи" ["Belye nochi"], 1848)
*"A Little Hero" ("Маленький герой" ["Malen'kij geroj"], 1849)
*"[[A Nasty Story|A Nasty Anecdote]]" ("Скверный анекдот" ["Skvernyj anekdot"], 1862)
*"[[The Crocodile (short story)|The Crocodile]]" ("Крокодил" ["Krokodil"], 1865)
*"[[Bobok]]" ("Бобок" ["Bobok"], 1873)
*"The Heavenly Christmas Tree" ("Мальчик у Христа на ёлке" ["Mal'chik u Hrista na elke"], 1876)
*"[[A Gentle Creature|The Meek One]]" ("Кроткая" ["Krotkaja"], 1876)
*"[[The Peasant Marey]]" ("Мужик Марей" ["Muzhik Marej"], 1876)
*"[[The Dream of a Ridiculous Man]]" ("Сон смешного человека" ["Son smeshnogo cheloveka"], 1877)

===Non-fiction===
*''[[A Writer's Diary]]'', collected essays
*''Winter Notes on Summer Impressions'' (1863)
*''[[A Writer's Diary]]'' (Дневник писателя [''Dnevnik pisatelya''], 1873–1881)
*''Letters'' (collected in English translations in five volumes of ''Complete Letters'')

==See also==
{{div col|colwidth=30em}}
*[[Albert Camus]]
*[[Aleksandr Solzhenitsyn]]
*[[Existentialism]]
*[[List of Russian philosophers]]
*[[Lev Shestov]]
*[[Nikolai Berdyaev]]
*[[Nikolay Strakhov]]
*[[Russian Orthodox Church]]
*[[Vasily Rozanov]]
{{div col end}}

==References==
{{reflist|colwidth=30em}}

==Bibliography==
{{refbegin||colwidth=30em}}
* [[Rowan Williams]], ''Dostoevsky Language, Faith and Fiction'' (2008)
*{{cite book
|title=[[Humiliated and Insulted]]
|last=Avsey
|first=Ignat
|others=Trans. Avsey
|year=2008
|publisher=Oneworld Classics
|location=[[London]]
|chapter=Extra Material on Fyodor Dostoevsky's ''Humiliated and Insulted''
|isbn=978-1847490452
|ref = harv}}
* M. Jones, ''Dostoevsky and the dynamics of religious experience'' (2005)
* W. J. Leatherbarrow, ''A Devil's Vaudeville: the demonic in Dostoevsky's major fiction'' (2005)
* ''The Cambridge Companion to Dostoevskii'', ed. W. J. Leatherbarrow (2002)
*{{cite book
|title=Dostoevsky: The Seeds of Revolt, 1821-1849
|last=Frank
|first=Joseph
|year=1979
|origyear=First published 1976
|publisher=[[Princeton University Press]]
|location=[[Princeton, New Jersey|Princeton]]
|isbn=978-0691013558
|url=http://books.google.com/books?id=pDEAXltygUIC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Years of Ordeal, 1850-1859
|last=Frank
|first=Joseph
|year=1987
|origyear=First published 1983
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691014227
|url=http://books.google.com/books?id=K98hhw0IEHgC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Stir of Liberation, 1860-1865
|last=Frank
|first=Joseph
|year=1988
|origyear=First published 1986
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691014524
|url=http://books.google.com/books?id=QJj6qb6Rh3AC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Miraculous Years, 1865-1871
|last=Frank
|first=Joseph
|year=1997
|origyear=First published 1995
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691015873
|url=http://books.google.com/books?id=iAs4Lz5yog0C
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Mantle of the Prophet, 1871-1881
|last=Frank
|first=Joseph
|year=2003
|origyear=First published 2002
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691115696
|url=http://books.google.com/books?id=mQqonU-pweEC
|ref = harv}}
* ''Dostoevsky and the Christian tradition'', ed. G. Pattison, D. O. Thompson (2001)
* P. Evdokimov, ''Gogol et Dostoievski'' (2nd. ed. 1984)
* ''New Essays on Dostoevsky'', ed. M. Jones, G. M. Terry (1983)
* S. Sutherland, ''Atheism and the rejection of God: contemporary philosophy and The Brothers Karamazov'' (1977)
* V. Seduro, ''Dostoevski's Image in Russia Today'' (1975)
*{{cite book
|title=Dostoevsky: His Life and Work
|last=Mochulsky
|first=Konstantin
|others=Trans. Minihan, Michael A
|year=1973
|origyear=First published 1967; Russian original 1947
|publisher=Princeton University Press
|location=Princeton
|isbn=0691012997
|url=http://books.google.com/books?id=mDKphT8_XLsC
|ref = harv}}
* D. Capetanakis, 'Dostoevsky', in ''Demetrios Capetanakis A Greek Poet In England'' (1947), p.103-116
* P. Evdokimov, ''Dostoevski et le probleme du mal'' (1942; repr. 1978)
* N. Berdyaev, ''Dostoevsky'' (1934; Russian original 1923)
* L. Shestov, ''Dostoevsky, Tolstoy and Nietzsche'' (1969; Russian original 1903)
{{refend}}

==External links==

{{Sister project links
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{{Fyodor Dostoyevsky}}

{{Authority control|PND=118527053|LCCN=n/79/029930|VIAF=104023256}}

{{Persondata
|NAME= Dostoyevsky, Fyodor Mikhailovich
|ALTERNATIVE NAMES= Dostoevsky, Fyodor Mikhailovich; Фёдор Миха́йлович Достое́вский (Russian)
|SHORT DESCRIPTION= Russian novelist
|DATE OF BIRTH= {{Birth date|1821|11|11|mf=y}}
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|DATE OF DEATH= {{Death date|1881|2|9|mf=y}}
|PLACE OF DEATH= Saint Petersburg
}}
{{DEFAULTSORT:Dostoyevsky, Fyodor}}
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[[mn:Фёдор Михайлович Достоевский]]
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[[zh:費奧多爾·陀思妥耶夫斯基]]

Fjodor Michailowitsch Dostojewski

2011-11-06T19:09:36Z

Paul August: revert : (these dates are, according to the note, the "new style" dates)

{{Redirect2|Dostoyevsky|Dostoevsky|other uses|Dostoyevsky (disambiguation)}}
{{Eastern Slavic name|Mikhaylovich|Dostoyevsky}}
{{Infobox writer 
|name = Fyodor Dostoyevsky
|image = Dostoevsky.jpg
|caption = 1879
|birth_name = Fyodor Mikhaylovich Dostoyevsky
|birth_date = {{Birth date|1821|11|11}}
|birth_place = [[Moscow]], [[Russian Empire]]
|death_date = {{Death date and age|1881|2|9|1821|11|11}}
|death_place = [[Saint Petersburg]], Russian Empire
|occupation = [[Novelist]], [[short story]] writer, [[essay]]ist
|language = [[Russian language|Russian]]
|nationality = [[Russians|Russian]]
|period = 1846–1881
|genre =
|subject =
|movement =
|religion = [[Russian Orthodox]]
|notableworks = ''[[Notes from Underground]]'' ''[[Crime and Punishment]]'' ''[[The Idiot]]'' ''[[The Brothers Karamazov]]''
|spouse = Mariya Dmitriyevna Isayeva (1857–64) [her death]
[[Anna Grigoryevna Snitkina]] (1867–1881) [his death]
|children = Sofiya (1868), [[Lyubov Dostoyevskaya|Lyubov]] (1869—1926), Fyodor (1871–1922), Alexei (1875—1878)
|relatives =
|signature = Fyodor Dostoyevsky Signature.svg
}}
'''Fyodor Mikhaylovich Dostoyevsky''' ({{lang-rus|Фёдор Михайлович Достоевский|p=ˈfʲodər mʲɪˈxajləvʲɪtɕ dəstɐˈjefskʲɪj|a=ru-Dostoevsky.ogg}}; November 11, 1821 – February 9, 1881<ref>[[Old Style and New Style dates|Old Style date]] October 30, 1821 – January 29, 1881.</ref>) was a [[Russia]]n writer of [[novel]]s, [[short story|short stories]] and [[essay]]s.<ref name="pravoslavye.org.ua">[http://www.pravoslavye.org.ua/index.php?r_type=article&action=fullinfo&id=13375 Ukrainian origin of Dostoyevsky (Українське коріння Достоєвського)]</ref> He is best known for his novels ''[[Crime and Punishment]]'', ''[[The Idiot]]'' and ''[[The Brothers Karamazov]]''. His name has been transcribed in to English using various spellings, with some early translations rendering his first name by its English equivalent, Theodore. This is because, before the post-revolutionary [[Reforms_of_Russian_orthography#The_post-revolution_reform|orthographic reform]] which, amongst other things, replaced the cyrillic letter Ѳ ('th') with the cyrillic letter Ф ('f'), Dostoyevsky's name was written Ѳеодоръ (Theodor) Михайловичъ Достоевскій.

Dostoyevsky's literary works explored human psychology in the troubled political, social and spiritual context of 19th-century Russian society. Considered by many as a founder or precursor of 20th-century [[existentialism]], Dostoyevsky wrote, with the embittered voice of the anonymous "underground man", '' [[Notes from Underground]]'' (1864), which was called the "best overture for existentialism ever written" by [[Walter Kaufmann (philosopher)|Walter Kaufmann]].<ref>Existentialism: from Dostoyevsky to Sartre, ed. Walter Kaufmann, Penguin Books, 1989 ISBN 0452009308 p. 12</ref> Dostoyevsky is often acknowledged by critics as one of the greatest and most prominent psychologists in [[world literature]].<ref name="BritannicaRussianLit">{{Cite web|url=http://www.britannica.com/EBchecked/topic/513793/Russian-literature|publisher=Encyclopedia Britannica|accessdate=2008-04-11|title=Russian literature|quote=Dostoyevsky, who is generally regarded as one of the supreme psychologists in world literature, sought to demonstrate the compatibility of Christianity with the deepest truths of the psyche.}}</ref>

==Biography==
===Early life===
[[File:Wki Dostoyevsky Street 2 Moscow Mariinsky Hospital.jpg|thumb|240px|Mariinsky Hospital in Moscow, Dostoyevsky's birthplace]]

Dostoyevsky's father Mikhail and grandfather, Andrey, were born in modern central [[Ukraine]]. <ref>[http://www.scribd.com/doc/56070816/1/ORIGIN-OF-THE-DOSTOYEVSKY-FAMILY ORIGIN OF THE DOSTOYEVSKY FAMILY ... become priests in Ukraine.]</ref><ref>[http://books.google.hr/books?id=mDKphT8_XLsC&pg=PA3&lpg=PA3&dq=dostoevsky+family+comes+from+ukraine&source=bl&ots=E6D5zSXZm3&sig=qPHtXSuAujHQu9l9vg4M-2huDps&hl=hr&ei=VJK2TriUO-L64QT43JD7Aw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBgQ6AEwAA#v=onepage&q&f=false ''Dostoevsky: his life and work'', Princeton University.]</ref> Mikhail was a doctor and a devout Christian, who practised at the Mariinsky Hospital for the Poor in Moscow.

Dostoyevsky was born in Moscow to Mikhail and Maria Dostoyevsky, the second of seven children.<ref>''The Best Short Stories of Dostoevsky'' Translated with an Introduction by David Magarshack. New York: The Modern Library, Random House; 1971.</ref> The family lived in a small apartment in the Mariinsky Hospital grounds. The hospital was located in one of the city's worst areas near a cemetery for criminals, a lunatic asylum, and an orphanage for abandoned infants. This urban landscape made a lasting impression on the young Dostoyevsky, whose interest in and compassion for the poor, oppressed and tormented was apparent in his life and works. Although it was forbidden by his parents, Dostoyevsky liked to wander out to the hospital garden, where the patients sat to catch a glimpse of the sun. The young Dostoyevsky appreciated spending time with these patients and listening to their stories.{{cn|date=November 2011}}

There are many stories of Dostoyevsky's father's despotic treatment of his children, but this despotism was tempered by his extreme care for his children and their upbringing. After returning home from work, he would take a nap while his children, ordered to keep absolutely silent, stood by their slumbering father in shifts and swatted the flies that came near his head. But the father was also careful to send his children to private schools where they would not be beaten. In the opinion of Joseph Frank, author of a definitive biography of Dostoyevsky, the father figure in ''[[The Brothers Karamazov]]'' is not based on Dostoyevsky's own father. Letters and personal accounts demonstrate that they did have a fairly loving relationship.

[[File:Dostoevskij 1847.jpg|thumb|upright|The young Dostoyevsky, in an 1847 portrait by [[Konstiantyn Trutovsky|Trutovsky]]]]

In 1837, shortly after his mother died of [[tuberculosis]], Dostoyevsky and his brother were sent to St Petersburg to attend the [[Military engineering-technical university|Nikolayev Military Engineering Institute]], now called the [[Military engineering-technical university|Military Engineering-Technical University]].<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]]</ref> Fyodor Dostoyevsky's father died in 1839. Though it has never been proven, it is believed by some that he was murdered by his own [[serf]]s.<ref>[http://worldebooklibrary.com/eBooks/Coradella_Collegiate_Bookshelf_Collection/Dostoevsky-notesfromtheunderground.pdf Notes from the Underground] Coradella Collegita Bookshelf edition, ''About the Author''.</ref> According to one account, the serfs became enraged during one of his drunken fits of violence, and after restraining him, poured [[vodka]] into his mouth until he drowned. A similar account appears in ''Notes from Underground''. Another story holds that Mikhail died of natural causes, and a neighboring landowner invented the story of his murder so that he could buy the estate at a cheaper price. Some, like Sigmund Freud in his 1928 article, "[[Dostoevsky and Parricide]]", have argued that his father's personality had influenced the character of Fyodor Pavlovich Karamazov, the "wicked and sentimental buffoon", father of the main characters in his 1880 novel ''[[The Brothers Karamazov]]'', but such claims fail to withstand the scrutiny of many critics{{Who|date=June 2009}}.

Dostoyevsky suffered from [[epilepsy]] and his first seizure occurred when he was nine years old.<ref>[http://www.epilepsy.com/epilepsy/famous_writers.html Epilepsy.com] Famous authors with epilepsy.</ref> Epileptic seizures recurred sporadically throughout his life, and Dostoyevsky's experiences are thought<ref>Dostoyevsky, Fyodor, Richard Pevear, and Larissa Volokhonsky. The Idiot. New York: Vintage, 2001. Print. Introduction pp. xix</ref> to have formed the basis for his description of Prince Myshkin's epilepsy in his novel ''[[The Idiot]]'' and that of Smerdyakov in ''The Brothers Karamazov'', among others.

At the [[Military engineering-technical university|Saint Petersburg Institute of Military Engineering]]<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]],</ref> Dostoyevsky was taught mathematics, a subject he despised. However, he also studied literature by [[Shakespeare]], [[Blaise Pascal|Pascal]], [[Victor Hugo]] and [[E.T.A. Hoffmann]]. Though he focused on areas different from mathematics, he did well in the exams and received a commission in 1841. That year, influenced by the German poet/playwright [[Friedrich Schiller]], he wrote two romantic plays: ''[[Maria Stuart (play)|Mary Stuart]]'' and ''[[Boris Godunov]]''. The plays have not been preserved. Dostoyevsky described himself as a "dreamer" when he was a young man. He also revered [[Schiller]] at that age. However, in the years during which he wrote his great masterpieces, his opinions changed and he sometimes made fun of Schiller.

Dostoyevsky was made a lieutenant in 1842, and left the Engineering Academy the following year. He completed a translation into Russian of [[Balzac]]'s novel ''[[Eugénie Grandet]]'' in 1843, but it brought him little to no attention. Dostoyevsky started to write his own fiction in late 1844 after leaving the army. In 1846, his first work, the epistolary short novel, ''[[Poor Folk]]'', printed in the almanac ''A Petersburg Collection'' (published by [[Nikolay Nekrasov|N. Nekrasov]]), was met with great acclaim. As legend has it, the editor of the magazine, poet [[Nikolai Nekrasov]], walked into the office of liberal critic [[Vissarion Belinsky]] and announced, "A new [[Nikolai Gogol|Gogol]] has arisen!" Belinsky, his followers, and many others agreed. After the novel was fully published in book form at the beginning of the next year, Dostoyevsky became a literary celebrity at the age of 24.

In 1846, Belinsky and many others reacted negatively to his novella, ''[[The Double: A Petersburg Poem|The Double]]'', a psychological study of a bureaucrat whose alter ego overtakes his life. Dostoyevsky's fame began to fade. Much of his work after ''[[Poor Folk]]'' received ambivalent reviews and it seemed that Belinsky's prediction that Dostoyevsky would be one of the greatest writers of Russia was mistaken.

===Exile in Siberia===
[[File:Omsk Dostoyevskiy Monument.jpg|thumb|upright|Statue of Dostoyevsky in [[Omsk]]]]
Dostoyevsky was incarcerated on 23 April 1849 for being part of the [[Liberalism|liberal]] intellectual group the [[Petrashevsky Circle]]. [[Tsar]] [[Nicholas I of Russia|Nicholas I]], after seeing the [[Revolutions of 1848]] in Europe, was harsh on any type of underground organization which he felt could put [[autocracy]] in jeopardy. On November 16 of that year, Dostoyevsky, along with other members of the Petrashevsky Circle, was [[death sentence|sentenced to death]]. After a [[mock execution]], in which he and other members of the group stood outside in freezing weather waiting to be shot by a firing squad, Dostoyevsky's sentence was commuted to four years of [[exile]] with hard labour at a [[katorga]] prison camp in [[Omsk]], [[Siberia]]. Later, Dostoyevsky described his years of suffering to his brother, as being, "shut up in a coffin." In describing the dilapidated barracks which "should have been torn down years ago", he wrote:
''{{quote|In summer, intolerable closeness; in winter, unendurable cold. All the floors were rotten. Filth on the floors an inch thick; one could slip and fall... We were packed like herrings in a barrel... There was no room to turn around. From dusk to dawn it was impossible not to behave like pigs... Fleas, lice, and black beetles by the bushel...''<ref>Frank 76. Quoted from Pisma, I: 135–37.</ref>}} This experience inspired him to write ''[[The House of the Dead (novel)|The House of the Dead]]''.

Dostoyevsky was released from prison in 1854, and was required to serve in the Siberian Regiment. He spent the following five years as a private (and later lieutenant) in the Regiment's Seventh Line Battalion, stationed at the fortress of [[Semey|Semipalatinsk]], now in [[Kazakhstan]]. While there, he began a relationship with Maria Dmitrievna Isayeva, the wife of an acquaintance in Siberia. After her husband's death, they married in February 1857.

===Post-prison maturation as a writer===
[[File:Valikhanov.jpg|thumb|upright|Dostoyevsky (right) and the [[Kazakhs|Kazakh]] scholar [[Shokan Walikhanuli]] in 1859]]

Dostoyevsky's experiences in prison and the army resulted in major changes in his political and religious convictions. First, his ordeal somehow caused him to become disillusioned with "Western" ideas; he repudiated the contemporary Western European philosophical movements, and instead paid greater tribute in his writings to traditional, rustic Russian values exemplified in the [[Slavophile]] concept of ''[[sobornost]]''. But even more significantly, he had what his biographer Joseph Frank describes as a [[Religious conversion|conversion]] experience in prison, which greatly strengthened his Christian, and specifically [[Russian Orthodox|Orthodox]], faith.{{sfn|Frank|1987|pp=124–27}} Dostoyevsky would later depict his conversion experience in the short story, ''[[The Peasant Marey]]'' (1876).

In his writings, Dostoyevsky started to extol the virtues of humility, submission, and suffering.<ref name="Nab81Censors">[[Vladimir Nabokov]] (1981) ''[[Lectures on Russian Literature]]'', lecture on ''Russian Writers, Censors, and Readers'', p.14</ref> He now displayed a much more critical stance on contemporary European philosophy and turned with intellectual rigour against the [[Nihilist movement|Nihilist]] and Socialist movements; and much of his post-prison work—particularly the novel, ''[[The Possessed (novel)|The Possessed]]'', and the essays, ''[[A Writer's Diary|The Diary of a Writer]]''—contains both criticism of socialist and nihilist ideas, as well as thinly veiled parodies of contemporary Western-influenced Russian intellectuals ([[Timofey Granovsky|Timofey Granovskiy]]), revolutionaries ([[Sergey Nechayev|Sergey Nyechayev]]), and even fellow novelists ([[Ivan Turgenev|Ivan Turgyenyev]]).<ref>Dostoevsky the Thinker James P. Scanlan. Dostoevsky the Thinker. Ithaca: Cornell University Press, 2002. xiii, p. 251</ref><ref>[http://ourworld.compuserve.com/homepages/jim_forest/pevear.htm Dostoevsky's View of Evil] Reprinted from ''In Communion'', April 1998.</ref> In social circles, Dostoyevsky allied himself with well-known conservatives, such as the statesman [[Konstantin Pobedonostsev|Konstantin Pobyedonostsyev]]. His post-prison essays praised the tenets of the [[Pochvennichestvo|Pochvyennichyestvo]] movement, a late-19th century Russian nativist ideology closely aligned with [[Slavophile|Slavophilism]].

Dostoyevsky's post-prison fiction abandoned the Western European-style domestic melodramas and quaint character studies of his youthful work in favor of dark, more complex storylines and situations, played-out by brooding, tortured characters—often styled partly on Dostoyevsky himself—who agonized over [[existentialism|existential]] themes of spiritual torment, religious awakening, and the psychological confusion caused by the conflict between traditional Russian culture and the influx of modern, Western philosophy. Nonetheless, this does not take from the debt which Dostoyevsky owed to earlier Western-influenced writers such as [[Gogol]] whose work grew from the irrational and anti-authoritarian spiritualist ideas contained within the [[Romantic movement]] which had immediately preceded Dostoyevsky in West Europe. However, Dostoyevsky's major novels focused on the idea that [[utopia]] and [[positivist]] ideas were unrealistic and unobtainable.<ref>{{Cite book|last = Sirotkina|first = Irina|title = Diagnosing Literary Genius: A Cultural History of Psychiatry in Russia, 1880|year = 1996|publisher = [[Johns Hopkins University Press]]|page = 55|isbn = 0801867827}}</ref>

===Later literary career===
[[File:Dostoevskij 1863.jpg|thumb|upright|Dostoyevsky in 1863]][[File:Fyodor Dostoevsky house.jpg|thumb|right|170px|Dostoyevsky's last address where he died, now a memorial and literary museum, St Petersburg.]]

In December 1859, Dostoyevsky returned to [[Saint Petersburg]], where he ran a series of unsuccessful literary journals, ''[[Vremya (magazine)|Vremya]]'' (Time) and ''[[Epoch (Russian magazine)|Epokha]]'' (Epoch), with his older brother [[Mikhail Dostoyevsky|Mikhail]].<ref>{{Cite book|title=F. M. Dostoyevsky. Collection of works in 15 volumes |volume=11|year=1993 |publisher=Nauka |location=Leningrad|pages=361–365 |chapter=A few words about Mikhail Mikhailovich Dostoyevsky}}</ref> The former was shut down as a consequence of its coverage of the [[January Uprising|Polish Uprising of 1863]]. That year Dostoyevsky traveled to Europe and frequented gambling casinos. There he met [[Polina Suslova|Apollinaria Suslova]], the model for Dostoyevsky's "proud women", such as the two characters named Katerina Ivanovna, in ''[[Crime and Punishment]]'' and in ''[[The Brothers Karamazov]]''.

Dostoyevsky was devastated by his wife's death in 1864, which was followed shortly thereafter by his brother's death. He was financially crippled by business debts; furthermore, he decided to assume the responsibility of his deceased brother's outstanding debts, as well providing for his wife's son from her earlier marriage and his brother's widow and children. Dostoyevsky sank into a deep depression, frequenting gambling parlors and accumulating massive losses at the tables.

Dostoyevsky suffered from an acute [[Problem gambling|gambling compulsion]] and its consequences. He completed ''Crime and Punishment'' in a mad hurry because he was in urgent need of an advance from his publisher. He had been left practically penniless after a gambling spree. Dostoyevsky wrote ''[[The Gambler (novel)|The Gambler]]'' simultaneously in order to satisfy an agreement with his publisher Stellovsky who, if he did not receive a new work, would have claimed the copyrights to all of Dostoyevsky's writings.<ref>{{cite web|title=Fyodor Dostoevsky||publisher=Russia Today (RT)|url=http://russiapedia.rt.com/prominent-russians/literature/fyodor-dostoevsky/|accessdate=12 July 2011}}</ref>

Motivated by the dual wish to escape his creditors at home and to visit the casinos abroad, Dostoyevsky traveled to Western Europe. There, he attempted to rekindle a love affair with Suslova, but she refused his marriage proposal. Dostoyevsky was heartbroken, but soon met [[Anna Dostoyevskaya|Anna Grigorevna Snitkina]], a twenty-year-old [[stenographer]]. Shortly before marrying her in 1867, he dictated ''The Gambler'' to her.<ref>{{cite book |last=Dostoevsky |first=Fyodor |others=Notes and Introduction by Maire Jaanus. Translated by [[Constance Garnett]] |title=The Brothers Karamazov |series=Barnes & Noble Classics |year=2004 |origyear=First published 1879–1880 |publisher=Barnes & Noble Books |location=New York, NY |isbn=978-1-59308-045-7 |oclc=34325193 |page=703 |chapter=Endnotes |quote=Anna Grigorievna Snitkina, Dostoyevsky's second wife, was a stenographer to whom Dostoyevsky dictated his novel ''The Gambler'' in 1866; they married the following year.}}</ref> From 1873 to 1881 he published the ''Writer's Diary'', a monthly journal of short stories, sketches, and articles on current events. The journal was an enormous success.

Dostoyevsky influenced, and was himself influenced, by the philosopher [[Vladimir Sergeyevich Solovyov]]. Solovyov was the inspiration for the characters [[Ivan Karamazov]] and [[Alyosha Karamazov]].<ref>Zouboff, Peter, Solovyov on Godmanhood: Solovyov’s Lectures on Godmanhood Harmon Printing House: Poughkeepsie, New York, 1944; see Czeslaw Milosz’s introduction to Solovyov’s War, Progress and the End of History. Lindisfarne Press: Hudson, New York 1990.</ref>

[[File:Fyodor Mikahailovich Dostoyevsky's Study in St Petersburg.jpg|right|thumb|180px|Dostoyevsky's study in [[Saint Petersburg]].]]
In 1877, Dostoyevsky gave the keynote [[eulogy]] at the funeral of his friend, the poet [[Nikolai Alekseevich Nekrasov|Nekrasov]], to much controversy{{Who|date=June 2009}}. On 8 June 1880, shortly before he died, he gave his famous [[Alexander Pushkin|Pushkin]] speech at the unveiling of the [[Pushkin Square|Pushkin monument in Moscow]].<ref>Dostoyevsky [http://az.lib.ru/d/dostoewskij_f_m/text_0340.shtml Az.lib.ru Пушкинская речь (Pushkin's style)] (in Russian)</ref> In his later years, Dostoyevsky lived for an extended period at the resort of [[Staraya Russa]] in northwestern Russia, which was closer to [[Saint Petersburg]] and less expensive than German resorts.

===Death===
Dostoyevsky died in St. Petersburg on {{OldStyleDate|9 February|1881|28 January}} of a lung hemorrhage associated with [[emphysema]] and an [[epileptic seizure]]. A copy of the New Testament Bible given to him in Siberia sat on his lap. He was interred in [[Tikhvin Cemetery]] at the [[Alexander Nevsky Monastery]] in [[Saint Petersburg]]. Forty thousand mourners attended his funeral.<ref>Dostoevsky, Fyodor; Introduction to The Idiot, Wordsworth Ed. Ltd, 1996.</ref>

His tombstone reads; ''Verily, Verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.'' (Excerpt from [[Gospel of John|John]] 12:24, which is also the [[Epigraph (literature)|epigraph]] of his final novel, ''The Brothers Karamazov''.)

The rented apartment where he died and spent the last few years of his life is where he wrote his final novel ''The Brothers Karamazov''. The apartment, situated in a building at 5 Kuznechnyi pereulok, has been restored with old photographs to how it looked when he lived there. It opened in 1971 as the Dostoyevsky House Museum and is a popular tourist attraction in the city.<ref>{{cite book |title=St Petersburg |last= Woodworth|first=Bradley |authorlink= |coauthors= Harold Bloom, Constance Richards|year=2005 |editor=Harold Bloom|publisher=Infobase Publishing |location= |isbn=0791083845, 9780791083840 |page=69 |pages= |url=http://books.google.com/books?id=tMn6qHyTIywC&dq=dostoevsky+house+museum,+St+Petersburg&source=gbs_navlinks_s |accessdate=19 November 2010}}</ref>

==Influence==
[[File:Vasily Perov - Портрет Ф.М.Достоевского - Google Art Project.jpg|thumb|upright|Portrait of Dostoyevsky in 1872 painted by [[Vasily Perov]].]]

Some, like journalist [[Otto Friedrich]],<ref>{{Cite news|publisher=Time Magazine|url=http://www.time.com/time/magazine/article/0,9171,943893,00.html?promoid=googlep|accessdate=2008-04-10|title=Freaking-Out with Fyodor|author=Otto Friedrich|date=6 September 1971}}</ref> consider Dostoyevsky to be one of Europe's major novelists, while others like [[Vladimir Nabokov]] maintain that from a point of view of enduring art and individual genius, he is a rather mediocre writer who produced wastelands of literary [[platitude]]s.<ref>Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 98</ref>

Dostoyevsky promoted in his novels religious moralities, particularly those of [[Eastern Orthodox Christianity]].<ref name="BritannicaRussianLit"/> Indeed, "Dostoyevsky and the Religion of Suffering," the essay devoted to Dostoyevsky in [[Eugène-Melchior de Vogüé]]'s ''Le roman russe'' (1886), is widely considered to be the most influential early analysis of the novelist's work, introducing Dostoyevsky and other Russian novelists to the West. Nabokov argued in his University courses at [[Cornell University|Cornell]], that such religious propaganda, rather than artistic qualities, was the main reason Dostoyevsky was praised and regarded as a 'Prophet' in Soviet Russia.<ref>Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 104</ref>{{Clarify|Why would the atheistic Soviets praise him for religious propaganda? Was the "propaganda" unconvincing?|date=July 2011}}

[[James Joyce]] and [[Virginia Woolf]] praised his prose. [[Ernest Hemingway]] cited Dostoyevsky as a major influence on his work, in his posthumous collection of sketches ''[[A Moveable Feast]]''. In a book of interviews with Arthur Power (''Conversations with James Joyce''), Joyce praised Dostoyevsky's prose:

''{{quote|...he is the man more than any other who has created modern prose, and intensified it to its present-day pitch. It was his explosive power which shattered the Victorian novel with its simpering maidens and ordered commonplaces; books which were without imagination or violence.}}''
In her essay ''The Russian Point of View'', Virginia Woolf said:

''{{quote|The novels of Dostoevsky are seething whirlpools, gyrating sandstorms, waterspouts which hiss and boil and suck us in. They are composed purely and wholly of the stuff of the soul. Against our wills we are drawn in, whirled round, blinded, suffocated, and at the same time filled with a giddy rapture. Out of [[Shakespeare]] there is no more exciting reading.''<ref>[http://etext.library.adelaide.edu.au/w/woolf/virginia/w91c/chapter16.html The Russian Point of View] Virginia Woolf.</ref>}}

[[File:Dostoevsky-Library Moscow Russia.jpg|thumb|upright|Dostoyevsky monument at the [[Russian State Library]] in Moscow.]]
Dostoyevsky displayed a nuanced understanding of human psychology in his major works. He created an opus of vitality and almost hypnotic power, characterized by feverishly dramatized scenes where his characters are frequently in scandalous and explosive atmospheres, passionately engaged in [[Socratic dialogue]]s. The quest for God, the [[problem of evil]] and suffering of the innocents haunt the majority of his novels.

His characters fall into a few distinct categories: humble and self-effacing Christians ([[Prince Myshkin]], [[Sonya Marmeladova]], [[Alyosha Karamazov]], [[Saint Ambrose of Optina]]), self-destructive [[nihilism|nihilists]] ([[Svidrigailov]], [[Smerdyakov]], [[Stavrogin]], [[Notes from Underground|the underground man]]){{Citation needed|date=May 2009}}, cynical debauchees ([[Fyodor Karamazov]], [[Dmitri Karamazov]]), and rebellious intellectuals ([[Raskolnikov]], [[Ivan Karamazov]], [[Ippolit]]); also, his characters are driven by ideas rather than by ordinary biological or social imperatives. In comparison with [[Leo Tolstoy|Tolstoy]], whose characters are [[Literary realism|realistic]], the characters of Dostoyevsky are usually more symbolic of the ideas they represent, thus Dostoyevsky is often cited as one of the forerunners of [[Symbolism (arts)|Literary Symbolism]], especially [[Russian Symbolism]] (see [[Alexander Blok]]).<ref>Dostoievsky by A. Steinberg p. 112</ref>
[[File:Dostoevsky MR280908.jpg|thumb|upright|Dostoyevsky statue, erected 1918, in front of [[Mariinsky Hospital]], the writer's birthplace in Moscow.]]
Dostoyevsky's novels are compressed in time (many cover only a few days) and this enables him to get rid of one of the dominant traits of [[realism (arts)|realist]] prose, the corrosion of human life in the process of the time flux; his characters primarily embody spiritual values, and these are, by definition, timeless. Other themes include suicide, wounded pride, collapsed family values, spiritual regeneration through suffering, rejection of the West and affirmation of the [[Russian Orthodox Church]] and of [[tsarism]]. Literary scholars such as [[Mikhail Bakhtin]] have characterized his work as "[[Polyphony (literature)|polyphonic]]": Dostoyevsky does not appear to aim for a "single vision", and beyond simply describing situations from various angles, Dostoyevsky engendered fully dramatic novels of ideas where conflicting views and characters are left to develop unevenly into unbearable crescendo.

Dostoyevsky and the other giant of late 19th century [[Russian literature]], [[Lev Nikolayevich Tolstoy]], never met in person, even though each praised, criticized, and influenced the other (Dostoyevsky remarked of Tolstoy's ''[[Anna Karenina]]'' that it was a "flawless work of art"; [[Henri Troyat]] reports that Tolstoy once remarked of ''[[Crime and Punishment]]'' that, "Once you read the first few chapters you know pretty much how the novel will end up").{{Citation needed|date=August 2007}} There was a meeting arranged, but there was a confusion about where the meeting was to take place and they never rescheduled. Tolstoy wept when he learned of Dostoyevsky's death.<ref>Letter from Leo Tolstoy to Nikolai Strakhov, from [http://www.archive.org/stream/lettersoffyodorm00dostuoft#page/n389/mode/2up Letters of Fyodor Michailovitch Dostoevsky to his Family and Friends, page 337], Chatto and Windus, London, 1914.</ref> A copy of ''[[The Brothers Karamazov]]'' was found on the nightstand next to Tolstoy's deathbed at the [[Lev Tolstoy (settlement)|Astapovo]] railway station.

[[File:450px-Grab-dostojewsky.jpg|thumb|upright|Dostoyevsky's tomb at the [[Alexander Nevsky Monastery]]]]

===Dostoyevsky on Jews in Russia===
{{Cleanup|section|date=May 2011}}
Several writers and critics (including Joseph Frank, [[Maxim D. Shrayer]]<ref>Shrayer, Maxim D. “The Jewish Question and The Brothers Karamazov.” In: A New Word on “The Brothers Karamazov.” Ed. Robert Louis Jackson. Evanston: Northwestern University Press, 2004. 210-233</ref>, Stephen Cassedy, David I. Goldstein, [[Gary Saul Morson]], and Felix Dreizin) have offered various insights and suppositions regarding Dostoyevsky’s views on [[Jews]] and organized [[Jewry in Russia]] — one such view is that Dostoyevsky perceived Jewish [[ethnocentrism]] and Jewish influence to be directly threatening the Russian peasantry in the border regions.{{Citation needed|date=May 2011}} In ''[[A Writer's Diary]]'', Dostoyevsky wrote:

<blockquote>Thus, Jewry is thriving precisely there where the people are still ignorant, or not free, or economically backward. It is there that Jewry has a champ libre. And instead of raising, by its influence, the level of education, instead of increasing knowledge, generating economic fitness in the native population—instead of this the Jew, wherever he has settled, has still more humiliated and debauched the people; there humaneness was still more debased and the educational level fell still lower; there inescapable, inhuman misery, and with it despair, spread still more disgustingly. Ask the native population in our border regions: What is propelling the Jew—and has been propelling him for centuries? You will receive a unanimous answer: mercilessness. He has been prompted so many centuries only by pitilessness to us, only by the thirst for our sweat and blood.

And, in truth, the whole activity of the Jews in these border regions of ours consisted of rendering the native population as much as possible inescapably dependent on them, taking advantage of the local laws. They have always managed to be on friendly terms with those upon whom the people were dependent. Point to any other tribe from among Russian aliens which could rival the Jew by his dreadful influence in this connection! You will find no such tribe. In this respect the Jew preserves all his originality as compared with other Russian aliens, and of course, the reason therefore is that status of status of his, that spirit of which specifically breathes pitilessness for everything that is not Jew, with disrespect for any people and tribe, for every human creature who is not a Jew...<ref name="M. Dostoevsky 1949">Dostoevsky, F. M. ''The Diary of a Writer'', trans. Boris Brasol (New York: Charles Scribner's Sons), 1949.</ref></blockquote>

Dostoyevsky has been noted as both having expressed [[antisemitic]] sentiments as well as standing up for the rights of the Jewish people. In a review of Joseph Frank's book, ''The Mantle of the Prophet'', [[Orlando Figes]] notes that ''A Writer's Diary'' is "filled with politics, literary criticism, and pan-[[Slav]] diatribes about the virtues of the Russian Empire, [and] represents a major challenge to the Dostoyevsky fan, not least on account of its frequent expressions of anti-semitism."<ref>Figes, Orlando. "Dostoevsky's leap of faith This volume concludes a magnificent biography which is also a cultural history", ''Sunday Telegraph'' (London), p.13. September 29, 2002.</ref> Frank, in his foreword for David I. Goldstein's book ''Dostoevsky and the Jews'', attempts to place Dostoyevsky as a product of his time. Frank notes that Dostoyevsky made antisemitic remarks, but that Dostoyevsky's writing and stance, by and large, was one where Dostoyevsky held a great deal of guilt for his comments and positions that were antisemitic.<ref>Frank, Joseph. "Foreword" p. xiv. in Goldstein, David I. ''Dostoevsky and the Jews'', University of Texas Press, 1981. ISBN 0292715285</ref>

Steven Cassedy alleges in his book, ''Dostoevsky's Religion'', that much of the depiction of Dostoyevsky's views as antisemitic omits that Dostoyevsky expressed support for the equal rights of the Russian Jewish population, an unpopular position in Russia at the time.<ref name="Cassedy1">{{Cite book|title= Dostoevsky's Religion |last= Cassedy |first= Steven |year= 2005 |publisher= [[Stanford University Press]] |isbn= 0804751374 |pages= 67–80}}</ref> Cassedy also notes that this criticism of Dostoyevsky also appears to deny his sincerity when he said that he was for equal rights for the Russian Jewish populace and the [[Russian serfdom|serf]]s of his own country (since neither group at that point in history had equal rights).<ref name=Cassedy1/> Cassedy again notes when Dostoyevsky stated that he did not hate Jewish people and was not antisemitic.<ref name=Cassedy1/> Even though Dostoyevsky spoke of the potential negative influence of Jewish people, Dostoyevsky advised Czar [[Alexander II of Russia]] to give them rights to positions of influence in Russian society, such as allowing them access to Professorships at Universities. According to Cassedy, labeling Dostoyevsky anti-Semitic does not take into consideration Dostoyevsky's expressed desire to peacefully reconcile Jews and Christians into a single universal brotherhood of all mankind.<ref name=Cassedy1 />

==Dostoyevsky and existentialism==
[[File:Fyodor Mikahailovich Dostoyevsky's Handwriting 1838.jpg|right|thumb|180px|Dostoyevsky's handwriting.]]
With the publication of ''[[Crime and Punishment]]'', in 1866, Dostoyevsky became one of Russia's most prominent authors. [[Will Durant]], in ''[[The Pleasures of Philosophy]]'' (1953), called Dostoyevsky one of the founding fathers of the philosophical movement known as [[existentialism]], and cited ''[[Notes from Underground]]'' in particular as a founding work of existentialism. For Dostoyevsky, war is the people's rebellion against the idea that [[reason]] guides everything, and thus, reason is not the ultimate guiding principle for either history or [[human|mankind]]. After his 1849 exile to the city of [[Omsk]], Siberia, Dostoyevsky focused heavily on notions of [[suffering]] and [[wiktionary:despair|despair]] in many of his works.

[[Friedrich Nietzsche]] referred to Dostoyevsky as "the only psychologist from whom I have something to learn: he belongs to the happiest windfalls of my life, happier even than the discovery of [[Stendhal]]." He said that ''Notes from Underground'' "cried truth from the blood." According to [[Kontinent|Mihajlo Mihajlov]]'s "The Great Catalyzer: Nietzsche and Russian Neo-Idealism", Nietzsche constantly refers to Dostoyevsky in his notes and drafts throughout the winter of 1886–1887. Nietzsche also wrote abstracts of several of Dostoyevsky's works.

[[Freud]] wrote an article titled ''[[Dostoevsky and Parricide]]'', asserting that the greatest works in world literature are all about [[parricide]]; though he is critical of Dostoyevsky's work overall, his inclusion of ''[[The Brothers Karamazov]]'' among the three greatest works of literature is remarkable.

==Bibliography==
===Fiction===
Dostoyevsky's works of fiction includes 2 translations, 15 novels and novellas, and 17 short stories. Many of his longer novels were first published in [[Serial (literature)|serialized form]] in various [[literary magazine]]s and [[journal]]s (see the individual articles). The years given below indicate the year in which the novel's final part or first complete book edition was published. Because English translations of Dostoyevsky's works have differentiated throughout the years, many of his novels and short stories are known by several titles.

===Translated books===
*''[[Eugénie Grandet]]', ([[Honore de Balzac]]) (1843)
* ''La dernière Aldini'' ([[George Sand]]) (1843)

====Novels and novellas====
*''[[Poor Folk]]'' (Бедные люди [''Bednye lyudi''], 1846)
*''[[The Double: A Petersburg Poem]]'' (Двойник: Петербургская поэма [''Dvoynik: Peterburgskaya poema''], 1846)
*''[[Netochka Nezvanova (novel)|Netochka Nezvanova]]'' (Неточка Незванова [''Netochka Nezvanova''], 1849)
*''[[Uncle's Dream]]'' (Дядюшкин сон [''Dyadyushkin son''], 1859)
*''[[The Village of Stepanchikovo]]'' (Село Степанчиково и его обитатели [''Selo Stepanchikovo i ego obitateli''], 1859)
*''[[Humiliated and Insulted]]'' (Униженные и оскорбленные [''Unizhennye i oskorblennye''], 1861)
*''[[The House of the Dead (novel)|The House of the Dead]]'' (Записки из мертвого дома [''Zapiski iz mertvogo doma''], 1862)
*''[[Notes from Underground]]'' (Записки из подполья [''Zapiski iz podpolya''], 1864)
*''[[Crime and Punishment]]'' (Преступление и наказание [''Prestuplenie i nakazanie''], 1866)
*''[[The Gambler (novel)|The Gambler]]'' (Игрок [''Igrok''], 1867)
*''[[The Idiot]]'' (Идиот [''Idiot''], 1869). Translated into English by [[Henry Carlisle]] and [[Olga Carlisle]].
*''[[The Eternal Husband]]'' (Вечный муж [''Vechnyj muzh''], 1870)
*''[[The Possessed (novel)|Demons]]'' (Бесы [''Besy''], 1872)
*''[[The Adolescent]]'' (Подросток [''Podrostok''], 1875)
*''[[The Brothers Karamazov]]'' (Братья Карамазовы [''Brat'ya Karamazovy''], 1880)

====Short stories====
*"[[Mr. Prokharchin]]" ("Господин Прохарчин" ["Gospodin Prokharchin"], 1846)
*"Novel in Nine Letters" ("Роман в девяти письмах" ["Roman v devyati pis'mah"], 1847)
*"The Landlady" ("Хозяйка" ["Hozyajka"], 1847)
*"The Jealous Husband" ("Чужая жена и муж под кроватью" ["Chuzhaya zhena i muzh pod krovat'yu"], 1848)
*"A Weak Heart" ("Слабое сердце" ["Slaboe serdze"], 1848)
*"Polzunkov" ("Ползунков" ["Polzunkov"], 1848)
*"[[An Honest Thief|The Honest Thief]]" ("Честный вор" ["Chestnyj vor"], 1848)
*"[[A Christmas Tree and a Wedding|The Christmas Tree and a Wedding]]" ("Елка и свадьба" ["Elka i svad'ba"], 1848)
*"[[White Nights (short story)|White Nights]]" ("Белые ночи" ["Belye nochi"], 1848)
*"A Little Hero" ("Маленький герой" ["Malen'kij geroj"], 1849)
*"[[A Nasty Story|A Nasty Anecdote]]" ("Скверный анекдот" ["Skvernyj anekdot"], 1862)
*"[[The Crocodile (short story)|The Crocodile]]" ("Крокодил" ["Krokodil"], 1865)
*"[[Bobok]]" ("Бобок" ["Bobok"], 1873)
*"The Heavenly Christmas Tree" ("Мальчик у Христа на ёлке" ["Mal'chik u Hrista na elke"], 1876)
*"[[A Gentle Creature|The Meek One]]" ("Кроткая" ["Krotkaja"], 1876)
*"[[The Peasant Marey]]" ("Мужик Марей" ["Muzhik Marej"], 1876)
*"[[The Dream of a Ridiculous Man]]" ("Сон смешного человека" ["Son smeshnogo cheloveka"], 1877)

===Non-fiction===
*''[[A Writer's Diary]]'', collected essays
*''Winter Notes on Summer Impressions'' (1863)
*''[[A Writer's Diary]]'' (Дневник писателя [''Dnevnik pisatelya''], 1873–1881)
*''Letters'' (collected in English translations in five volumes of ''Complete Letters'')

==See also==
{{div col|colwidth=30em}}
*[[Albert Camus]]
*[[Aleksandr Solzhenitsyn]]
*[[Existentialism]]
*[[List of Russian philosophers]]
*[[Lev Shestov]]
*[[Nikolai Berdyaev]]
*[[Nikolay Strakhov]]
*[[Russian Orthodox Church]]
*[[Vasily Rozanov]]
{{div col end}}

==References==
{{reflist|colwidth=30em}}

===Bibliography===
{{refbegin||colwidth=30em}}
*{{cite book
|title=[[Humiliated and Insulted]]
|last=Avsey
|first=Ignat
|others=Trans. Avsey
|year=2008
|publisher=Oneworld Classics
|location=[[London]]
|chapter=Extra Material on Fyodor Dostoevsky's ''Humiliated and Insulted''
|isbn=978-1847490452
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Seeds of Revolt, 1821-1849
|last=Frank
|first=Joseph
|year=1979
|origyear=First published 1976
|publisher=[[Princeton University Press]]
|location=[[Princeton, New Jersey|Princeton]]
|isbn=978-0691013558
|url=http://books.google.com/books?id=pDEAXltygUIC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Years of Ordeal, 1850-1859
|last=Frank
|first=Joseph
|year=1987
|origyear=First published 1983
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691014227
|url=http://books.google.com/books?id=K98hhw0IEHgC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Stir of Liberation, 1860-1865
|last=Frank
|first=Joseph
|year=1988
|origyear=First published 1986
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691014524
|url=http://books.google.com/books?id=QJj6qb6Rh3AC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Miraculous Years, 1865-1871
|last=Frank
|first=Joseph
|year=1997
|origyear=First published 1995
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691015873
|url=http://books.google.com/books?id=iAs4Lz5yog0C
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Mantle of the Prophet, 1871-1881
|last=Frank
|first=Joseph
|year=2003
|origyear=First published 2002
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691115696
|url=http://books.google.com/books?id=mQqonU-pweEC
|ref = harv}}
*{{cite book
|title=Dostoevsky: His Life and Work
|last=Mochulsky
|first=Konstantin
|others=Trans. Minihan, Michael A
|year=1973
|origyear=First published 1967
|publisher=Princeton University Press
|location=Princeton
|isbn=0691012997
|url=http://books.google.com/books?id=mDKphT8_XLsC
|ref = harv}}
{{refend}}

==External links==

{{Sister project links
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*{{gutenberg author| id=Fyodor+Dostoyevsky|name=Fyodor Dostoyevsky}}
*{{worldcat id|id=lccn-n79-29930}}
*{{IBList |type=author|id=96|name=Fyodor Dostoevsky}}
*{{imdb_name|id=0234502|name=Fyodor Dostoevsky}}
*

{{Fyodor Dostoyevsky}}

{{Authority control|PND=118527053|LCCN=n/79/029930|VIAF=104023256}}

{{Persondata
|NAME= Dostoyevsky, Fyodor Mikhailovich
|ALTERNATIVE NAMES= Dostoevsky, Fyodor Mikhailovich; Фёдор Миха́йлович Достое́вский (Russian)
|SHORT DESCRIPTION= Russian novelist
|DATE OF BIRTH= {{Birth date|1821|11|11|mf=y}}
|PLACE OF BIRTH= Moscow
|DATE OF DEATH= {{Death date|1881|2|9|mf=y}}
|PLACE OF DEATH= Saint Petersburg
}}
{{DEFAULTSORT:Dostoyevsky, Fyodor}}
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[[Category:Russian prisoners and detainees]]

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In [[mathematics]], the [[repeating decimal]] '''0.999...''' (which may also be written as '''0.9''', <math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math>, '''0.(9)''', or as "0." followed by any number of 9s in the repeating decimal) denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a [[Cantor's diagonal argument|classic proof]] of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks to students. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument from authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification.

The equality of 0.999... and 1 is closely related to the absence of nonzero [[infinitesimal]] real numbers. Some alternative number systems, such as the [[hyperreals]], do contain nonzero infinitesimals. In these systems, unlike in the reals, there can be numbers whose difference from 1 is less than any rational number. Other systems, known as the [[p-adic numbers|''p''-adic numbers]], have a different form of "decimal expansions" which behave quite differently than expansions of real numbers. Although the real numbers are the most common object of study in the field of [[mathematical analysis]], the hyperreals and ''p''-adics both have applications in that area.

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.

===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111…, in which the digits repeat without end. This decimal yields a quick proof for {{nowrap|1=0.999… = 1}}. Multiplication of 9 times 1 produces 9 in each digit, so {{nowrap|9 × 0.111…}} equals 0.999… and {{nowrap|9 × {{frac|1|9}}}} equals 1, so {{nowrap|1=0.999… = 1}}:

:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
\end{align}
</math>

Another form of this proof multiplies {{nowrap|1=⅓ = 0.333…}} by 3.

===Digit manipulation===
When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number; it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It should be noted that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p. 170</ref>

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

===Dedekind cuts===
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form
:<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p. 399</ref>
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
:<math>\begin{align}\tfrac{a}{b}<1\end{align},</math>
which implies
:<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp. 388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (exponentiation)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by the smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of – rather than independent alternatives to – the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to the real numbers, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note however that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> [[A. H. Lightstone]] developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

:<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1}\dots,</math>

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>\underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}.</math><ref>Katz & Katz 2010</ref>
All such interpretations of "0.999..." are infinitely close to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111...2 follows directly from Berlekamp's Rule.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + ''x'' = 1"
has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the [[sign and magnitude]] or [[ones' complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}
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*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
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* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
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* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |jstor=2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |jstor=2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |jstor=2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi= |accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |jstor=2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{Cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |jstor=493261 |doi=10.2307/493261 |postscript={{inconsistent citations}}}}
*{{Cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |jstor=40248503 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936 |postscript={{inconsistent citations}}}}
*{{Cite book |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student’s understanding and use of mathematical definitions in real analysis |editor=Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22 |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146 |postscript={{inconsistent citations}}}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
*{{Cite journal |last=Ferrini-Mundy |first=J. |last2=Graham |first2=K. |year=1994 |chapter=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky |title=Research issues in undergraduate mathematics learning |postscript={{inconsistent citations}}}}
* {{cite arxiv | eprint=math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006 | class=math.NT }}
*{{Cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501 |issue=3 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=Maria Angeles |last=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999…=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf |postscript={{inconsistent citations}}}}
*{{Cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |jstor=2975103 |doi=10.2307/2975103 |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |jstor=749807 |doi=10.2307/749807 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33 |postscript={{inconsistent citations}}}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2011-11-05T16:30:04Z

Paul August: Undid revision 459154105 by 99.181.142.31 (talk)

{{Portal:Mathematics/Featured article template}}
[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' (which may also be written as '''0.9''', <math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math>, '''0.(9)''', or as 0. followed by any number of 9s in the repeating decimal) denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a [[Cantor's diagonal argument|classic proof]] of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks to students. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument from authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification.

The equality of 0.999... and 1 is closely related to the absence of nonzero [[infinitesimal]] real numbers. Some alternative number systems, such as the [[hyperreals]], do contain nonzero infinitesimals. In these systems, unlike in the reals, there can be numbers whose difference from 1 is less than any rational number. Other systems, known as the [[p-adic numbers|''p''-adic numbers]], have a different form of "decimal expansions" which behave quite differently than expansions of real numbers. Although the real numbers are the most common object of study in the field of [[mathematical analysis]], the hyperreals and ''p''-adics both have applications in that area.

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.

===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111…, in which the digits repeat without end. This decimal yields a quick proof for {{nowrap|1=0.999… = 1}}. Multiplication of 9 times 1 produces 9 in each digit, so {{nowrap|9 × 0.111…}} equals 0.999… and {{nowrap|9 × {{frac|1|9}}}} equals 1, so {{nowrap|1=0.999… = 1}}:

:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
\end{align}
</math>

Another form of this proof multiplies {{nowrap|1=⅓ = 0.333…}} by 3.

===Digit manipulation===
When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number; it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It should be noted that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p. 170</ref>

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

===Dedekind cuts===
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form
:<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p. 399</ref>
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
:<math>\begin{align}\tfrac{a}{b}<1\end{align},</math>
which implies
:<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp. 388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (exponentiation)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by the smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of – rather than independent alternatives to – the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to the real numbers, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note however that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> [[A. H. Lightstone]] developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

:<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1}\dots,</math>

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>\underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}.</math><ref>Katz & Katz 2010</ref>
All such interpretations of "0.999..." are infinitely close to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111...2 follows directly from Berlekamp's Rule.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + ''x'' = 1"
has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the [[sign and magnitude]] or [[ones' complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
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*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
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* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
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* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |accessdate=2009-05-03}}
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*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi= |accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |jstor=2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{Cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |jstor=493261 |doi=10.2307/493261 |postscript={{inconsistent citations}}}}
*{{Cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |jstor=40248503 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936 |postscript={{inconsistent citations}}}}
*{{Cite book |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student’s understanding and use of mathematical definitions in real analysis |editor=Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22 |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146 |postscript={{inconsistent citations}}}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
*{{Cite journal |last=Ferrini-Mundy |first=J. |last2=Graham |first2=K. |year=1994 |chapter=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky |title=Research issues in undergraduate mathematics learning |postscript={{inconsistent citations}}}}
* {{cite arxiv | eprint=math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006 | class=math.NT }}
*{{Cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501 |issue=3 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=Maria Angeles |last=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999…=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf |postscript={{inconsistent citations}}}}
*{{Cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |jstor=2975103 |doi=10.2307/2975103 |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |jstor=749807 |doi=10.2307/749807 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33 |postscript={{inconsistent citations}}}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Fjodor Michailowitsch Dostojewski

2011-08-25T20:55:34Z

Paul August: Short story titles are enclosed in quotes (See MOS:T)

{{Redirect2|Dostoyevsky|Dostoevsky|other uses|Dostoyevsky (disambiguation)}}
{{Eastern Slavic name|Mikhaylovich|Dostoyevsky}}
{{Infobox writer 
| name = Fyodor Dostoyevsky
| image = Dostoevsky.jpg
| caption = 1879
| birth_name = Fyodor Mikhaylovich Dostoyevsky
| birth_date = {{Birth date|1821|11|11}}
| birth_place = [[Moscow]], [[Russian Empire]]
| death_date = {{Death date and age|1881|2|9|1821|11|11}}
| death_place = [[Saint Petersburg]], Russian Empire
| occupation = [[Novelist]], [[short story]] writer, [[essay]]ist
| language = [[Russian language|Russian]]
| nationality = [[Russians|Russian]]
| period = 1846–1881
| genre =
| subject =
| movement =
| religion = [[Russian Orthodox]]
| notableworks = ''[[Notes from Underground]]'' ''[[Crime and Punishment]]'' ''[[The Idiot]]'' ''[[The Brothers Karamazov]]''
| spouse = Mariya Dmitriyevna Isayeva (1857–64) [her death]
[[Anna Grigoryevna Snitkina]] (1867–1881) [his death]
| children = Sofiya (1868), [[Lyubov Dostoyevskaya|Lyubov]] (1869—1926), Fyodor (1875–1878)
| relatives =
| signature = Fyodor Dostoyevsky Signature.svg
}}
'''Fyodor Mikhaylovich Dostoyevsky'''<ref>({{lang-ru|Фёдор Миха́йлович Достое́вский}}, {{IPA-ru|ˈfʲodər mʲɪˈxajləvʲɪtɕ dəstɐˈjefskʲɪj|pron|ru-Dostoevsky.ogg}}; loose phonetic pronunciation: fyo-der mi-(k)hail-a-vitch das-ta-yef-skee. His last name has been transcribed in English as Dostoevski, Dostoevsky, Dostoievski, Dostoievsky, Dostoyevski and Dostoyevsky. Some early translations render his first name by its English equivalent, Theodore.</ref> (November 11, 1821 – February 9, 1881<ref>[[Old Style and New Style dates|Old Style date]] October 30, 1821 – January 29, 1881.</ref>) was a [[Russia]]n writer of [[novel]]s, [[short story|short stories]] and [[essay]]s.<ref name="pravoslavye.org.ua">[http://www.pravoslavye.org.ua/index.php?r_type=article&action=fullinfo&id=13375 Ukrainian origin of Dostoyevsky (Українське коріння Достоєвського)]</ref> He is best known for his novels ''[[Crime and Punishment]]'', ''[[The Idiot]]'' and ''[[The Brothers Karamazov]]''.

Dostoyevsky's literary works explored human psychology in the troubled political, social and spiritual context of 19th-century Russian society. Considered by many as a founder or precursor of 20th-century [[existentialism]], Dostoyevsky wrote, with the embittered voice of the anonymous "underground man", '' [[Notes from Underground]]'' (1864), which was called the "best overture for existentialism ever written" by [[Walter Kaufmann (philosopher)|Walter Kaufmann]].<ref>Existentialism: from Dostoyevsky to Sartre, ed. Walter Kaufmann, Penguin Books, 1989 ISBN 0452009308 p. 12</ref> Dostoyevsky is often acknowledged by critics as one of the greatest and most prominent psychologists in [[world literature]].<ref name="BritannicaRussianLit">{{Cite web|url=http://www.britannica.com/EBchecked/topic/513793/Russian-literature|publisher=Encyclopedia Britannica|accessdate=2008-04-11|title=Russian literature|quote=Dostoyevsky, who is generally regarded as one of the supreme psychologists in world literature, sought to demonstrate the compatibility of Christianity with the deepest truths of the psyche.}}</ref>

==Biography==
===Early life===
[[File:Wki Dostoyevsky Street 2 Moscow Mariinsky Hospital.jpg|thumb|240px|Mariinsky Hospital in Moscow, Dostoyevsky's birthplace]]Dostoyevsky was born in Moscow, the second of seven children born to Mikhail and Maria Dostoyevsky.<ref>The Best Short Stories of Dostoevsky: Translated with an Introduction by David Magarshack. New York: The Modern Library, Random House; 1971.</ref> Dostoyevsky's father Mikhail was a doctor and a devout Christian, who practiced at the Mariinsky Hospital for the Poor in Moscow. The family lived in a small apartment in the hospital grounds, and it wasn't until he was 16 years old, that Dostoyevsky moved to [[St Petersburg]] to attend a [[Military engineering-technical university|Military Engineering Institute]]. The hospital was located in one of the city's worst areas; local landmarks included a cemetery for criminals, a lunatic asylum, and an orphanage for abandoned infants. This urban landscape made a lasting impression on the young Dostoyevsky, whose interest in and compassion for the poor, oppressed and tormented was apparent in his life and works. Although it was forbidden by his parents, Dostoyevsky liked to wander out to the hospital garden, where the patients sat to catch a glimpse of the sun. The young Dostoyevsky appreciated spending time with these patients and listening to their stories.

There are many stories of Dostoyevsky's father's despotic treatment of his children, but this despotism was tempered by his extreme care for his children and their upbringing. After returning home from work, he would take a nap while his children, ordered to keep absolutely silent, stood by their slumbering father in shifts and swatted the flies that came near his head. But the father was also careful to send his children to private schools where they would not be beaten. In the opinion of Joseph Frank, author of a definitive biography of Dostoyevsky, the father figure in ''[[The Brothers Karamazov]]'' is not based on Dostoyevsky's own father. Letters and personal accounts demonstrate that they did have a fairly loving relationship.

[[File:Dostoevskij 1847.jpg|thumb|upright|The young Dostoyevsky, in an 1847 portrait by [[Konstiantyn Trutovsky|Trutovsky]]]]

In 1837, shortly after his mother died of [[tuberculosis]], Dostoyevsky and his brother were sent to St Petersburg to attend the [[Military engineering-technical university|Nikolayev Military Engineering Institute]], nowadays called the [[Military engineering-technical university|Military Engineering-Technical University]].<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]]</ref> Fyodor's father died in 1839. Though it has never been proven, it is believed by some that he was murdered by his own [[serf]]s.<ref>[http://worldebooklibrary.com/eBooks/Coradella_Collegiate_Bookshelf_Collection/Dostoevsky-notesfromtheunderground.pdf Notes from the Underground] Coradella Collegita Bookshelf edition, ''About the Author''.</ref> According to one account, the serfs became enraged during one of his drunken fits of violence, and after restraining him, poured [[vodka]] into his mouth until he drowned. A similar account appears in ''Notes from Underground''. Another story holds that Mikhail died of natural causes, and a neighboring landowner invented the story of his murder so that he could buy the estate at a cheaper price. Some, like Sigmund Freud in his 1928 article, "[[Dostoevsky and Parricide]]", have argued that his father's personality had influenced the character of Fyodor Pavlovich Karamazov, the "wicked and sentimental buffoon", father of the main characters in his 1880 novel ''[[The Brothers Karamazov]]'', but such claims fail to withstand the scrutiny of many critics{{Who|date=June 2009}}.

Dostoyevsky suffered from [[epilepsy]] and his first seizure occurred when he was nine years old.<ref>[http://www.epilepsy.com/epilepsy/famous_writers.html Epilepsy.com] Famous authors with epilepsy.</ref> Epileptic seizures recurred sporadically throughout his life, and Dostoyevsky's experiences are thought<ref>Dostoyevsky, Fyodor, Richard Pevear, and Larissa Volokhonsky. The Idiot. New York: Vintage, 2001. Print. Introduction pp. xix</ref> to have formed the basis for his description of Prince Myshkin's epilepsy in his novel ''[[The Idiot]]'' and that of Smerdyakov in ''The Brothers Karamazov'', among others.

At the [[Military engineering-technical university|Saint Petersburg Institute of Military Engineering]]<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]],</ref> Dostoyevsky was taught mathematics, a subject he despised. However, he also studied literature by [[Shakespeare]], [[Blaise Pascal|Pascal]], [[Victor Hugo]] and [[E.T.A. Hoffmann]]. Though he focused on areas different from mathematics, he did well in the exams and received a commission in 1841. That year, influenced by the German poet/playwright [[Friedrich Schiller]], he wrote two romantic plays: ''[[Maria Stuart (play)|Mary Stuart]]'' and ''[[Boris Godunov]]''. The plays have not been preserved. Dostoyevsky described himself as a "dreamer" when he was a young man. He also revered [[Schiller]] at that age. However, in the years during which he wrote his great masterpieces, his opinions changed and he sometimes made fun of Schiller.

Dostoyevsky was made a lieutenant in 1842, and left the Engineering Academy the following year. He completed a translation into Russian of [[Balzac]]'s novel ''[[Eugénie Grandet]]'' in 1843, but it brought him little to no attention. Dostoyevsky started to write his own fiction in late 1844 after leaving the army. In 1846, his first work, the epistolary short novel, ''[[Poor Folk]]'', printed in the almanac ''A Petersburg Collection'' (published by [[Nikolay Nekrasov|N. Nekrasov]]), was met with great acclaim. As legend has it, the editor of the magazine, poet [[Nikolai Nekrasov]], walked into the office of liberal critic [[Vissarion Belinsky]] and announced, "A new [[Nikolai Gogol|Gogol]] has arisen!" Belinsky, his followers, and many others agreed. After the novel was fully published in book form at the beginning of the next year, Dostoyevsky became a literary celebrity at the age of 24.

In 1846, Belinsky and many others reacted negatively to his novella, ''[[The Double: A Petersburg Poem|The Double]]'', a psychological study of a bureaucrat whose alter ego overtakes his life. Dostoyevsky's fame began to fade. Much of his work after ''[[Poor Folk]]'' received ambivalent reviews and it seemed that Belinsky's prediction that Dostoyevsky would be one of the greatest writers of Russia was mistaken.

===Exile in Siberia===
[[File:Omsk Dostoyevskiy Monument.jpg|thumb|upright|Statue of Dostoyevsky in [[Omsk]]]]
Dostoyevsky was incarcerated on 23 April 1849 for being part of the [[Liberalism|liberal]] intellectual group the [[Petrashevsky Circle]]. [[Tsar]] [[Nicholas I of Russia|Nicholas I]], after seeing the [[Revolutions of 1848]] in Europe, was harsh on any type of underground organization which he felt could put [[autocracy]] in jeopardy. On November 16 of that year, Dostoyevsky, along with other members of the Petrashevsky Circle, was [[death sentence|sentenced to death]]. After a [[mock execution]], in which he and other members of the group stood outside in freezing weather waiting to be shot by a firing squad, Dostoyevsky's sentence was commuted to four years of [[exile]] with hard labour at a [[katorga]] prison camp in [[Omsk]], [[Siberia]]. Later, Dostoyevsky described his years of suffering to his brother, as being, "shut up in a coffin." In describing the dilapidated barracks which "should have been torn down years ago", he wrote:
''{{quote|In summer, intolerable closeness; in winter, unendurable cold. All the floors were rotten. Filth on the floors an inch thick; one could slip and fall... We were packed like herrings in a barrel... There was no room to turn around. From dusk to dawn it was impossible not to behave like pigs... Fleas, lice, and black beetles by the bushel...''<ref>Frank 76. Quoted from Pisma, I: 135–37.</ref>}} This experience inspired him to write ''[[The House of the Dead (novel)|The House of the Dead]]''.

Dostoyevsky was released from prison in 1854, and was required to serve in the Siberian Regiment. He spent the following five years as a private (and later lieutenant) in the Regiment's Seventh Line Battalion, stationed at the fortress of [[Semey|Semipalatinsk]], now in [[Kazakhstan]]. While there, he began a relationship with Maria Dmitrievna Isayeva, the wife of an acquaintance in Siberia. After her husband's death, they married in February 1857.

===Post-prison maturation as a writer===
[[File:Valikhanov.jpg|thumb|upright|Dostoyevsky (right) and the [[Kazakhs|Kazakh]] scholar [[Shokan Walikhanuli]] in 1859]]

Dostoyevsky's experiences in prison and the army resulted in major changes in his political and religious convictions. First, his ordeal somehow caused him to become disillusioned with "Western" ideas; he repudiated the contemporary Western European philosophical movements, and instead paid greater tribute in his writings to traditional, rustic Russian values exemplified in the [[Slavophile]] concept of ''[[sobornost]]''. But even more significantly, he had what his biographer Joseph Frank describes as a [[Religious conversion|conversion]] experience in prison, which greatly strengthened his Christian, and specifically [[Russian Orthodox|Orthodox]], faith.{{sfn|Frank|1987|pp=124–27}} Dostoyevsky would later depict his conversion experience in the short story, ''[[The Peasant Marey]]'' (1876).

In his writings, Dostoyevsky started to extol the virtues of humility, submission, and suffering.<ref name="Nab81Censors">[[Vladimir Nabokov]] (1981) ''[[Lectures on Russian Literature]]'', lecture on ''Russian Writers, Censors, and Readers'', p.14</ref> He now displayed a much more critical stance on contemporary European philosophy and turned with intellectual rigour against the [[Nihilist movement|Nihilist]] and Socialist movements; and much of his post-prison work—particularly the novel, ''[[The Possessed (novel)|The Possessed]]'', and the essays, ''[[A Writer's Diary|The Diary of a Writer]]''—contains both criticism of socialist and nihilist ideas, as well as thinly veiled parodies of contemporary Western-influenced Russian intellectuals ([[Timofey Granovsky|Timofey Granovskiy]]), revolutionaries ([[Sergey Nechayev|Sergey Nyechayev]]), and even fellow novelists ([[Ivan Turgenev|Ivan Turgyenyev]]).<ref>Dostoevsky the Thinker James P. Scanlan. Dostoevsky the Thinker. Ithaca: Cornell University Press, 2002. xiii, p. 251</ref><ref>[http://ourworld.compuserve.com/homepages/jim_forest/pevear.htm Dostoevsky's View of Evil] Reprinted from ''In Communion'', April 1998.</ref> In social circles, Dostoyevsky allied himself with well-known conservatives, such as the statesman [[Konstantin Pobedonostsev|Konstantin Pobyedonostsyev]]. His post-prison essays praised the tenets of the [[Pochvennichestvo|Pochvyennichyestvo]] movement, a late-19th century Russian nativist ideology closely aligned with [[Slavophile|Slavophilism]].

Dostoyevsky's post-prison fiction abandoned the Western European-style domestic melodramas and quaint character studies of his youthful work in favor of dark, more complex storylines and situations, played-out by brooding, tortured characters—often styled partly on Dostoyevsky himself—who agonized over [[existentialism|existential]] themes of spiritual torment, religious awakening, and the psychological confusion caused by the conflict between traditional Russian culture and the influx of modern, Western philosophy. Nonetheless, this does not take from the debt which Dostoyevsky owed to earlier Western-influenced writers such as [[Gogol]] whose work grew from the irrational and anti-authoritarian spiritualist ideas contained within the [[Romantic movement]] which had immediately preceded Dostoyevsky in West Europe. However, Dostoyevsky's major novels focused on the idea that [[utopia]] and [[positivist]] ideas were unrealistic and unobtainable.<ref>{{Cite book|last = Sirotkina|first = Irina|title = Diagnosing Literary Genius: A Cultural History of Psychiatry in Russia, 1880|year = 1996|publisher = [[Johns Hopkins University Press]]|page = 55|isbn = 0801867827}}</ref>

===Later literary career===
[[File:Dostoevskij 1863.jpg|thumb|upright|Dostoyevsky in 1863]][[File:Fyodor Dostoevsky house.jpg|thumb|right|170px|Dostoyevsky's last address where he died, now a memorial and literary museum, St Petersburg.]]

In December 1859, Dostoyevsky returned to [[Saint Petersburg]], where he ran a series of unsuccessful literary journals, ''[[Vremya (magazine)|Vremya]]'' (Time) and ''[[Epoch (Russian magazine)|Epokha]]'' (Epoch), with his older brother [[Mikhail Dostoyevsky|Mikhail]].<ref>{{Cite book|title=F. M. Dostoyevsky. Collection of works in 15 volumes |volume=11|year=1993 |publisher=Nauka |location=Leningrad|pages=361–365 |chapter=A few words about Mikhail Mikhailovich Dostoyevsky}}</ref> The former was shut down as a consequence of its coverage of the [[January Uprising|Polish Uprising of 1863]]. That year Dostoyevsky traveled to Europe and frequented gambling casinos. There he met [[Polina Suslova|Apollinaria Suslova]], the model for Dostoyevsky's "proud women", such as the two characters named Katerina Ivanovna, in ''[[Crime and Punishment]]'' and in ''[[The Brothers Karamazov]]''.

Dostoyevsky was devastated by his wife's death in 1864, which was followed shortly thereafter by his brother's death. He was financially crippled by business debts; furthermore, he decided to assume the responsibility of his deceased brother's outstanding debts, as well providing for his wife's son from her earlier marriage and his brother's widow and children. Dostoyevsky sank into a deep depression, frequenting gambling parlors and accumulating massive losses at the tables.

Dostoyevsky suffered from an acute [[Problem gambling|gambling compulsion]] and its consequences. He completed ''Crime and Punishment'', possibly his best known novel, in a mad hurry because he was in urgent need of an advance from his publisher. He had been left practically penniless after a gambling spree. Dostoyevsky wrote ''[[The Gambler (novel)|The Gambler]]'' simultaneously in order to satisfy an agreement with his publisher Stellovsky who, if he did not receive a new work, would have claimed the copyrights to all of Dostoyevsky's writings.<ref>{{cite web|title=Fyodor Dostoevsky||publisher=Russia Today (RT)|url=http://russiapedia.rt.com/prominent-russians/literature/fyodor-dostoevsky/|accessdate=12 July 2011}}</ref>

Motivated by the dual wish to escape his creditors at home and to visit the casinos abroad, Dostoyevsky traveled to Western Europe. There, he attempted to rekindle a love affair with Suslova, but she refused his marriage proposal. Dostoyevsky was heartbroken, but soon met [[Anna Dostoyevskaya|Anna Grigorevna Snitkina]], a twenty-year-old [[stenographer]]. Shortly before marrying her in 1867, he dictated ''The Gambler'' to her.<ref>{{cite book |last=Dostoevsky |first=Fyodor |others=Notes and Introduction by Maire Jaanus. Translated by [[Constance Garnett]] |title=The Brothers Karamazov |series=Barnes & Noble Classics |year=2004 |origyear=First published 1879–1880 |publisher=Barnes & Noble Books |location=New York, NY |isbn=978-1-59308-045-7 |oclc=34325193 |page=703 |chapter=Endnotes |quote=Anna Grigorievna Snitkina, Dostoyevsky's second wife, was a stenographer to whom Dostoyevsky dictated his novel ''The Gambler'' in 1866; they married the following year.}}</ref> From 1873 to 1881 he published the ''Writer's Diary'', a monthly journal of short stories, sketches, and articles on current events. The journal was an enormous success.

Dostoyevsky influenced, and was himself influenced, by the philosopher [[Vladimir Sergeyevich Solovyov]]. Solovyov was the inspiration for the characters [[Ivan Karamazov]] and [[Alyosha Karamazov]].<ref>Zouboff, Peter, Solovyov on Godmanhood: Solovyov’s Lectures on Godmanhood Harmon Printing House: Poughkeepsie, New York, 1944; see Czeslaw Milosz’s introduction to Solovyov’s War, Progress and the End of History. Lindisfarne Press: Hudson, New York 1990.</ref>

[[File:Fyodor Mikahailovich Dostoyevsky's Study in St Petersburg.jpg|right|thumb|180px|Dostoyevsky's study in [[Saint Petersburg]].]]
In 1877, Dostoyevsky gave the keynote [[eulogy]] at the funeral of his friend, the poet [[Nikolai Alekseevich Nekrasov|Nekrasov]], to much controversy{{Who|date=June 2009}}. On 8 June 1880, shortly before he died, he gave his famous [[Alexander Pushkin|Pushkin]] speech at the unveiling of the [[Pushkin Square|Pushkin monument in Moscow]].<ref>Dostoyevsky [http://az.lib.ru/d/dostoewskij_f_m/text_0340.shtml Az.lib.ru Пушкинская речь (Pushkin's style)] (in Russian)</ref> In his later years, Dostoyevsky lived for an extended period at the resort of [[Staraya Russa]] in northwestern Russia, which was closer to [[Saint Petersburg]] and less expensive than German resorts.

===Death===
Dostoyevsky died in St. Petersburg on {{OldStyleDate|9 February|1881|28 January}} of a lung hemorrhage associated with [[emphysema]] and an [[epileptic seizure]]. A copy of the New Testament Bible given to him in Siberia sat on his lap. He was interred in [[Tikhvin Cemetery]] at the [[Alexander Nevsky Monastery]] in [[Saint Petersburg]]. Forty thousand mourners attended his funeral.<ref>Dostoevsky, Fyodor; Introduction to The Idiot, Wordsworth Ed. Ltd, 1996.</ref>

His tombstone reads; ''Verily, Verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.'' (Excerpt from [[Gospel of John|John]] 12:24, which is also the [[Epigraph (literature)|epigraph]] of his final novel, ''The Brothers Karamazov''.)

The rented apartment where he died and spent the last few years of his life is where he wrote his final novel ''The Brothers Karamazov''. The apartment, situated in a building at 5 Kuznechnyi pereulok, has been restored with old photographs to how it looked when he lived there. It opened in 1971 as the Dostoyevsky House Museum and is a popular tourist attraction in the city.<ref>{{cite book |title=St Petersburg |last= Woodworth|first=Bradley |authorlink= |coauthors= Harold Bloom, Constance Richards|year=2005 |editor=Harold Bloom|publisher=Infobase Publishing |location= |isbn=0791083845, 9780791083840 |page=69 |pages= |url=http://books.google.com/books?id=tMn6qHyTIywC&dq=dostoevsky+house+museum,+St+Petersburg&source=gbs_navlinks_s |accessdate=19 November 2010}}</ref>

==Influence==
[[File:Vasily Perov - Портрет Ф.М.Достоевского - Google Art Project.jpg|thumb|upright|Portrait of Dostoyevsky in 1872 painted by [[Vasily Perov]].]]

Some, like journalist [[Otto Friedrich]],<ref>{{Cite news|publisher=Time Magazine|url=http://www.time.com/time/magazine/article/0,9171,943893,00.html?promoid=googlep|accessdate=2008-04-10|title=Freaking-Out with Fyodor|author=Otto Friedrich|date=6 September 1971}}</ref> consider Dostoyevsky to be one of Europe's major novelists, while others like [[Vladimir Nabokov]] maintain that from a point of view of enduring art and individual genius, he is a rather mediocre writer who produced wastelands of literary [[platitude]]s.<ref>Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 98</ref>

Dostoyevsky promoted in his novels religious moralities, particularly those of [[Eastern Orthodox Christianity]].<ref name="BritannicaRussianLit"/> Indeed, "Dostoyevsky and the Religion of Suffering," the essay devoted to Dostoyevsky in [[Eugène-Melchior de Vogüé]]'s ''Le roman russe'' (1886), is widely considered to be the most influential early analysis of the novelist's work, introducing Dostoyevsky and other Russian novelists to the West. Nabokov argued in his University courses at [[Cornell University|Cornell]], that such religious propaganda, rather than artistic qualities, was the main reason Dostoyevsky was praised and regarded as a 'Prophet' in Soviet Russia.<ref>Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 104</ref>{{Clarify|Why would the atheistic Soviets praise him for religious propaganda? Was the "propaganda" unconvincing?|date=July 2011}}

[[James Joyce]] and [[Virginia Woolf]] praised his prose. [[Ernest Hemingway]] cited Dostoyevsky as a major influence on his work, in his posthumous collection of sketches ''[[A Moveable Feast]]''. In a book of interviews with Arthur Power (''Conversations with James Joyce''), Joyce praised Dostoyevsky's prose:

''{{quote|...he is the man more than any other who has created modern prose, and intensified it to its present-day pitch. It was his explosive power which shattered the Victorian novel with its simpering maidens and ordered commonplaces; books which were without imagination or violence.}}''
In her essay ''The Russian Point of View'', Virginia Woolf said:

''{{quote|The novels of Dostoevsky are seething whirlpools, gyrating sandstorms, waterspouts which hiss and boil and suck us in. They are composed purely and wholly of the stuff of the soul. Against our wills we are drawn in, whirled round, blinded, suffocated, and at the same time filled with a giddy rapture. Out of [[Shakespeare]] there is no more exciting reading.''<ref>[http://etext.library.adelaide.edu.au/w/woolf/virginia/w91c/chapter16.html The Russian Point of View] Virginia Woolf.</ref>}}

[[File:Dostoevsky-Library Moscow Russia.jpg|thumb|upright|Dostoyevsky monument at the [[Russian State Library]] in Moscow.]]
Dostoyevsky displayed a nuanced understanding of human psychology in his major works. He created an opus of vitality and almost hypnotic power, characterized by feverishly dramatized scenes where his characters are frequently in scandalous and explosive atmospheres, passionately engaged in [[Socratic dialogue]]s. The quest for God, the [[problem of evil]] and suffering of the innocents haunt the majority of his novels.

His characters fall into a few distinct categories: humble and self-effacing Christians ([[Prince Myshkin]], [[Sonya Marmeladova]], [[Alyosha Karamazov]], [[Saint Ambrose of Optina]]), self-destructive [[nihilism|nihilists]] ([[Svidrigailov]], [[Smerdyakov]], [[Stavrogin]], [[Notes from Underground|the underground man]]){{Citation needed|date=May 2009}}, cynical debauchees ([[Fyodor Karamazov]], [[Dmitri Karamazov]]), and rebellious intellectuals ([[Raskolnikov]], [[Ivan Karamazov]], [[Ippolit]]); also, his characters are driven by ideas rather than by ordinary biological or social imperatives. In comparison with [[Leo Tolstoy|Tolstoy]], whose characters are [[Literary realism|realistic]], the characters of Dostoyevsky are usually more symbolic of the ideas they represent, thus Dostoyevsky is often cited as one of the forerunners of [[Symbolism (arts)|Literary Symbolism]], especially [[Russian Symbolism]] (see [[Alexander Blok]]).<ref>Dostoievsky by A. Steinberg p. 112</ref>
[[File:Dostoevsky MR280908.jpg|thumb|upright|Dostoyevsky statue, erected 1918, in front of [[Mariinsky Hospital]], the writer's birthplace in Moscow.]]
Dostoyevsky's novels are compressed in time (many cover only a few days) and this enables him to get rid of one of the dominant traits of [[realism (arts)|realist]] prose, the corrosion of human life in the process of the time flux; his characters primarily embody spiritual values, and these are, by definition, timeless. Other themes include suicide, wounded pride, collapsed family values, spiritual regeneration through suffering, rejection of the West and affirmation of the [[Russian Orthodox Church]] and of [[tsarism]]. Literary scholars such as [[Mikhail Bakhtin]] have characterized his work as "[[Polyphony (literature)|polyphonic]]": Dostoyevsky does not appear to aim for a "single vision", and beyond simply describing situations from various angles, Dostoyevsky engendered fully dramatic novels of ideas where conflicting views and characters are left to develop unevenly into unbearable crescendo.

Dostoyevsky and the other giant of late 19th century [[Russian literature]], [[Lev Nikolayevich Tolstoy]], never met in person, even though each praised, criticized, and influenced the other (Dostoyevsky remarked of Tolstoy's ''[[Anna Karenina]]'' that it was a "flawless work of art"; [[Henri Troyat]] reports that Tolstoy once remarked of ''[[Crime and Punishment]]'' that, "Once you read the first few chapters you know pretty much how the novel will end up").{{Citation needed|date=August 2007}} There was a meeting arranged, but there was a confusion about where the meeting was to take place and they never rescheduled. Tolstoy wept when he learned of Dostoyevsky's death.<ref>Letter from Leo Tolstoy to Nikolai Strakhov, from [http://www.archive.org/stream/lettersoffyodorm00dostuoft#page/n389/mode/2up Letters of Fyodor Michailovitch Dostoevsky to his Family and Friends, page 337], Chatto and Windus, London, 1914.</ref> A copy of ''[[The Brothers Karamazov]]'' was found on the nightstand next to Tolstoy's deathbed at the [[Lev Tolstoy (settlement)|Astapovo]] railway station.

[[File:450px-Grab-dostojewsky.jpg|thumb|upright|Dostoyevsky's tomb at the [[Alexander Nevsky Monastery]]]]

===Dostoyevsky on Jews in Russia===
{{Cleanup|section|date=May 2011}}
Several writers (including Joseph Frank, Stephen Cassedy, David I. Goldstein, [[Gary Saul Morson]], and Felix Dreizin) have offered various insights and suppositions regarding Dostoyevsky’s views on [[Jews]] and organized [[Jewry in Russia]] — one such view is that Dostoyevsky perceived Jewish [[ethnocentrism]] and Jewish influence to be directly threatening the Russian peasantry in the border regions.{{Citation needed|date=May 2011}} In ''[[A Writer's Diary]]'', Dostoyevsky wrote:

<blockquote>Thus, Jewry is thriving precisely there where the people are still ignorant, or not free, or economically backward. It is there that Jewry has a champ libre. And instead of raising, by its influence, the level of education, instead of increasing knowledge, generating economic fitness in the native population—instead of this the Jew, wherever he has settled, has still more humiliated and debauched the people; there humaneness was still more debased and the educational level fell still lower; there inescapable, inhuman misery, and with it despair, spread still more disgustingly. Ask the native population in our border regions: What is propelling the Jew—and has been propelling him for centuries? You will receive a unanimous answer: mercilessness. He has been prompted so many centuries only by pitilessness to us, only by the thirst for our sweat and blood.

And, in truth, the whole activity of the Jews in these border regions of ours consisted of rendering the native population as much as possible inescapably dependent on them, taking advantage of the local laws. They have always managed to be on friendly terms with those upon whom the people were dependent. Point to any other tribe from among Russian aliens which could rival the Jew by his dreadful influence in this connection! You will find no such tribe. In this respect the Jew preserves all his originality as compared with other Russian aliens, and of course, the reason therefore is that status of status of his, that spirit of which specifically breathes pitilessness for everything that is not Jew, with disrespect for any people and tribe, for every human creature who is not a Jew...<ref name="M. Dostoevsky 1949">Dostoevsky, F. M. ''The Diary of a Writer'', trans. Boris Brasol (New York: Charles Scribner's Sons), 1949.</ref></blockquote>

Dostoyevsky has been noted as both having expressed [[antisemitic]] sentiments as well as standing up for the rights of the Jewish people. In a review of Joseph Frank's book, ''The Mantle of the Prophet'', [[Orlando Figes]] notes that ''A Writer's Diary'' is "filled with politics, literary criticism, and pan-[[Slav]] diatribes about the virtues of the Russian Empire, [and] represents a major challenge to the Dostoyevsky fan, not least on account of its frequent expressions of anti-semitism."<ref>Figes, Orlando. "Dostoevsky's leap of faith This volume concludes a magnificent biography which is also a cultural history", ''Sunday Telegraph'' (London), p.13. September 29, 2002.</ref> Frank, in his foreword for David I. Goldstein's book ''Dostoevsky and the Jews'', attempts to place Dostoyevsky as a product of his time. Frank notes that Dostoyevsky made antisemitic remarks, but that Dostoyevsky's writing and stance, by and large, was one where Dostoyevsky held a great deal of guilt for his comments and positions that were antisemitic.<ref>Frank, Joseph. "Foreword" p. xiv. in Goldstein, David I. ''Dostoevsky and the Jews'', University of Texas Press, 1981. ISBN 0292715285</ref>

Steven Cassedy alleges in his book, ''Dostoevsky's Religion'', that much of the depiction of Dostoyevsky's views as antisemitic omits that Dostoyevsky expressed support for the equal rights of the Russian Jewish population, an unpopular position in Russia at the time.<ref name="Cassedy1">{{Cite book|title= Dostoevsky's Religion |last= Cassedy |first= Steven |year= 2005 |publisher= [[Stanford University Press]] |isbn= 0804751374 |pages= 67–80}}</ref> Cassedy also notes that this criticism of Dostoyevsky also appears to deny his sincerity when he said that he was for equal rights for the Russian Jewish populace and the [[Russian serfdom|serf]]s of his own country (since neither group at that point in history had equal rights).<ref name=Cassedy1/> Cassedy again notes when Dostoyevsky stated that he did not hate Jewish people and was not antisemitic.<ref name=Cassedy1/> Even though Dostoyevsky spoke of the potential negative influence of Jewish people, Dostoyevsky advised Czar [[Alexander II of Russia]] to give them rights to positions of influence in Russian society, such as allowing them access to Professorships at Universities. According to Cassedy, labeling Dostoyevsky anti-Semitic does not take into consideration Dostoyevsky's expressed desire to peacefully reconcile Jews and Christians into a single universal brotherhood of all mankind.<ref name=Cassedy1 />

==Dostoyevsky and existentialism==
[[File:Fyodor Mikahailovich Dostoyevsky's Handwriting 1838.jpg|right|thumb|180px|Dostoyevsky's handwriting.]]
With the publication of ''[[Crime and Punishment]]'', in 1866, Dostoyevsky became one of Russia's most prominent authors. [[Will Durant]], in ''[[The Pleasures of Philosophy]]'' (1953), called Dostoyevsky one of the founding fathers of the philosophical movement known as [[existentialism]], and cited ''[[Notes from Underground]]'' in particular as a founding work of existentialism. For Dostoyevsky, war is the people's rebellion against the idea that [[reason]] guides everything, and thus, reason is not the ultimate guiding principle for either history or [[human|mankind]]. After his 1849 exile to the city of [[Omsk]], Siberia, Dostoyevsky focused heavily on notions of [[suffering]] and [[wiktionary:despair|despair]] in many of his works.

[[Frederich Nietzsche]] referred to Dostoyevsky as "the only psychologist from whom I have something to learn: he belongs to the happiest windfalls of my life, happier even than the discovery of [[Stendhal]]." He said that ''Notes from Underground'' "cried truth from the blood." According to [[Kontinent|Mihajlo Mihajlov]]'s "The Great Catalyzer: Nietzsche and Russian Neo-Idealism", Nietzsche constantly refers to Dostoyevsky in his notes and drafts throughout the winter of 1886–1887. Nietzsche also wrote abstracts of several of Dostoyevsky's works.

[[Freud]] wrote an article titled ''[[Dostoevsky and Parricide]]'', asserting that the greatest works in world literature are all about [[parricide]]; though he is critical of Dostoyevsky's work overall, his inclusion of ''[[The Brothers Karamazov]]'' among the three greatest works of literature is remarkable.

==Bibliography==
===Fiction===
Dostoyevsky's works of fiction includes 15 novels and novellas and 17 short stories. Many of his longer novels were first published in [[Serial (literature)|serialized form]] in various [[literary magazine]]s and [[journal]]s (see the individual articles). The years given below indicate the year in which the novel's final part or first complete book edition was published. Because English translations of Dostoyevsky's works have differentiated throughout the years, many of his novels and short stories are known by several titles.

====Novels and novellas====
*''[[Poor Folk]]'' (Бедные люди [''Bednye lyudi''], 1846)
*''[[The Double: A Petersburg Poem]]'' (Двойник: Петербургская поэма [''Dvoynik: Peterburgskaya poema''], 1846)
*''[[Netochka Nezvanova (novel)|Netochka Nezvanova]]'' (Неточка Незванова [''Netochka Nezvanova''], 1849)
*''[[Uncle's Dream]]'' (Дядюшкин сон [''Dyadyushkin son''], 1859)
*''[[The Village of Stepanchikovo]]'' (Село Степанчиково и его обитатели [''Selo Stepanchikovo i ego obitateli''], 1859)
*''[[Humiliated and Insulted]]'' (Униженные и оскорбленные [''Unizhennye i oskorblennye''], 1861)
*''[[The House of the Dead (novel)|The House of the Dead]]'' (Записки из мертвого дома [''Zapiski iz mertvogo doma''], 1862)
*''[[Notes from Underground]]'' (Записки из подполья [''Zapiski iz podpolya''], 1864)
*''[[Crime and Punishment]]'' (Преступление и наказание [''Prestuplenie i nakazanie''], 1866)
*''[[The Gambler (novel)|The Gambler]]'' (Игрок [''Igrok''], 1867)
*''[[The Idiot]]'' (Идиот [''Idiot''], 1869). Translated into English by [[Henry Carlisle]] and [[Olga Carlisle]].
*''[[The Eternal Husband]]'' (Вечный муж [''Vechnyj muzh''], 1870)
*''[[The Possessed (novel)|Demons]]'' (Бесы [''Besy''], 1872)
*''[[The Adolescent]]'' (Подросток [''Podrostok''], 1875)
*''[[The Brothers Karamazov]]'' (Братья Карамазовы [''Brat'ya Karamazovy''], 1880)

====Short stories====
*"[[Mr. Prokharchin]]" ("Господин Прохарчин" ["Gospodin Prokharchin"], 1846)
*"Novel in Nine Letters" ("Роман в девяти письмах" ["Roman v devyati pis'mah"], 1847)
*"The Landlady" ("Хозяйка" ["Hozyajka"], 1847)
*"The Jealous Husband" ("Чужая жена и муж под кроватью" ["Chuzhaya zhena i muzh pod krovat'yu"], 1848)
*"A Weak Heart" ("Слабое сердце" ["Slaboe serdze"], 1848)
*"Polzunkov" ("Ползунков" ["Polzunkov"], 1848)
*"[[An Honest Thief|The Honest Thief]]" ("Честный вор" ["Chestnyj vor"], 1848)
*"[[A Christmas Tree and a Wedding|The Christmas Tree and a Wedding]]" ("Елка и свадьба" ["Elka i svad'ba"], 1848)
*"[[White Nights (short story)|White Nights]]" ("Белые ночи" ["Belye nochi"], 1848)
*"A Little Hero" ("Маленький герой" ["Malen'kij geroj"], 1849)
*"[[A Nasty Story|A Nasty Anecdote]]" ("Скверный анекдот" ["Skvernyj anekdot"], 1862)
*"[[The Crocodile (short story)|The Crocodile]]" ("Крокодил" ["Krokodil"], 1865)
*"[[Bobok]]" ("Бобок" ["Bobok"], 1873)
*"The Heavenly Christmas Tree" ("Мальчик у Христа на ёлке" ["Mal'chik u Hrista na elke"], 1876)
*"[[A Gentle Creature|The Meek One]]" ("Кроткая" ["Krotkaja"], 1876)
*"[[The Peasant Marey]]" ("Мужик Марей" ["Muzhik Marej"], 1876)
*"[[The Dream of a Ridiculous Man]]" ("Сон смешного человека" ["Son smeshnogo cheloveka"], 1877)

===Non-fiction===
In addition to fiction, Dostoyevsky wrote occasional [[essay]]s, most of which have been collected in ''[[A Writer's Diary]]''.
*''Winter Notes on Summer Impressions'' (1863)
*''[[A Writer's Diary]]'' (Дневник писателя [''Dnevnik pisatelya''], 1873–1881)
*Letters (collected in English translations in five volumes of ''Complete Letters'')

==See also==
{{div col|colwidth=30em}}
*[[Albert Camus]]
*[[Aleksandr Solzhenitsyn]]
*[[Existentialism]]
*[[List of Russian philosophers]]
*[[Lev Shestov]]
*[[Nikolai Berdyaev]]
*[[Nikolay Strakhov]]
*[[Russian Orthodox Church]]
*[[Vasily Rozanov]]
{{div col end}}

==References==
{{reflist|colwidth=30em}}

===Bibliography===
{{refbegin||colwidth=30em}}
*{{cite book
|title=[[Humiliated and Insulted]]
|last=Avsey
|first=Ignat
|others=Trans. Avsey
|year=2008
|publisher=Oneworld Classics
|location=[[London]]
|chapter=Extra Material on Fyodor Dostoevsky's ''Humiliated and Insulted''
|isbn=978-1847490452
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Seeds of Revolt, 1821-1849
|last=Frank
|first=Joseph
|year=1979
|origyear=First published 1976
|publisher=[[Princeton University Press]]
|location=[[Princeton, New Jersey|Princeton]]
|isbn=978-0691013558
|url=http://books.google.com/books?id=pDEAXltygUIC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Years of Ordeal, 1850-1859
|last=Frank
|first=Joseph
|year=1987
|origyear=First published 1983
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691014227
|url=http://books.google.com/books?id=K98hhw0IEHgC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Stir of Liberation, 1860-1865
|last=Frank
|first=Joseph
|year=1988
|origyear=First published 1986
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691014524
|url=http://books.google.com/books?id=QJj6qb6Rh3AC
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Miraculous Years, 1865-1871
|last=Frank
|first=Joseph
|year=1997
|origyear=First published 1995
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691015873
|url=http://books.google.com/books?id=iAs4Lz5yog0C
|ref = harv}}
*{{cite book
|title=Dostoevsky: The Mantle of the Prophet, 1871-1881
|last=Frank
|first=Joseph
|year=2003
|origyear=First published 2002
|publisher=Princeton University Press
|location=Princeton
|isbn=978-0691115696
|url=http://books.google.com/books?id=mQqonU-pweEC
|ref = harv}}
*{{cite book
|title=Dostoevsky: His Life and Work
|last=Mochulsky
|first=Konstantin
|others=Trans. Minihan, Michael A
|year=1973
|origyear=First published 1967
|publisher=Princeton University Press
|location=Princeton
|isbn=0691012997
|url=http://books.google.com/books?id=mDKphT8_XLsC
|ref = harv}}
{{refend}}

==External links==

{{Sister project links
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|commons=Fyodor Dostoevsky
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|q=Fyodor Dostoevsky
|s=Author:Fyodor Dostoevsky
|v=no
|species=no}}
*{{gutenberg author| id=Fyodor+Dostoyevsky|name=Fyodor Dostoyevsky}}
*{{worldcat id|id=lccn-n79-29930}}
*{{IBList |type=author|id=96|name=Fyodor Dostoevsky}}
*

{{Fyodor Dostoyevsky}}
{{Lists of Russians}}

{{Persondata
|NAME= Dostoyevsky, Fyodor Mikhailovich
|ALTERNATIVE NAMES= Dostoevsky, Fyodor Mikhailovich; Фёдор Миха́йлович Достое́вский (Russian)
|SHORT DESCRIPTION= Russian novelist
|DATE OF BIRTH= {{Birth date|1821|11|11|mf=y}}
|PLACE OF BIRTH= Moscow
|DATE OF DEATH= {{Death date|1881|2|9|mf=y}}
|PLACE OF DEATH= Saint Petersburg
}}
{{DEFAULTSORT:Dostoyevsky, Fyodor}}
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Benutzer:JonskiC/Kleinste-Quadrate-Schätzung

2011-05-21T21:03:01Z

Paul August: Reverted edits by 92.90.23.27 (talk) to last version by Kiefer.Wolfowitz

The method of '''least squares''' is a standard approach to the approximate solution of [[overdetermined system]]s, i.e. sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every single equation.

The most important application is in [[curve fitting|data fitting]]. The best fit in the least-squares sense minimizes the sum of squared [[errors and residuals in statistics|residuals]], a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the [[independent variable]] (the 'x' variable), then simple regression and least squares methods have problems; in such cases, [[errors in variable]] models may be considered.

Least squares problems fall into two categories: linear or [[ordinary least squares]] and [[non-linear least squares]], depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical [[regression analysis]]; it has a closed-form solution. The non-linear problem has no closed-form solution and is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, thus the core calculation is similar in both cases.

The least-squares method was first described by [[Carl Friedrich Gauss]] around 1794.<ref name=brertscher>{{cite book|author = Bretscher, Otto|title = Linear Algebra With Applications, 3rd ed.|publisher = Prentice Hall|year = 1995|location = Upper Saddle River NJ}}</ref> Least squares corresponds to the [[maximum likelihood]] criterion if the experimental errors have a [[normal distribution]] and can also be derived as a [[method of moments (statistics)|method of moments]] estimator.

The following discussion is mostly presented in terms of [[linear]] functions but the use of least-squares is valid and practical for more general families of functions. For example, the [[Fourier series]] approximation of degree ''n'' is optimal in the least-squares sense, amongst all approximations in terms of [[trigonometric polynomial]]s of degree ''n''. Also, by iteratively applying local [[quadratic approximation]] to the likelihood (through the [[Fisher information]]), the least-squares method may be used to fit a [[generalized linear model]].

[[Image:Linear least squares2.png|right|thumb|The result of fitting a set of data points with a quadratic function.]]

==History==
===Context===

The method of least squares grew out of the fields of [[astronomy]] and [[geodesy]] as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the [[Age of Exploration]]. The accurate description of the behavior of celestial bodies was key to enabling ships to sail in open seas where before sailors had relied on land sightings to determine the positions of their ships.

The method was the culmination of several advances that took place during the course of the eighteenth century<ref name=stigler>{{cite book
| author = Stigler, Stephen M.
| title = The History of Statistics: The Measurement of Uncertainty Before 1900
| publisher = Belknap Press of Harvard University Press
| year = 1986
| location = Cambridge, MA
| isbn = 0674403401
}}</ref>:

*The combination of different observations taken under the ''same'' conditions contrary to simply trying one's best to observe and record a single observation accurately. This approach was notably used by [[Tobias Mayer]] while studying the [[libration]]s of the moon.
*The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, perhaps first expressed by [[Roger Cotes]].
*The combination of different observations taken under ''different'' conditions as notably performed by [[Roger Joseph Boscovich]] in his work on the shape of the earth and [[Pierre-Simon Laplace]] in his work in explaining the differences in motion of [[Jupiter]] and [[Saturn]].
*The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved, developed by Laplace in his ''Method of Least Squares''.

===The method===

[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|[[Carl Friedrich Gauss]]]]

[[Carl Friedrich Gauss]] is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen. [[Adrien-Marie Legendre|Legendre]] was the first to publish the method, however.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid [[Ceres (asteroid)|Ceres]]. On January 1, 1801, the Italian astronomer [[Giuseppe Piazzi]] discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, astronomers desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated [[Kepler's laws of planetary motion|Kepler's nonlinear equations]] of planetary motion. The only predictions that successfully allowed Hungarian astronomer [[Franz Xaver von Zach]] to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium''.
In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. This result is known as the [[Gauss–Markov theorem]].

The idea of least-squares analysis was also independently formulated by the Frenchman [[Adrien-Marie Legendre]] in 1805 and the American [[Robert Adrain]] in 1808. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares.<ref>See {{ cite journal|doi=10.1111/j.1751-5823.1998.tb00406.x|author=J. Aldrich|year=1998|title=Doing Least Squares: Perspectives from Gauss and Yule|journal=International Statistical Review|volume=66|issue=1|pages= 61–81}}</ref>

==Problem statement==

The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of ''n'' points (data pairs) <math>(x_i,y_i)\!</math>, ''i'' = 1, ..., ''n'', where <math>x_i\!</math> is an [[independent variable]] and <math>y_i\!</math> is a [[dependent variable]] whose value is found by observation. The model function has the form <math>f(x,\beta)</math>, where the ''m'' adjustable parameters are held in the vector <math>\boldsymbol \beta</math>. The goal is to find the parameter values for the model which "best" fits the data. The least squares method finds its optimum when the sum, ''S'', of squared residuals
:<math>S=\sum_{i=1}^{n}{r_i}^2</math>
is a minimum. A [[errors and residuals in statistics|residual]] is defined as the difference between the value predicted by the model and the actual value of the dependent variable.

:<math>r_i=y_i-f(x_i,\boldsymbol \beta)</math>.

An example of a model is that of the straight line. Denoting the intercept as <math>\beta_0</math> and the slope as <math>\beta_1</math>, the model function is given by <math>f(x,\boldsymbol \beta)=\beta_0+\beta_1 x</math>. See [[linear least squares#Example_with_real_data|linear least squares]] for a fully worked out example of this model.

A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, ''x'' and ''z'', say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point. 

== Limitations ==
This regression formulation considers only residuals in the dependent variable. There is an implicit assumption that errors in the [[independent variable]] are zero or strictly controlled so as to be negligible. When errors in the [[independent variable]] are non-negligible, models of measurement error can be used; such methods are more [[robust statistics|robust]] for [[parameter estimation]] than for [[hypothesis testing]] or for computing [[confidence interval]]s.<ref> Along with standard statistical methods for estimating measurement-error models, one may fit a model by [[total least squares]].</ref>

==Solving the least squares problem==

The [[Maxima and minima|minimum]] of the sum of squares is found by setting the [[gradient]] to zero. Since the model contains ''m'' parameters there are ''m'' gradient equations.

:<math>\frac{\partial S}{\partial \beta_j}=2\sum_i r_i\frac{\partial r_i}{\partial \beta_j}=0,\ j=1,\ldots,m</math>

and since <math>r_i=y_i-f(x_i,\boldsymbol \beta)\,</math> the gradient equations become

:<math>-2\sum_i \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} r_i=0,\ j=1,\ldots,m</math>.

The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.

=== Linear least squares ===

{{main|Linear_least_squares_(mathematics)|l1=Linear least squares}}

A regression model is a linear one when the model comprises a [[linear combination]] of the parameters, i.e.

:<math> f(x_i, \beta) = \sum_{j = 1}^{m} \beta_j \phi_j(x_{i})</math>

where the coefficients, <math>\phi_{j}</math>, are functions of <math> x_{i} </math>.

Letting

:<math> X_{ij}= \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j}= \phi_j(x_{i}) . \, </math>

we can then see that in that case the least square estimate (or estimator, in the context of a random sample), <math> \boldsymbol \beta</math> is given by

:<math> \boldsymbol{\hat\beta} =( X ^TX)^{-1}X^{T}\boldsymbol y.</math>

For a derivation of this estimate see [[Linear least squares]].

====Functional analysis====

{{See also|Fourier series|Generalized Fourier series}}

A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an [[orthogonal functions|orthogonal set]]:<ref name=Lanczos>

{{cite book |title=Applied analysis |author=Cornelius Lanczos |pages =212–213 |isbn=048665656X |publisher=Dover Publications |year=1988 |edition=Reprint of 1956 Prentice-Hall |url=http://books.google.com/books?id=6E85hExIqHYC&pg=PA212}}

</ref>

:<math>f(x) \approx f_n (x) = a_1 \phi _1 (x) + a_2 \phi _2(x) + \cdots + a_n \phi _n (x), \ </math>

with the set of functions {<math>\ \phi _j (x) </math>} an [[Orthonormal_set#Real-valued_functions|orthonormal set]] over the interval of interest, {{nowrap|say [a, b]}}: see also [[Fejér's theorem]]. The coefficients {<math>\ a_j </math>} are selected to make the magnitude of the difference ||{{nowrap|''f − f ''''n'' }}||2 as small as possible. For example, the magnitude, or norm, of a function {{nowrap|''g'' (''x'' )}} over the {{nowrap|interval [a, b]}} can be defined by:<ref name=Folland>

{{cite book |title=Fourier analysis and its application |page =69 |chapter=Equation 3.14 |author=Gerald B Folland |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0821847902 |publisher=American Mathematical Society Bookstore |year=2009 |edition=Reprint of Wadsworth and Brooks/Cole 1992}}

</ref>

:<math> \|g\| = \left(\int_a^b g^*(x)g(x) \, dx \right)^{1/2} </math>

where the ‘*’ denotes complex conjugate in the case of complex functions. The extension of Pythagoras' theorem in this manner leads to [[function space]]s and the notion of [[Lebesgue measure]], an idea of “space” more general than the original basis of Euclidean geometry. The {{nowrap|{ <math>\phi_j (x)\ </math> } }} satisfy [[Orthogonal#Orthogonal_functions|orthonormality relations]]:<ref name=Folland2>

{{cite book |title=cited work |page =69 |chapter=Equation 3.17 |author=Gerald B Folland |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0821847902 |date=2009-01-13 }}

</ref>

:<math> \int_a^b \phi _i^* (x)\phi _j (x) \, dx =\delta_{ij},</math>

where ''δ''''ij'' is the [[Kronecker delta]]. Substituting function {{nowrap|''f''''n''}} into these equations then leads to
the ''n''-dimensional [[Pythagorean theorem]]:<ref name=Wood>

{{cite book |title=Statistical methods: the geometric approach |author= David J. Saville, Graham R. Wood |chapter=§2.5 Sum of squares |page=30 |url=http://books.google.com/books?id=8ummgMVRev0C&pg=PA30 |isbn=0387975179 |year=1991 |edition=3rd |publisher=Springer}}

</ref>

:<math>\|f_n\|^2 = |a_1|^2 + |a_2|^2 + \cdots + |a_n|^2. \, </math>

The coefficients {''aj''} making {{nowrap begin}}||''f − fn''||2{{nowrap end}} as small as possible are found to be:<ref name=Lanczos/>

:<math>a_j = \int_a^b \phi _j^* (x)f (x) \, dx. </math>

The generalization of the ''n''-dimensional Pythagorean theorem to ''infinite-dimensional '' [[real number|real]] inner product spaces is known as [[Parseval's identity]] or Parseval's equation.<ref name=Folland3>

{{cite book |title=cited work |page =77 |chapter=Equation 3.22 |author=Gerald B Folland |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA77 |isbn=0821847902 |date=2009-01-13 }}

</ref> Particular examples of such a representation of a function are the [[Fourier series]] and the [[generalized Fourier series]].

=== Non-linear least squares ===

{{main|Non-linear least squares}}

There is no closed-form solution to a non-linear least squares problem. Instead, numerical algorithms are used to find the value of the parameters <math>\beta</math> which minimize the objective. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation.
:<math>{\beta_j}^{k+1}={\beta_j}^k+\Delta \beta_j</math>
''k'' is an iteration number and the vector of increments, <math>\Delta \beta_j\,</math> is known as the shift vector. In some commonly used algorithms, at each iteration the model may be linearized by approximation to a first-order [[Taylor series]] expansion about <math> \boldsymbol \beta^k\!</math>

:<math>
\begin{align}
f(x_i,\boldsymbol \beta) & = f^k(x_i,\boldsymbol \beta) +\sum_j \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} \left(\beta_j-{\beta_j}^k \right) \\
& = f^k(x_i,\boldsymbol \beta) +\sum_j J_{ij} \Delta\beta_j.
\end{align}
</math>

The [[Jacobian matrix and determinant|Jacobian]], '''J''', is a function of constants, the independent variable ''and'' the parameters, so it changes from one iteration to the next. The residuals are given by

:<math>r_i=y_i- f^k(x_i,\boldsymbol \beta)- \sum_{j=1}^{m} J_{ij}\Delta\beta_j=\Delta y_i- \sum_{j=1}^{m} J_{ij}\Delta\beta_j</math>.

To minimize the sum of squares of <math>r_i</math>, the gradient equation is set to zero and solved for <math> \Delta \beta_j\!</math>

:<math>-2\sum_{i=1}^{n}J_{ij} \left( \Delta y_i-\sum_{j=1}^{m} J_{ij}\Delta \beta_j \right)=0</math>

which, on rearrangement, become ''m'' simultaneous linear equations, the '''normal equations'''.

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} J_{ij}J_{ik}\Delta \beta_k=\sum_{i=1}^{n} J_{ij}\Delta y_i \qquad (j=1,\ldots,m)\,</math>

The normal equations are written in matrix notation as

:<math>\mathbf{\left(J^TJ\right)\Delta \boldsymbol \beta=J^T\Delta y}.\,</math>


These are the defining equations of the [[Gauss–Newton algorithm]].

=== Differences between linear and non-linear least squares ===

* The model function, ''f'', in LLSQ (linear least squares) is a linear combination of parameters of the form <math>f = X_{i1}\beta_1 + X_{i2}\beta_2 +\cdots</math> The model may represent a straight line, a parabola or any other linear combination of functions. In NLLSQ (non-linear least squares) the parameters appear as functions, such as <math>\beta^2, e^{\beta x}</math> and so forth. If the derivatives <math>\partial f /\partial \beta_j</math> are either constant or depend only on the values of the independent variable, the model is linear in the parameters. Otherwise the model is non-linear.
*Algorithms for finding the solution to a NLLSQ problem require initial values for the parameters, LLSQ does not.
*Like LLSQ, solution algorithms for NLLSQ often require that the Jacobian be calculated. Analytical expressions for the partial derivatives can be complicated. If analytical expressions are impossible to obtain either the partial derivatives must be calculated by numerical approximation or an estimate must be made of the Jacobian.
*In NLLSQ non-convergence (failure of the algorithm to find a minimum) is a common phenomenon whereas the LLSQ is globally concave so non-convergence is not an issue.
*NLLSQ is usually an iterative process. The iterative process has to be terminated when a convergence criterion is satisfied. LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the [[Gauss–Seidel]] method.
*In LLSQ the solution is unique, but in NLLSQ there may be multiple minima in the sum of squares.
*Under the condition that the errors are uncorrelated with the predictor variables, LLSQ yields unbiased estimates, but even under that condition NLLSQ estimates are generally biased.
These differences must be considered whenever the solution to a non-linear least squares problem is being sought.

==Least squares, regression analysis and statistics==

The methods of least squares and [[regression analysis]] are conceptually different. However, the method of least squares is often used to generate estimators and other statistics in regression analysis.

Consider a simple example drawn from physics. A spring should obey [[Hooke's law]] which states that the extension of a spring is proportional to the force, ''F'', applied to it.
:<math>f(F_i,k)=kF_i\!</math>
constitutes the model, where ''F'' is the independent variable. To estimate the [[force constant]], ''k'', a series of ''n'' measurements with different forces will produce a set of data, <math>(F_i, y_i), i=1,n\!</math>, where ''yi'' is a measured spring extension. Each experimental observation will contain some error. If we denote this error <math>\varepsilon</math>, we may specify an empirical model for our observations,

: <math> y_i = kF_i + \varepsilon_i. \, </math>

There are many methods we might use to estimate the unknown parameter ''k''. Noting that the ''n'' equations in the ''m'' variables in our data comprise an [[overdetermined system]] with one unknown and ''n'' equations, we may choose to estimate ''k'' using least squares. The sum of squares to be minimized is

:<math> S = \sum_{i=1}^{n} \left(y_i - kF_i\right)^2. </math>

The least squares estimate of the force constant, ''k'', is given by

:<math>\hat k=\frac{\sum_i F_i y_i}{\sum_i {F_i}^2}.</math>

Here it is assumed that application of the force '''''causes''''' the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.

In regression analysis the researcher specifies an empirical model. For example, a very common model is the straight line model which is used to test if there is a linear relationship between dependent and independent variable. If a linear relationship is found to exist, the variables are said to be [[correlated]]. However, [[Correlation_does_not_imply_causation|correlation does not prove causation]], as both variables may be correlated with other, hidden, variables, or the dependent variable may "reverse" cause the independent variables, or the variables may be otherwise spuriously correlated. For example, suppose there is a correlation between deaths by drowning and the volume of ice cream sales at a particular beach. Yet, both the number of people going swimming and the volume of ice cream sales increase as the weather gets hotter, and presumably the number of deaths by drowning is correlated with the number of people going swimming. Perhaps an increase in swimmers causes both the other variables to increase.

In order to make statistical tests on the results it is necessary to make assumptions about the nature of the experimental errors. A common (but not necessary) assumption is that the errors belong to a [[Normal distribution]]. The [[central limit theorem]] supports the idea that this is a good approximation in many cases.
* The [[Gauss–Markov theorem]]. In a linear model in which the errors have [[expectation]] zero conditional on the independent variables, are [[uncorrelated]] and have equal [[variance]]s, the best linear [[unbiased]] estimator of any linear combination of the observations, is its least-squares estimator. "Best" means that the least squares estimators of the parameters have minimum variance. The assumption of equal variance is valid when the errors all belong to the same distribution.
*In a linear model, if the errors belong to a [[Normal distribution]] the least squares estimators are also the [[maximum likelihood estimator]]s.

However, if the errors are not normally distributed, a [[central limit theorem]] often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large. For this reason, given the important property that the error is mean independent of the independent variables, the distribution of the error term is not an important issue in regression analysis. Specifically, it is not typically important whether the error term follows a normal distribution.

In a least squares calculation with unit weights, or in linear regression, the variance on the ''j''th parameter,
denoted <math>\text{var}(\hat{\beta}_j)</math>, is usually estimated with

:<math>\text{var}(\hat{\beta}_j)= \sigma^2\left( \left[X^TX\right]^{-1}\right)_{jj} \approx \frac{S}{n-m}\left( \left[X^TX\right]^{-1}\right)_{jj},</math>
where the true residual variance σ2 is replaced by an estimate based on the minimised value of the sum of squares objective function ''S''.

[[Confidence limits]] can be found if the [[probability distribution]] of the parameters is known, or an asymptotic approximation is made, or assumed. Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed. The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed. Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.

==Weighted least squares==

{{see also|Weighted mean}}

The expressions given above are based on the implicit assumption that the errors are uncorrelated with each other and with the independent variables and have equal variance. The [[Gauss–Markov theorem]] shows that, when this is so, <math>\hat{\boldsymbol{\beta}}</math> is a [[best linear unbiased estimator]] (BLUE). If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted. [[Alexander Aitken|Aitken]] showed that when a weighted sum of squared residuals is minimized, <math>\hat{\boldsymbol{\beta}}</math> is BLUE if each weight is equal to the reciprocal of the variance of the measurement.
:<math> S = \sum_{i=1}^{n} W_{ii}{r_i}^2,\qquad W_{ii}=\frac{1}{{\sigma_i}^2} </math>
The gradient equations for this sum of squares are

:<math>-2\sum_i W_{ii}\frac{\partial f(x_i,\boldsymbol {\beta})}{\partial \beta_j} r_i=0,\qquad j=1,\ldots,n</math>

which, in a linear least squares system give the modified normal equations,

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} X_{ij}W_{ii}X_{ik}\hat{ \beta}_k=\sum_{i=1}^{n} X_{ij}W_{ii}y_i, \qquad j=1,\ldots,m\,.</math>

When the observational errors are uncorrelated and the weight matrix, '''W''', is diagonal, these may be written as

:<math>\mathbf{\left(X^TWX\right)\hat {\boldsymbol {\beta}}=X^TWy}.</math>

If the errors are correlated, the resulting estimator is BLUE if the weight matrix is equal to the inverse of the [[variance-covariance matrix]] of the observations.

When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as <math>w_{ii}=\sqrt W_{ii}</math>. The normal equations can then be written as

:<math>\mathbf{\left(X'^TX'\right)\hat{\boldsymbol{\beta}}=X'^Ty'}\,</math>

where

: <math>\mathbf{X'}=\mathbf{wX}, \mathbf{y'}=\mathbf{wy}.\,</math>

For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.

:<math>\mathbf{\left(J^TWJ\right)\boldsymbol \Delta \beta=J^TW \boldsymbol\Delta y}.\,</math>

Note that for empirical tests, the appropriate '''W''' is not known for sure and must be
estimated. For this [[Feasible Generalized Least Squares]] (FGLS) techniques may be used.

==Relationship to principal components==

The first [[Principal component analysis|principal component]] about the mean of a set of points can be represented by that line which most closely approaches the data points (as measured by squared distance of closest approach, i.e. perpendicular to the line). In contrast, linear least squares tries to minimize the distance in the <math>y</math> direction only. Thus, although the two use a similar error metric, linear least squares is a method that treats one dimension of the data preferentially, while PCA treats all dimensions equally.

==LASSO method==

In some contexts a [[Regularization (machine learning)|regularized]] version of the least squares solution may be preferable. The ''LASSO'' (least absolute shrinkage and selection operator) algorithm, for example, finds a least-squares solution with the constraint that <math>|\beta|_1</math>, the [[L1-norm|L1-norm]] of the parameter vector, is no greater than a given value. Equivalently, it may solve an unconstrained minimization of the least-squares penalty with <math>\alpha|\beta|_1</math> added, where <math>\alpha</math> is a constant (this is the [[Lagrange multipliers|Lagrangian]] form of the constrained problem.) This problem may be solved using [[quadratic programming]] or more general [[convex optimization]] methods, as well as by specific algorithms such as the [[least angle regression]] algorithm. The L1-regularized formulation is useful in some contexts due to its tendency to prefer solutions with fewer nonzero parameter values, effectively reducing the number of variables upon which the given solution is dependent.<ref>{{ cite journal|author=Tibshirani, R. |year=1996|title=Regression shrinkage and selection via the lasso. |journal=J. Royal. Statist. Soc B.|volume= 58|issue= 1| pages =267–288}}</ref> For this reason, the LASSO and its variants are fundamental to the field of [[compressed sensing]].

==See also==
* [[Best linear unbiased prediction]] (BLUP)
* [[L2 norm|''L''2 norm]]
* [[Least absolute deviation]]
* [[Measurement uncertainty]]
* [[Root mean square]]
* [[Squared deviations]]
* [[Quadratic loss function]]
* [[Statistical meaningfulness test]]

==Notes==

<references />

==References==

*{{ cite book|author=Å. Björck|isbn=978-0-898713-60-2|title=Numerical Methods for Least Squares Problems|publisher=SIAM|year=1996|url=http://www.ec-securehost.com/SIAM/ot51.html}}
*{{cite book| author=C.R. Rao, H. Toutenburg, A. Fieger, C. Heumann, T. Nittner and S. Scheid| title=Linear Models: Least Squares and Alternatives| series=Springer Series in Statistics|year=1999}}
*{{cite book|author=T. Kariya and H. Kurata |title=Generalized Least Squares|publisher= Wiley|year= 2004}}
*{{cite book|author=J. Wolberg|title=Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments|publisher= Springer|year=2005|isbn=3540256741}}
*{{ cite book|author=T. Strutz| title=Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond) |publisher=Vieweg+Teubner | isbn= 978-3-8348-1022-9}}

==External links==
* [http://www.personal.psu.edu/faculty/j/h/jhm/f90/lectures/lsq2.html Derivation of quadratic least squares] (Penn State)
* [http://www2.uta.edu/infosys/baker/STATISTICS/Keller7/Keller%20PP%20slides-7/Chapter17.ppt Power Point Statistics Book] -- Excellent slides providing an introductory regression example (University of Texas at Arlington)
* [http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-16-projection-matrices-and-least-squares/ MIT Lecture on Least Squares and Projection Matrices]

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Fjodor Michailowitsch Dostojewski

2011-03-23T20:43:39Z

Paul August: /* Exile in Siberia */ dubious

{{Eastern Slavic name|Mikhaylovich|Dostoyevsky}}
{{Infobox Writer 
| name = Fyodor Dostoyevsky
| image = Vasily Perov - Портрет Ф.М.Достоевского - Google Art Project.jpg
|caption =Dostoyevsky in 1872
| birth_name = Fyodor Mikhaylovich Dostoevsky
| birthdate = {{Birth date|1821|11|11|df=y}}
| birthplace = [[Moscow]], [[Russian Empire]]
| deathdate = {{Death date and age|1881|2|9|1821|11|11|df=y}}
| deathplace = [[Saint Petersburg]], [[Russian Empire]]
| occupation = Novelist
| language = [[Russian language|Russian]]
| nationality =
| period=
| genre= Literary fiction
| subject=
| movement=
| religion = [[Russian Orthodox]]
| notableworks= ''[[Notes from Underground]]'' ''[[Crime and Punishment]]'' ''[[The Idiot (novel)|The Idiot]]'' ''[[The Brothers Karamazov]]'' ''[[The Possessed (novel)|The Possessed]]''
| spouse= Mariya Dmitriyevna Isayeva m. (1857–64) [her death]
[[Anna Grigoryevna Snitkina]] (1867–1918) [his death]
| children=Sofiya (1868), [[Lyubov_Dostoyevskaya|Lyubov]] (1869—1926), Fyodor (1875–1878)
| relatives=
| influences= Writers: [[Alexander Pushkin]], [[Nikolai Gogol]], [[Miguel de Cervantes]],<ref>[http://goliath.ecnext.com/coms2/gi_0199-5055894/Dostoevsky-s-other-Quixote-influence.html Dostoevsky's other Quixote (influence of Miguel de Cervantes' Don Quixote on Fyodor Dostoevsky's The Idiot)]. Fambrough, Preston</ref> [[Charles Dickens]], [[Edgar Allan Poe]], [[Mikhail Lermontov]], [[Friedrich Schiller]], [[Honoré de Balzac]], [[Vasily Zhukovsky]], [[Victor Hugo]], [[E.T.A. Hoffmann]], [[Adam Mickiewicz]], Philosophers: [[Mikhail Bakunin]], [[Vissarion Belinsky]], [[Georg Wilhelm Friedrich Hegel]], [[Aleksandr Herzen]], [[Konstantin Leontyev]], [[Sergei Nechaev]], [[Mikhail Petrashevsky]], [[Vladimir Solovyov (philosopher)|Vladimir Solovyov]], [[Tikhon of Zadonsk]]
| influenced = [[Knut Hamsun]], [[Richard Brautigan]], [[Charles Bukowski]], [[Albert Camus]], [[Orhan Pamuk]],<ref>{{Cite book
| first = Orhan
| last = Pamuk
| authorlink = Orhan Pamuk
| title = [[Istanbul: Memories of a City]]
| publisher = [[Vintage Books]]
| year = 2006
| isbn = 978-1400033881
}}</ref><ref>{{Cite book
| first = Orhan
| last = Pamuk
| authorlink = Orhan Pamuk
| title = [[Other Colors: Essays and a Story]]
| publisher = [[Vintage Books]]
| year = 2008
| isbn = 978-0307386236
}}</ref> [[Sigmund Freud]], [[Witold Gombrowicz]], [[Franz Kafka]], [[Jack Kerouac]], [[James Joyce]], [[Czesław Miłosz]], [[Yukio Mishima]], [[Alberto Moravia]], [[Iris Murdoch]], [[Friedrich Nietzsche]], [[Marcel Proust]], [[Ayn Rand]], [[Jean-Paul Sartre]], [[Aleksandr Solzhenitsyn]], [[Wisława Szymborska]], [[Irvine Welsh]], [[Ludwig Wittgenstein]], [[Cormac McCarthy]], [[Ken Kesey]], [[Albert Einstein]], [[Leonid Leonov]]
| signature = Fyodor Dostoyevsky Signature.svg
}}
'''Fyodor Mikhaylovich Dostoyevsky'''<ref>{{lang-ru|Фёдор Миха́йлович Достое́вский}}, {{IPA-ru|ˈfʲodər mʲɪˈxajləvʲɪtɕ dəstɐˈjɛfskʲɪj|pron|ru-Dostoevsky.ogg}}; loose phonetic pronunciation: fyo-der mi-(k)hail-a-vitch das-ta-yef-skee. His last name is also commonly transcribed in English as ''Dostoevsky'' or ''Dostoevski'', and some early translations render his first name by its English equivalent, ''Theodore''.</ref> (11 November 1821<ref>[[Old Style and New Style dates|Old Style date]] 30 October 1821</ref> – 9 February<ref>[[Old Style and New Style dates|Old Style date]] 29 January.</ref> 1881) was a [[Russia]]n writer of [[realist fiction]] and [[essay]]s.<ref name="pravoslavye.org.ua">[http://www.pravoslavye.org.ua/index.php?r_type=article&action=fullinfo&id=13375 Ukrainian origin of Dostoyevsky (Українське коріння Достоєвського)]</ref> He is best known for his novels ''[[Crime and Punishment]]'', ''[[The Idiot (novel)|The Idiot]]'' and ''[[The Brothers Karamazov]]''.

Dostoyevsky's literary works explored human psychology in the troubled political, social and spiritual context of 19th-century Russian society. Considered by many as a founder or precursor of 20th-century [[existentialism]], Dostoyevsky wrote, with the embittered voice of the anonymous "underground man", '' [[Notes from Underground]]'' (1864), which was called the "best overture for existentialism ever written" by [[Walter Kaufmann (philosopher)|Walter Kaufmann]].<ref>Existentialism: from Dostoyevsky to Sartre, ed. Walter Kaufmann, Penguin Books, 1989 ISBN 0452009308 p. 12</ref> Dostoyevsky is often acknowledged by critics as one of the greatest and most prominent psychologists in [[world literature]].<ref name="BritannicaRussianLit">{{Cite web|url=http://www.britannica.com/EBchecked/topic/513793/Russian-literature|publisher=Encyclopedia Britannica|accessdate=2008-04-11|title=Russian literature|quote=Dostoyevsky, who is generally regarded as one of the supreme psychologists in world literature, sought to demonstrate the compatibility of Christianity with the deepest truths of the psyche.}}</ref>

==Biography==
===Family origins===
Dostoyevsky's paternal ancestors were from a town called Dostoyev in [[Belarus]], in the [[Guberniya]] (province) of [[Minsk]], not far from [[Pinsk]]. Dostoyevsky's mother was Russian. The stress on the family name was originally on the second syllable, matching that of the town (Dostóev). However, in the nineteenth century, the stress was shifted to the third syllable.<ref>B.O. Unbegaun, ''Russkie familii'' (Moscow: "Univers"), pp. 28, 345.</ref> According to one account, Dostoyevsky's paternal ancestors were Polonized nobles ([[szlachta]]) of [[Ruthenians|Ruthenian]] origin who went to war bearing Polish [[Radwan Coat of Arms]]. Dostoyevsky (Polish ''Dostojewski'') Radwan armorial bearings were drawn for the Dostoyevsky Museum in Moscow.<ref>Aimée Dostoyevsky, [http://www.worldcat.org/title/fyodor-dostoyevsky-a-study/oclc/61397936 "FYODOR DOSTOYEVSKY: A STUDY"] (Honolulu, HAWAII: University Press of the Pacific, 2001), [http://books.google.com/books?id=n7fb7eH6nRUC&pg=PA6#v=onepage&q&f=false p. 6].</ref>

===Early life===
[[File:Wki Dostoyevsky Street 2 Moscow Mariinsky Hospital.jpg|thumb|240px|Mariinsky Hospital in Moscow, Dostoyevsky's birthplace]]Dostoyevsky was born in Moscow, the second of seven children born to Mikhail and Maria Dostoyevsky.<ref>The Best Short Stories of Dostoevsky: Translated with an Introduction by David Magarshack. New York: The Modern Library, Random House; 1971.</ref> Dostoyevsky's father Mikhail was a doctor and a devout Christian, who practiced at the Mariinsky Hospital for the Poor in Moscow. The family lived in a small apartment in the hospital grounds, and it wasn't until he was 16 years old, that Dostoyevsky moved to [[St Petersburg]] to attend a [[Military engineering-technical university|Military Engineering Institute]]. The hospital was located in one of the city's worst areas; local landmarks included a cemetery for criminals, a lunatic asylum, and an orphanage for abandoned infants. This urban landscape made a lasting impression on the young Dostoyevsky, whose interest in and compassion for the poor, oppressed and tormented was apparent in his life and works. Although it was forbidden by his parents, Dostoyevsky liked to wander out to the hospital garden, where the patients sat to catch a glimpse of the sun. The young Dostoyevsky loved to spend time with these patients and listen to their stories.

There are many stories of Dostoyevsky's father's despotic treatment of his children, but this despotism was tempered by his extreme care for his children and their upbringing. After returning home from work, he would take a nap while his children, ordered to keep absolutely silent, stood by their slumbering father in shifts and swatted the flies that came near his head. But the father was also careful to send his children to private schools where they would not be beaten. In the opinion of Joseph Frank, author of a definitive biography of Dostoyevsky, the father figure in ''[[The Brothers Karamazov]]'' is not based on Dostoyevsky's own father. Letters and personal accounts demonstrate that they did have a fairly loving relationship.

[[File:Dostoevskij 1847.jpg|thumb|upright|The young Dostoyevsky, in an 1847 portrait by [[Konstiantyn Trutovsky|Trutovsky]]]]

In 1837, shortly after his mother died of [[tuberculosis]], Dostoyevsky and his brother were sent to St Petersburg to attend the [[Military engineering-technical university|Nikolayev Military Engineering Institute]], nowadays called the [[Military engineering-technical university|Military Engineering-Technical University]] (Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]]). Fyodor's father died in 1839. Though it has never been proven, it is believed by some that he was murdered by his own [[serf]]s.<ref>[http://worldebooklibrary.com/eBooks/Coradella_Collegiate_Bookshelf_Collection/Dostoevsky-notesfromtheunderground.pdf Notes from the Underground] Coradella Collegita Bookshelf edition, ''About the Author''.</ref> According to one account, the serfs became enraged during one of his drunken fits of violence, and after restraining him, poured [[vodka]] into his mouth until he drowned. A similar account appears in ''Notes from Underground''. Another story holds that Mikhail died of natural causes, and a neighboring landowner invented the story of his murder so that he could buy the estate at a cheaper price. Some, like Sigmund Freud in his 1928 article, "[[Dostoevsky and Parricide]]", have argued that his father's personality had influenced the character of Fyodor Pavlovich Karamazov, the "wicked and sentimental buffoon", father of the main characters in his 1880 novel ''[[The Brothers Karamazov]]'', but such claims fail to withstand the scrutiny of many critics{{Who|date=June 2009}}.

Dostoyevsky suffered from [[epilepsy]] and his first seizure occurred when he was nine years old.<ref>[http://www.epilepsy.com/epilepsy/famous_writers.html Epilepsy.com] Famous authors with epilepsy.</ref> Epileptic seizures recurred sporadically throughout his life, and Dostoyevsky's experiences are thought<ref>Dostoyevsky, Fyodor, Richard Pevear, and Larissa Volokhonsky. The Idiot. New York: Vintage, 2001. Print. Introduction pp. xix</ref> to have formed the basis for his description of Prince Myshkin's epilepsy in his novel ''[[The Idiot (novel)|The Idiot]]'' and that of Smerdyakov in ''The Brothers Karamazov'', among others.

At the [[Military engineering-technical university|Saint Petersburg Institute of Military Engineering]]<ref>Russian: [[:ru:Военный инженерно-технический университет|Военный инженерно-технический университет]],</ref> Dostoyevsky was taught mathematics, a subject he despised. However, he also studied literature by [[Shakespeare]], [[Blaise Pascal|Pascal]], [[Victor Hugo]] and [[E.T.A. Hoffmann]]. Though he focused on areas different from mathematics, he did well in the exams and received a commission in 1841. That year, influenced by the German poet/playwright [[Friedrich Schiller]], he wrote two romantic plays: ''[[Maria Stuart (play)|Mary Stuart]]'' and ''[[Boris Godunov]]''. The plays have not been preserved. Dostoyevsky described himself as a "dreamer" when he was a young man. He also revered [[Schiller]] at that age. However, in the years during which he wrote his great masterpieces, his opinions changed and he sometimes made fun of Schiller.

===Beginnings of a literary career===
[[File:Fjodor signatur.jpg|thumb|right|210px|Dostoyevsky's signature]]
Dostoyevsky was made a lieutenant in 1842, and left the Engineering Academy the following year. He completed a translation into Russian of [[Balzac]]'s novel ''[[Eugénie Grandet]]'' in 1843, but it brought him little to no attention. Dostoyevsky started to write his own fiction in late 1844 after leaving the army. In 1846, his first work, the epistolary short novel, ''[[Poor Folk]]'', printed in the almanac ''A Petersburg Collection'' (published by [[Nikolay Nekrasov|N. Nekrasov]]), was met with great acclaim. As legend has it, the editor of the magazine, poet [[Nikolai Nekrasov]], walked into the office of liberal critic [[Vissarion Belinsky]] and announced, "A new [[Nikolai Gogol|Gogol]] has arisen!" Belinsky, his followers, and many others agreed. After the novel was fully published in book form at the beginning of the next year, Dostoyevsky became a literary celebrity at the age of 24.

In 1846, Belinsky and many others reacted negatively to his novella, ''[[The Double: A Petersburg Poem|The Double]]'', a psychological study of a bureaucrat whose alter ego overtakes his life. Dostoyevsky's fame began to fade. Much of his work after ''[[Poor Folk]]'' received ambivalent reviews and it seemed that Belinsky's prediction that Dostoyevsky would be one of the greatest writers of Russia was mistaken.

===Exile in Siberia===
[[File:Omsk Dostoyevskiy Monument.jpg|thumb|upright|Statue of Dostoyevsky in [[Omsk]]]]
Dostoyevsky was incarcerated on 23 April 1849 for being part of the [[Liberalism|liberal]] intellectual group the [[Petrashevsky Circle]]. [[Tsar]] [[Nicholas I of Russia|Nicholas I]], after seeing the [[Revolutions of 1848]] in Europe, was harsh on any type of underground organization which he felt could put [[autocracy]] in jeopardy. On November 16 of that year, Dostoyevsky, along with other members of the Petrashevsky Circle, was [[death sentence|sentenced to death]]. After a [[mock execution]], in which he and other members of the group stood outside in freezing weather waiting to be shot by a firing squad, Dostoyevsky's sentence was commuted to four years of [[exile]] with hard labour at a [[katorga]] prison camp in [[Omsk]], [[Siberia]]. Later, Dostoyevsky described his years of suffering to his brother, as being, "shut up in a coffin." In describing the dilapidated barracks which "should have been torn down years ago", he wrote:
''{{quote|In summer, intolerable closeness; in winter, unendurable cold. All the floors were rotten. Filth on the floors an inch thick; one could slip and fall... We were packed like herrings in a barrel... There was no room to turn around. From dusk to dawn it was impossible not to behave like pigs... Fleas, lice, and black beetles by the bushel...''<ref>Frank 76. Quoted from Pisma, I: 135–37.</ref>}} This experience inspired him to write ''[[The House of the Dead (novel)|The House of the Dead]]''.

Dostoyevsky was released from prison in 1854, and was required to serve in the Siberian Regiment. He spent the following five years as a private (and later lieutenant) in the Regiment's Seventh Line Battalion, stationed at the fortress of [[Semey|Semipalatinsk]], now in [[Kazakhstan]]. While there, he began a relationship with Maria Dmitrievna Isayeva, the wife of an acquaintance in Siberia. After her husband's death, they married in February 1857.

===Post-prison maturation as a writer===
[[File:Valikhanov.jpg|thumb|upright|Dostoyevsky (right) and the [[Kazakhs|Kazakh]] scholar [[Shokan Walikhanuli]] in 1859]]

Dostoyevsky's experiences in prison and the army resulted in major changes in his political and religious convictions. First, his ordeal somehow caused him to become disillusioned with "Western" ideas; he repudiated the contemporary Western European philosophical movements, and instead paid greater tribute in his writings to traditional, rustic Russian values exemplified in the [[Slavophile]] concept of ''[[sobornost]]''. But even more significantly, he had what his biographer Joseph Frank describes as a [[Religious conversion|conversion]] experience in prison, which greatly strengthened his Christian, and specifically [[Russian Orthodox|Orthodox]], faith.<ref>Frank (1987) ''Dostoevsky: The Years of Ordeal, 1850–1859'', pp. 124–127.</ref> Dostoyevsky would later depict his conversion experience in the short story, ''[[The Peasant Marey]]'' (1876).

In his writings, Dostoyevsky started to extol the virtues of humility, submission, and suffering.<ref name="Nab81Censors">[[Vladimir Nabokov]] (1981) ''[[Lectures on Russian Literature]]'', lecture on ''Russian Writers, Censors, and Readers'', p.14</ref> He now displayed a much more critical stance on contemporary European philosophy and turned with intellectual rigour against the [[Nihilist movement|Nihilist]] and Socialist movements; and much of his post-prison work—particularly the novel, ''[[The Possessed (novel)|The Possessed]]'', and the essays, ''[[A Writer's Diary|The Diary of a Writer]]''—contains both criticism of socialist and nihilist ideas, as well as thinly veiled parodies of contemporary Western-influenced Russian intellectuals ([[Timofey Granovsky|Timofey Granovskiy]]), revolutionaries ([[Sergey Nechayev|Sergey Nyechayev]]), and even fellow novelists ([[Ivan Turgenev|Ivan Turgyenyev]]).<ref>Dostoevsky the Thinker James P. Scanlan. Dostoevsky the Thinker. Ithaca: Cornell University Press, 2002. xiii, p. 251</ref><ref>[http://ourworld.compuserve.com/homepages/jim_forest/pevear.htm Dostoevsky's View of Evil] Reprinted from ''In Communion'', April 1998.</ref> In social circles, Dostoyevsky allied himself with well-known conservatives, such as the statesman [[Konstantin Pobedonostsev|Konstantin Pobyedonostsyev]]. His post-prison essays praised the tenets of the [[Pochvennichestvo|Pochvyennichyestvo]] movement, a late-19th century Russian nativist ideology closely aligned with [[Slavophile|Slavophilism]].

Dostoyevsky's post-prison fiction abandoned the West-European-style domestic melodramas and quaint character studies of his youthful work in favor of dark, more complex storylines and situations, played-out by brooding, tortured characters—often styled partly on Dostoyevsky himself—who agonized over [[existentialism|existential]] themes of spiritual torment, religious awakening, and the psychological confusion caused by the conflict between traditional Russian culture and the influx of modern, Western philosophy. Nonetheless, this does not take from the debt which Dostoyevsky owed to earlier Western-influenced writers such as [[Gogol]] whose work grew from the irrational and anti-authoritarian spiritualist ideas contained within the [[Romantic movement]] which had immediately preceded Dostoyevsky in West Europe. However, Dostoyevsky's major novels focused on the idea that [[utopia]] and [[positivist]] ideas were unrealistic and unobtainable.<ref>{{Cite book|last = Sirotkina|first = Irina|title = Diagnosing Literary Genius: A Cultural History of Psychiatry in Russia, 1880|year = 1996|publisher = [[Johns Hopkins University Press]]|page = 55|isbn = 0801867827}}</ref>

===Later literary career===
[[File:Dostoevskij 1863.jpg|thumb|upright|Dostoyevsky in 1863]][[File:Fyodor Dostoevsky house.jpg|thumb|right|170px|Dostoyevsky's last address where he died, now a memorial and literary museum, St Petersburg.]]

In December 1859, Dostoyevsky returned to [[Saint Petersburg]], where he ran a series of unsuccessful literary journals, ''[[Vremya (magazine)|Vremya]]'' (Time) and ''[[Epoch (Russian magazine)|Epokha]]'' (Epoch), with his older brother [[Mikhail Dostoyevsky|Mikhail]].<ref>{{Cite book|title=F. M. Dostoyevsky. Collection of works in 15 volumes |volume=11|year=1993 |publisher=Nauka |location=Leningrad|pages=361–365 |chapter=A few words about Mikhail Mikhailovich Dostoyevsky}}</ref> The former was shut down as a consequence of its coverage of the [[January Uprising|Polish Uprising of 1863]]. That year Dostoyevsky traveled to Europe and frequented gambling casinos. There he met [[Polina Suslova|Apollinaria Suslova]], the model for Dostoyevsky's "proud women", such as the two characters named Katerina Ivanovna, in ''[[Crime and Punishment]]'' and in ''[[The Brothers Karamazov]]''.

Dostoyevsky was devastated by his wife's death in 1864, which was followed shortly thereafter by his brother's death. He was financially crippled by business debts; furthermore, he decided to assume the responsibility of his deceased brother's outstanding debts, as well providing for his wife's son from her earlier marriage and his brother's widow and children. Dostoyevsky sank into a deep depression, frequenting gambling parlors and accumulating massive losses at the tables.

Dostoyevsky suffered from an acute [[Problem gambling|gambling compulsion]] and its consequences. By one account{{Who|date=June 2009}} he completed ''Crime and Punishment'', possibly his best known novel, in a mad hurry because he was in urgent need of an advance from his publisher. He had been left practically penniless after a gambling spree. Dostoyevsky wrote ''[[The Gambler (novel)|The Gambler]]'' simultaneously in order to satisfy an agreement with his publisher Stellovsky who, if he did not receive a new work, would have claimed the copyrights to all of Dostoyevsky's writings.

Motivated by the dual wish to escape his creditors at home and to visit the casinos abroad, Dostoyevsky traveled to Western Europe. There, he attempted to rekindle a love affair with Suslova, but she refused his marriage proposal. Dostoyevsky was heartbroken, but soon met [[Anna Dostoyevskaya|Anna Grigorevna Snitkina]], a twenty-year-old [[stenographer]]. Shortly before marrying her in 1867, he dictated ''The Gambler'' to her.<ref>{{cite book |last=Dostoevsky |first=Fyodor |others=Notes and Introduction by Maire Jaanus. Translated by [[Constance Garnett]] |title=The Brothers Karamazov |series=Barnes & Noble Classics |year=2004 |origyear=First published 1879–1880 |publisher=Barnes & Noble Books |location=New York, NY |isbn=978-1-59308-045-7 |oclc=34325193 |page=703 |chapter=Endnotes |quote=Anna Grigorievna Snitkina, Dostoevsky's second wife, was a stenographer to whom Dostoevsky dictated his novel ''The Gambler'' in 1866; they married the following year.}}</ref> From 1873 to 1881 he published the ''Writer's Diary'', a monthly journal of short stories, sketches, and articles on current events. The journal was an enormous success.

Dostoyevsky influenced, and was himself influenced, by the philosopher [[Vladimir Sergeyevich Solovyov]]. Solovyov was the inspiration for the characters [[Ivan Karamazov]] and [[Alyosha Karamazov]].<ref>Zouboff, Peter, Solovyov on Godmanhood: Solovyov’s Lectures on Godmanhood Harmon Printing House: Poughkeepsie, New York, 1944; see Czeslaw Milosz’s introduction to Solovyov’s War, Progress and the End of History. Lindisfarne Press: Hudson, New York 1990.</ref>

In 1877, Dostoyevsky gave the keynote [[eulogy]] at the funeral of his friend, the poet [[Nikolai Alekseevich Nekrasov|Nekrasov]], to much controversy{{Who|date=June 2009}}. On 8 June 1880, shortly before he died, he gave his famous [[Alexander Pushkin|Pushkin]] speech at the unveiling of the [[Pushkin Square|Pushkin monument in Moscow]].<ref>Dostoyevsky [http://az.lib.ru/d/dostoewskij_f_m/text_0340.shtml Az.lib.ru Пушкинская речь (Pushkin's style)] (in Russian)</ref> In his later years, Dostoyevsky lived for an extended period at the resort of [[Staraya Russa]] in northwestern Russia, which was closer to [[Saint Petersburg]] and less expensive than German resorts.

===Death===
Dostoyevsky died in St. Petersburg on {{OldStyleDate|9 February|1881|28 January}} of a lung hemorrhage associated with [[emphysema]] and an [[epileptic seizure]]. A copy of the New Testament Bible given to him in Siberia sat on his lap. He was interred in [[Tikhvin Cemetery]] at the [[Alexander Nevsky Monastery]] in [[Saint Petersburg]]. Forty thousand mourners attended his funeral.<ref>Dostoevsky, Fyodor; Introduction to The Idiot, Wordsworth Ed. Ltd, 1996.</ref>

His tombstone reads; ''Verily, Verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.'' (Excerpt from [[Gospel of John|John]] 12:24, which is also the [[Epigraph (literature)|epigraph]] of his final novel, ''The Brothers Karamazov''.)

The rented apartment where he died and spent the last few years of his life is where he wrote his final novel ''The Brothers Karamazov''. The apartment, situated in a building at 5 Kuznechnyi pereulok, has been restored with old photographs to how it looked when he lived there. It opened in 1971 as the Dostoyevsky House Museum and is a popular tourist attraction in the city.<ref>{{cite book |title=St Petersburg |last= Woodworth|first=Bradley |authorlink= |coauthors= Harold Bloom, Constance Richards|year=2005 |editor=Harold Bloom|publisher=Infobase Publishing |location= |isbn=0791083845, 9780791083840 |page=69 |pages= |url=http://books.google.com/books?id=tMn6qHyTIywC&dq=dostoevsky+house+museum,+St+Petersburg&source=gbs_navlinks_s |accessdate=19 November, 2010}}</ref>

==Influence==
[[File:Dostoevsky.jpg|thumb|upright|Dostoyevsky in 1879]]

Some, like journalist [[Otto Friedrich]],<ref>{{Cite news|publisher=Time Magazine|url=http://www.time.com/time/magazine/article/0,9171,943893,00.html?promoid=googlep|accessdate=2008-04-10|title=Freaking-Out with Fyodor|author=Otto Friedrich|date=6 September 1971}}</ref> consider Dostoyevsky to be one of Europe's major novelists, while others like [[Vladimir Nabokov]] maintain that from a point of view of enduring art and individual genius, he is a rather mediocre writer who produced wastelands of literary [[platitude]]s.<ref> Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 98</ref> Dostoyevsky promoted in his novels religious moralities, particularly those of [[Eastern Orthodox Christianity]]. <ref name="BritannicaRussianLit"/> Indeed, "Dostoyevsky and the Religion of Suffering," the essay devoted to Dostoevsky in Eugène [[Melchior de Vogüé]]'s ''Le roman russe'' (1886), is widely considered to be the most influential early analysis of the novelist's work, introducing Dostoevsky and other Russian novelists to the West. Nabokov argued in his University courses at [[Cornell University|Cornell]], that such religious propaganda, rather than artistic qualities, was the main reason Dostoyevsky was praised and regarded as a 'Prophet' in Soviet Russia.<ref> Nabokov, Vladimir. “Lectures on Russian Literature”. Harcourt, 1981, p. 104</ref>

Dostoyevsky influenced American novelist [[Ernest Hemingway]]. [[James Joyce]] and [[Virginia Woolf]] praised his prose. Hemingway cited Dostoyevsky as a major influence on his work, in his posthumous collection of sketches ''[[A Moveable Feast]]''. In a book of interviews with Arthur Power (''Conversations with James Joyce''), Joyce praised Dostoyevsky's prose:

''{{quote|...he is the man more than any other who has created modern prose, and intensified it to its present-day pitch. It was his explosive power which shattered the Victorian novel with its simpering maidens and ordered commonplaces; books which were without imagination or violence.}}''

In her essay ''The Russian Point of View'', Virginia Woolf said:

''{{quote|The novels of Dostoevsky are seething whirlpools, gyrating sandstorms, waterspouts which hiss and boil and suck us in. They are composed purely and wholly of the stuff of the soul. Against our wills we are drawn in, whirled round, blinded, suffocated, and at the same time filled with a giddy rapture. Out of [[Shakespeare]] there is no more exciting reading.''<ref>[http://etext.library.adelaide.edu.au/w/woolf/virginia/w91c/chapter16.html The Russian Point of View] Virginia Woolf.</ref>}}

[[File:Dostoevsky-Library Moscow Russia.jpg|thumb|upright|Dostoyevsky monument at the [[Russian State Library]] in Moscow.]]

Dostoyevsky displayed a nuanced understanding of human psychology in his major works. He created an opus of vitality and almost hypnotic power, characterized by feverishly dramatized scenes where his characters are frequently in scandalous and explosive atmospheres, passionately engaged in [[Socratic dialogue]]s. The quest for God, the [[problem of Evil]] and suffering of the innocents haunt the majority of his novels.

His characters fall into a few distinct categories: humble and self-effacing Christians ([[Prince Myshkin]], [[Sonya Marmeladova]], [[Alyosha Karamazov]], [[Saint Ambrose of Optina|Starets Zosima]]), self-destructive [[nihilism|nihilists]] ([[Svidrigailov]], [[Smerdyakov]], [[Stavrogin]], [[Notes from Underground|the underground man]]){{Citation needed|date=May 2009}}, cynical debauchees ([[Fyodor Karamazov]], [[Dmitri Karamazov]]), and rebellious intellectuals ([[Raskolnikov]], [[Ivan Karamazov]], [[Ippolit]]); also, his characters are driven by ideas rather than by ordinary biological or social imperatives. In comparison with [[Leo Tolstoy|Tolstoy]], whose characters are [[Literary realism|realistic]], the characters of Dostoyevsky are usually more symbolic of the ideas they represent, thus Dostoyevsky is often cited as one of the forerunners of [[Symbolism (arts)|Literary Symbolism]], especially [[Russian Symbolism]] (see [[Alexander Blok]]).<ref>Dostoievsky by A. Steinberg p. 112</ref>

[[File:Dostoevsky MR280908.jpg|thumb|upright|Dostoyevsky statue, erected 1918, in front of [[Mariinsky Hospital]], the writer's birthplace in Moscow.]]

Dostoyevsky's novels are compressed in time (many cover only a few days) and this enables him to get rid of one of the dominant traits of [[realism (arts)|realist]] prose, the corrosion of human life in the process of the time flux; his characters primarily embody spiritual values, and these are, by definition, timeless. Other themes include suicide, wounded pride, collapsed family values, spiritual regeneration through suffering, rejection of the West and affirmation of [[Russian Orthodox Church|Russian Orthodoxy]] and [[Tsarism]]. Literary scholars such as [[Mikhail Bakhtin|Bakhtin]] have characterized his work as "[[Polyphony (literature)|polyphonic]]": Dostoyevsky does not appear to aim for a "single vision", and beyond simply describing situations from various angles, Dostoyevsky engendered fully dramatic novels of ideas where conflicting views and characters are left to develop unevenly into unbearable crescendo.

Dostoyevsky and the other giant of late 19th century [[Russian literature]], [[Lev Nikolayevich Tolstoy]], never met in person, even though each praised, criticized, and influenced the other (Dostoyevsky remarked of Tolstoy's ''[[Anna Karenina]]'' that it was a "flawless work of art"; [[Henri Troyat]] reports that Tolstoy once remarked of ''[[Crime and Punishment]]'' that, "Once you read the first few chapters you know pretty much how the novel will end up").{{Citation needed|date=August 2007}} There was a meeting arranged, but there was a confusion about where the meeting place was to take place and they never rescheduled. Tolstoy reportedly{{Who|date=June 2009}} burst into tears when he learned of Dostoyevsky's death. A copy of ''[[The Brothers Karamazov]]'' was found on the nightstand next to Tolstoy's deathbed at the [[Lev Tolstoy (settlement)|Astapovo]] railway station.

[[File:450px-Grab-dostojewsky.jpg|thumb|upright|Dostoyevsky's tomb at the [[Alexander Nevsky Monastery]]]]

===Dostoyevsky on Jews in Russia===
Notable writers, e.g. [[Joseph N. Frank]], Stephen Cassedy, David I. Goldstein, [[Gary Saul Morson]], and Felix Dreizin, have offered various insights and unique suppositions regarding Dostoyevsky’s views on Jews and organized Jewry in Russia – specifically, that Dostoyevsky perceived Jewish ethnocentrism and Jewish influence to be directly threatening the Russian peasantry in the border regions. For example, in ''[[A Writer's Diary]]'', Dostoyevsky wrote:

<blockquote>''Thus, Jewry is thriving precisely there where the people are still ignorant, or not free, or economically backward. It is there that Jewry has a champ libre. And instead of raising, by its influence, the level of education, instead of increasing knowledge, generating economic fitness in the native population—instead of this the Jew, wherever he has settled, has still more humiliated and debauched the people; there humaneness was still more debased and the educational level fell still lower; there inescapable, inhuman misery, and with it despair, spread still more disgustingly. Ask the native population in our border regions: What is propelling the Jew—and has been propelling him for centuries? You will receive a unanimous answer: mercilessness. He has been prompted so many centuries only by pitilessness to us, only by the thirst for our sweat and blood.''

''And, in truth, the whole activity of the Jews in these border regions of ours consisted of rendering the native population as much as possible inescapably dependent on them, taking advantage of the local laws. They have always managed to be on friendly terms with those upon whom the people were dependent. Point to any other tribe from among Russian aliens which could rival the Jew by his dreadful influence in this connection! You will find no such tribe. In this respect the Jew preserves all his originality as compared with other Russian aliens, and of course, the reason therefore is that status of status of his, that spirit of which specifically breathes pitilessness for everything that is not Jew, with disrespect for any people and tribe, for every human creature who is not a Jew...''<ref name="M. Dostoevsky 1949">F. M. Dostoevsky, ''The Diary of a Writer'', trans. Boris Brasol (New York: Charles Scribner's Sons, 1949)</ref></blockquote>

Dostoyevsky has been noted as both having expressed [[antisemitic]] sentiments as well as standing up for the rights of the Jewish people. In the recent biography by [[Joseph Frank (academic)|Joseph Frank]], ''The Mantle of the Prophet,'' Frank spent much time on ''A Writer's Diary''—a regular column which Dostoyevsky wrote in the periodical ''The Citizen'' from 1873 to the year before his death in 1881. Frank notes that the ''Diary'' is "filled with politics, literary criticism, and pan-Slav diatribes about the virtues of the Russian Empire, [and] represents a major challenge to the Dostoyevsky fan, not least on account of its frequent expressions of antisemitism."<ref>''Dostoevsky's leap of faith This volume concludes a magnificent biography which is also a cultural history.'' Orlando Figes. ''Sunday Telegraph'' (London) p. 13. September 29, 2002.</ref> Frank, in his foreword for the book ''Dostoevsky and the Jews'', attempts to place Dostoyevsky as a product of his time. Frank notes that Dostoyevsky ''did'' make antisemitic remarks, but that Dostoyevsky's writing and stance by and large was one where Dostoyevsky held a great deal of guilt for his comments and positions that were antisemitic.<ref>Dostoevsky and the Jews (University of Texas Press Slavic series) (Hardcover) 2 Joseph Frank, "Foreword" p. xiv. by David I. Goldstein ISBN 0292715285</ref> Steven Cassedy, for example, alleges in his book, ''Dostoevsky's Religion'', that much of the depiction of Dostoyevsky’s views as antisemitic omits that Dostoyevsky expressed support for the equal rights of the Russian Jewish population, a position that was not widely supported in Russia at the time.<ref name="Cassedy1">{{Cite book|title= Dostoevsky's Religion |last= Cassedy |first= Steven |year= 2005 |publisher= [[Stanford University Press]] |isbn= 0804751374 |pages= 67–80}}</ref> Cassedy also notes that this criticism of Dostoyevsky also appears to deny his sincerity when he said that he was for equal rights for the Russian Jewish populace and the [[Russian serfdom|Serf]]s of his own country (since neither group at that point in history had equal rights).<ref name=Cassedy1/> Cassedy again notes when Dostoyevsky stated that he did not hate Jewish people and was not antisemitic.<ref name=Cassedy1/> Even though Dostoyevsky spoke of the potential negative influence of Jewish people, Dostoyevsky advised Czar [[Alexander II of Russia|Alexander II]] to give them rights to positions of influence in Russian society, such as allowing them access to Professorships at Universities. According to Cassedy, labeling Dostoyevsky antisemitic does not take into consideration Dostoyevsky's expressed desire to peacefully reconcile Jews and Christians into a single universal brotherhood of all mankind.<ref name=Cassedy1 />

==Dostoyevsky and existentialism==
With the publication of ''[[Crime and Punishment]]'' in 1866, Dostoyevsky became one of Russia's most prominent authors. [[Will Durant]], in ''[[The Pleasures of Philosophy]]'' (1953), called Dostoyevsky one of the founding fathers of the philosophical movement known as [[existentialism]], and cited ''[[Notes from Underground]]'' in particular as a founding work of existentialism. For Dostoyevsky, war is the people's rebellion against the idea that [[reason]] guides everything, and thus, reason is not the ultimate guiding principle for either history or [[human|mankind]]. After his 1849 exile to the city of [[Omsk]], Siberia, Dostoyevsky focused heavily on notions of [[suffering]] and [[wiktionary:despair|despair]] in many of his works.

[[Nietzsche]] referred to Dostoyevsky as "the only psychologist from whom I have something to learn: he belongs to the happiest windfalls of my life, happier even than the discovery of [[Stendhal]]." He said that ''Notes from Underground'' "cried truth from the blood." According to [[Kontinent|Mihajlo Mihajlov]]'s "The Great Catalyzer: Nietzsche and Russian Neo-Idealism", [[Nietzsche]] constantly refers to Dostoyevsky in his notes and drafts throughout the winter of 1886–1887. Nietzsche also wrote abstracts of several Dostoyevsky works.

[[Freud]] wrote an article titled ''[[Dostoevsky and Parricide]]'', asserting that the greatest works in world literature are all about [[parricide]]; though he is critical of Dostoyevsky's work overall, his inclusion of ''[[The Brothers Karamazov]]'' among the three greatest works of literature is remarkable.

==Bibliography==
[[File:Fyodor Mikahailovich Dostoyevsky's Study in St Petersburg.jpg|right|thumb|250px|Dostoyevsky's study in [[Saint Petersburg]].]]
[[File:Fyodor Mikahailovich Dostoyevsky's Handwriting 1838.jpg|right|thumb|250px|Dostoyevsky's handwriting.]]
===Fiction===
Dostoyevsky's oeuvre includes 15 novels and novellas and 17 short stories.
====Novels and novellas====
*''[[Poor Folk]]'' (Бедные люди [''Bednye lyudi''], 1846)
*''[[The Double: A Petersburg Poem]]'' (Двойник: Петербургская поэма [''Dvoynik: Peterburgskaya poema''], 1846)
*''[[Netochka Nezvanova (novel)|Netochka Nezvanova]]'' (Неточка Незванова [''Netochka Nezvanova''], 1849)
*''[[The Uncle's Dream]]'' (Дядюшкин сон [''Dyadyushkin son''], 1859)
*''[[The Village of Stepanchikovo]]'' (Село Степанчиково и его обитатели [''Selo Stepanchikovo i ego obitateli''], 1859)
*''[[Humiliated and Insulted]]'' (Униженные и оскорбленные [''Unizhennye i oskorblennye''], 1861)
*''[[The House of the Dead (novel)|The House of the Dead]]'' (Записки из мертвого дома [''Zapiski iz mertvogo doma''], 1862)
*''[[Notes from Underground]]'' (Записки из подполья [''Zapiski iz podpolya''], 1864)
*''[[Crime and Punishment]]'' (Преступление и наказание [''Prestuplenie i nakazanie''], 1866)
*''[[The Gambler (novel)|The Gambler]]'' (Игрок [''Igrok''], 1867)
*''[[The Idiot (novel)|The Idiot]]'' (Идиот [''Idiot''], 1869)
*''[[The Eternal Husband]]'' (Вечный муж [''Vechnyj muzh''], 1870)
*''[[The Possessed (novel)|Demons]]'' (Бесы [''Besyi''], 1872)
*''[[The Adolescent]]'' (Подросток [''Podrostok''], 1875)
*''[[The Brothers Karamazov]]'' (Братья Карамазовы [''Brat'ya Karamazovy''], 1880)

====Short stories====
*"[[Mr. Prokharchin]]" (Господин Прохарчин ["Gospodin Prokharchin"], 1846)
*"Novel in Nine Letters" (Роман в девяти письмах ["Roman v devyati pis'mah"], 1847)
*"[[The Landlady]]" (Хозяйка ["Hozyajka"], 1847)
*"[[The Jealous Husband]]" (Чужая жена и муж под кроватью ["Chuzhaya zhena i muzh pod krovat'yu"], 1848)
*"A Weak Heart" (Слабое сердце ["Slaboe serdze"], 1848)
*"Polzunkov" (Ползунков ["Polzunkov"], 1848)
*"[[An Honest Thief]]" (Честный вор ["Chestnyj vor"], 1848)
*"[[A Christmas Tree and a Wedding]]" (Елка и свадьба ["Elka i svad'ba"], 1848)
*"[[White Nights (short story)|White Nights]]" (Белые ночи ["Belye nochi"], 1848)
*"A Little Hero" (Маленький герой ["Malen'kij geroj"], 1849)
*"[[A Nasty Story]]" (Скверный анекдот ["Skvernyj anekdot"], 1862)
*"[[The Crocodile (short story)|The Crocodile]]" (Крокодил ["Krokodil"], 1865)
*"[[Bobok]]" (Бобок ["Bobok"], 1873)
*"The Heavenly Christmas Tree" (Мальчик у Христа на ёлке ["Mal'chik u Hrista na elke"], 1876)
*"[[A Gentle Creature]]" (Кроткая ["Krotkaja"], 1876)
*"[[The Peasant Marey]]" (Мужик Марей ["Muzhik Marej"], 1876)
*"[[The Dream of a Ridiculous Man]]" (Сон смешного человека ["Son smeshnogo cheloveka"], 1877)

===Non-fiction===
*''Winter Notes on Summer Impressions'' (1863)
*''[[A Writer's Diary]]'' (Дневник писателя [''Dnevnik pisatelya''], 1873–1881)
*Assorted letters

==See also==
{{col-begin}}
{{col-break}}
*[[Albert Camus]]
*[[Aleksandr Solzhenitsyn]]
*[[Anti-Catholicism]]
*[[Determinism]]
*[[Free will]]
*[[Hesychasm]]
*[[Existentialism]]
*[[History of Eastern Christianity]]
*[[History of the Eastern Orthodox Church]]
*[[History of the Russian Orthodox Church]]
*[[Ivan Ilyin]]
*[[Jean-Paul Sartre]]
*[[List of Russian philosophers]]
{{col-break}}
*[[Lev Shestov]]
*[[Mikhail Epstein]]
*[[Negative theology]]
*[[Nikolai Berdyaev]]
*[[Nikolai Lossky]]
*[[Nikolay Strakhov]]
*[[Philokalia]]
*[[Russian Orthodox Church]]
*[[Søren Kierkegaard]]
*[[Voluntarism (philosophy)|Voluntarism]]
*[[Vasily Rozanov]]
{{col-end}}

==References==
{{Reflist|2}}
* {{imdb|0234502}}

==Biography==
*Frank, Joseph: 5 biographical volumes, Princeton University Press, c1976–2002.
**''Seeds of Revolt, 1821–1849''. (c1976). ISBN 0691062609 (v. 1)
**''Years of Ordeal, 1850–1859''. ISBN 0691065764 (v. 2)
**''Stir of Liberation, 1860–1865''. ISBN 0691066523 (v. 3)
**''Miraculous Years, 1865–1871'' ISBN 0691043647 (v. 4)
**''Mantle of the Prophet, 1871–1881'' (2002). ISBN 0691086656 (v. 5)
*Frank, Joseph, ''Dostoevsky: A Writer in His Time''. Princeton University Press. 2009. ISBN 9780691128191

==External links==
{{Wikisource|Author:Fyodor Dostoevsky|Fyodor Dostoevsky}}
{{Wikiquote}}
{{Commons|Фёдор Михайлович Достоевский|Fyodor Dostoyevsky}}
*{{gutenberg author| id=Fyodor+Dostoyevsky|name=Fyodor Dostoevsky}}
*{{worldcat id|id=lccn-n79-29930}}
*[http://fortnightlyreview.co.uk/2010/05/dostoyevski-and-the-religion-of-suffering-part-i/ "Dostoyevsky and the Religion of Suffering"] from ''Le roman russe'', Eugène Melchior de Vogüé's influential essay introducing Dostoevsky to the West.
*[http://www.FyodorDostoevsky.com FyodorDostoevsky.com] – Fan site: discussion forum, bookstore, essays, e-texts, photos, biography, quotes, and links
*[http://Dostoyevsky.thefreelibrary.com/ Fyodor Dostoevsky's brief biography and works]
*[http://digital.library.upenn.edu/webbin/book/search?author=Dostoyevsky%2c+ Selected Dostoevsky e-texts from Penn Librarys digital library project]
*[http://www.ytayta.com/artists/dostoyevsky_fyodor Sketches and drawings made by Dostoevsky]
*[http://ilibrary.ru/author/dostoevski/ Full texts of some Dostoevsky's works in the original Russian]
*[http://www.magister.msk.ru/library/dostoevs/ Another site with full texts of Dostoevsky's works in Russian]
*[http://www.fmdostoyevsky.com Fyodor Dostoyevsky] – Biography, ebooks, quotations, and other resources
*[http://www.kiosek.com/dostoevsky/contents.html Dostoevsky Research Station]
*[http://people.emich.edu/wmoss/publications/ Alexander II and his times: A Narrative History of Russia in the Age of Alexander II, Tolstoy, and Dostoevsky]
*{{IBList |type=author|id=96|name=Fyodor Dostoevsky}}
*[http://www.the-ledge.com/flash/ledge.php?book=75&lan=UK Dostoyevsky 'Bookweb' on literary website The Ledge, with suggestions for further reading.]
*{{worldcat id|id=lccn-n79-29930}}
*[http://www.mootnotes.com/literature/dostoevsky/index.html Dostoevsky works (HTML/PDF), media gallery & interactive timeline]
*[http://forums.toketastic.com/User/Discussion.aspx?id=186088 Crime and Punishment Review]
*[http://www.tanais.info/art/en/inquisitor.html Legend of the Grand Inquisitor] from ''Brothers Karamazov'' by Fyodor Mikhailovich Dostoevsky.
*[http://www.telegram.com/article/20080409/NEWS/804090488/1008/NEWS02 Recent Dostoevsky exhibit]
*[http://www.russianart.dk/exhibition.asp?e=346 Illustrations to Dostoyevsky tales "[[Netochka Nezvanova]]" and "[[A Gentle Creature]]" by Soviet artist [[Mikhail Rojter]]]

{{Fyodor Dostoevsky}}
{{Persondata
|NAME= Dostoevsky, Fyodor Mikhailovich
|ALTERNATIVE NAMES= Dostoyevsky, Fyodor Mikhailovich; Фёдор Миха́йлович Достое́вский (Russian)
|SHORT DESCRIPTION= Russian novelist
|DATE OF BIRTH= {{Birth date|1821|11|11|mf=y}}
|PLACE OF BIRTH= Moscow
|DATE OF DEATH= {{Death date|1881|2|9|mf=y}}
|PLACE OF DEATH= Saint Petersburg
}}
{{DEFAULTSORT:Dostoevsky, Fyodor}}
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[[Category:Existentialists]]
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[[Category:People with epilepsy]]
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[[Category:Russian-language writers]]
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[[Category:Russian people of Belarusian descent]]
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0,999…

2011-02-04T23:14:18Z

Paul August: Undid revision 412062161 by Paul August (talk)

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', <math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math> or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternative number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.

===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
\end{align}
</math>

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===
When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p. 170</ref>

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

===Dedekind cuts===
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form
:<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p. 399</ref>
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
:<math>\begin{align}\tfrac{a}{b}<1\end{align},</math>
which implies
:<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp. 388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infintesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

:<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1}\dots,</math>

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> All such interpretations of "0.999..." are [[adequality|adequal]] to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111...2 follows directly from Berlekamp's Rule.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + ''x'' = 1"
has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp. 9–11)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |url=http://jstor.org/stable/2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |url=http://jstor.org/stable/2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |url=http://jstor.org/stable/2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |url=http://jstor.org/stable/2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |url=http://jstor.org/stable/2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |url=http://jstor.org/stable/2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |url=http://jstor.org/stable/2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |url=http://jstor.org/stable/2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2011-02-04T23:13:47Z

Paul August: Reverted edits by 28bytes (talk) to last version by 95.102.237.71

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', <math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math> or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternative number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==2011 New Year - New Way of Looking at Infinite-Recurring 0.9==

OK! The Way I would like to put forward is...to find The Negative,The Root,and The Positive.

Below is an Example Using 100

100 Negative = 99.9

100 Root = 100 ( THE ROOT IS THE START! THE ORIGIN! ) (The Root) is Not The Square Root)

100 Positive = 100.1

A Better way of Looking at the above is 100N(99.9) 100R(100) 100P(100.1)

---------------------------------------------------------------------------------------------------- --------

So Now all we have to find is...

Infinite-Recurring 0.9 Negative = ?

Infinite-Recurring 0.9 Root = ?

Infinite-Recurring 0.9 Positive = ?

A Better way of Looking at the above is IR0.9N( ? ) IR0.9R( ? ) IR0.9P( ? )

Infinite-Recurring 0.9 Negative = ? Put Here I/R 0.9 as a Negative Value

Infinite-Recurring 0.9 Root = ? Put Here what you think I/R 0.9 is ?

Infinite-Recurring 0.9 Positive = ? Put Here I/R 0.9 as a Positive Value

Meanwhile, can you find a number between 0.9999... and 1? = 0.9(.9)1

I have seen Many Example Calculations for Infinite-Recurring 0.9
But no One is Showing the Math from Root as Explained in my Example to it Becoming or is equal to 1

Below is an Example Using 100

100 Negative = 99.9

100 Root = 100

100 Positive = 100.1

Can the above be Shown in the same way for Infinite-Recurring 0.9

Now we are Getting somewhere! Let me show what is wrong with the Examples Below...

0.99999... - 0.1 = 0.8999999... = 0.8 + 0.09999... = 0.8 + 0.1 = 0.9
0.99999... = 1
0.99999... + 0.1 = 1.0999999... = 1 + 0.09999... = 1 + 0.1 = 1.1

First of all you Cannot Add or Subtract Etc. from Something that is Continuous! In this case Infinite-Recurring 0.9 As soon as you do one or more of the .9s is no longer Continuous!

In the Negative Example Below a .9 has become .8

0.99999... - 0.1 = 0.8999999... = 0.8 + 0.09999... = 0.8 + 0.1 = 0.9

In the Root Example Below All the .9s Have Stopped being Continuous! By Assuming They All = 1

0.99999... = 1

In the Positive Example Below...Which is the Worst case! All the .9s Have Stopped being Continuous! Plus There is Now a .1

0.99999... + 0.1 = 1.0999999... = 1 + 0.09999... = 1 + 0.1 = 1.1

What the Above shows is that you Cannot Apply Calculations to Something that is Continuous!
This is the Mistake everyone is doing when trying to Prove Infinite-Recurring 0.9 = 1

The Next Main Mistake everyone is doing is Trying to Go Against the Above...by Calculating the Ten Times Calculation 10 x Etc.

There are Three types of Ten Times Calculation Which Concern the Infinite-Recurring 0.9 Problem! Only One is ever being used None of the other Two are mentioned as Possibilities!?

The First is the Normal 10 x 0.9 = 9 This is OK if the 0.9 is a Single 0.9 But I have seen many Example where it is Trying to be Applied to 0.999...Etc. Against the Above.

The Second is The 10 x 0.9 Which could Equal (.9999999999) That is Ten of the Continuous .9s But How can you Separate The Ten From the Rest? Again Against the Above.

And the Third is 10 x All the .9s Again Not Possible Because of the Above.

So to End this... Infinite-Recurring 0.9 Must always be Known and Shown as Having Continuous .9s

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The Philosopher Quotes: “ I Think Therefore I am “

The Mathematician answers “ I Continue Therefore I’m Recurring “

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GeniusIsBack.

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.

===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
\end{align}
</math>

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===
When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p. 170</ref>

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

===Dedekind cuts===
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form
:<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p. 399</ref>
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
:<math>\begin{align}\tfrac{a}{b}<1\end{align},</math>
which implies
:<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp. 388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infintesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

:<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1}\dots,</math>

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> All such interpretations of "0.999..." are [[adequality|adequal]] to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111...2 follows directly from Berlekamp's Rule.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + ''x'' = 1"
has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp. 9–11)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |url=http://jstor.org/stable/2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |url=http://jstor.org/stable/2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |url=http://jstor.org/stable/2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |url=http://jstor.org/stable/2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |url=http://jstor.org/stable/2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |url=http://jstor.org/stable/2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |url=http://jstor.org/stable/2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |url=http://jstor.org/stable/2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Liste deutscher Erfinder und Entdecker

2011-01-26T20:30:31Z

Paul August: revert myself (for consistency)

__NOTOC__
This is a list of '''German inventors and discoverers'''. The following list comprises people from Germany or German-speaking Europe, also of people of predominantly German heritage, in alphabetical order of the surname. The main section includes existing articles, indicated by blue links, and possibly non-existing, indicated by red links. '''Please read the [[#Notes|Notes]] section below for more information BEFORE adding an inventor or discoverer.'''

----
{| align=center
! bgcolor=#dddddd align=left | '''Existing''': 
| bgcolor=#dddddd | [[#A|A]]
| bgcolor=#dddddd | [[#B|B]]
| bgcolor=#dddddd | [[#C|C]]
| bgcolor=#dddddd | [[#D|D]]
| bgcolor=#dddddd | [[#E|E]]
| bgcolor=#dddddd | [[#F|F]]
| bgcolor=#dddddd | [[#G|G]]
| bgcolor=#dddddd | [[#H|H]]
| bgcolor=#dddddd |  [[#I|I]] 
| bgcolor=#dddddd |  [[#J|J]] 
| bgcolor=#dddddd | [[#K|K]]
| bgcolor=#dddddd | [[#L|L]]
| bgcolor=#dddddd | [[#M|M]]
| bgcolor=#dddddd | [[#N|N]]
| bgcolor=#dddddd | [[#O|O]]
| bgcolor=#dddddd | [[#P|P]]
| bgcolor=#dddddd | [[#Q|Q]]
| bgcolor=#dddddd | [[#R|R]]
| bgcolor=#dddddd | [[#S|S]]
| bgcolor=#dddddd | [[#T|T]]
| bgcolor=#dddddd | [[#U|U]]
| bgcolor=#dddddd | [[#V|V]]
| bgcolor=#dddddd | [[#W|W]]
| bgcolor=#dddddd | [[#X|X]]
| bgcolor=#dddddd | [[#Y|Y]]
| bgcolor=#dddddd | [[#Z|Z]]
|-
|-
| colspan=9 align=center bgcolor=#bbbbbb | [[#Notes|Notes]]
| colspan=9 align=center bgcolor=#bbbbbb | [[#See also|See also]]
| colspan=10 align=center bgcolor=#bbbbbb | [[#External links|External links]]
|}
----

[[Image:Bundesarchiv Bild 183-K0917-501, Prof. Manfred v. Ardenne.jpg|thumb| right|175px| Ardenne in 1933]]
==A==
*[[Ernst Abbe]]: Invented the first [[refractometer]], and many other devices.
*[[Robert Adler]]: Invented a better television remote control.
*[[Alois Alzheimer]]: Psychiatrist who discovered [[Alzheimer]]´s disease, a degeneration of the brain in old age.
*[[Manfred von Ardenne]]: Self taught researcher, applied physicist and inventor. Inventor of television among other things. 600 patents in fields including electron microscopy, medical technology, nuclear technology, plasma physics, and radio and television technology.
*[[Leopold Auerbach]]: Discovery of Plexus myentericus Auerbachi, or Auerbach's plexus.

==B==
*[[Karl Ernst von Baer]]: Discovered mammal ovum.
*[[Oscar Barnack]]: The father of the first mass marketed 35mm camera and [[Leica]].
*[[Wilhelm Bauer]]: Inventor and engineer, who built several hand-powered submarines.
*[[Hans Beck]]: Inventor of the toy [[Playmobil]].
*[[Martin Behaim]]: Inventor of the first globe of the world ([[Erdapfel]]).
*[[Emil Adolf von Behring]]: Discoverer of diphtheria antitoxin. He was awarded histories first Nobel Prize in Physiology or Medicine in 1901.
*[[Melitta Bentz]]: Inventor of the coffee filter, 1908.
*[[Karl Benz]]: Father and inventor of the gasoline-powered automobile, 1885, and pioneering founder of automobile manufacturing.
[[Image:S-IC engines and Von Braun.jpg|right|thumb|200px|[[Wernher von Braun]]]]
*[[Johann Elert Bode]] Discovered the Titus-Bode Law
*[[Carl Bosch]]: Chemist and Nobel laureate, discovered the processes of industrial high pressure chemistry.
*[[Robert Bosch]]: He invented, engineered and launched various innovations for the motor vehicle.
*[[Karlheinz Brandenburg]]: Inventor and audio engineer; father of audio compression format MPEG Audio Layer 3, more commonly known as MP3.
*[[Wernher von Braun]]: The preeminent rocket engineer of the 20th century. Developed the [[V2 rocket]] for Germany. Built [[Saturn V rocket]] in USA which put the man on the moon.
*[[Carl Friedrich Bückling]]: Inventor of the first German steam-powered machine.
*[[Robert Bunsen]]: Chemist who developed the [[Bunsen burner]], and with [[Gustav Kirchhoff]] he discovered caesium (1860) and rubidium (1861).
*[[Wilhelm Busch]]: Caricaturist, painter and poet; father of comics.

==C==

[[Image:gottliebdaimler1.jpg|thumb|right|175px|Gottlieb Daimler]]
*[[Georg Cantor]]: Mathematician, discoverer of the [[set theory]] (1870s), which has become a fundamental theory in mathematics.
*[[Carl von Clausewitz]]: The father of modern military theory.

==D==
*[[Gottlieb Daimler]]: He invented the first high-speed petrol engine and the first four-wheel automobile, also the first gasoline-powered motorcycle.
*[[Adolf Dassler|Adolf "Adi" Dassler]]: Sport sneakers with and without spikes. ([[Addidas]]..)
*[[Rudolf Dassler]]: First sport shoes with screw-in shoe spikes, 1949. ([[Puma AG|Puma]]..)
*[[Johann Friedrich Dieffenbach]]: Pioneer of skin transplantation and cosmetic surgery.
*[[Rudolf Diesel]]: Inventor of the [[diesel engine]].
*[[Christian Doppler]]: Discovered the [[Doppler effect]].
*[[Walter Robert Dornberger]]: Co-inventor of the [[V2 rocket]].
*[[Karl Drais]]: Inventor of the bicycle and typewriter (1821) among other things.
*[[Peter Ferdinand Drucker]]: Invented the science of modern management.

==E==
[[Image:Einstein1921 by F Schmutzer 4.jpg|thumb|right|175px|[[Albert Einstein]] in [[List of Nobel Laureates in Physics|1921]], the year he was awarded the [[Nobel Prize in Physics]]]]
*[[Paul Ehrlich]]: Scientist in the fields of hematology, immunology, and chemotherapy, and Nobel laureate. Developed an effective treatment against syphilis.
*[[Albert Einstein]]: Godfather of Theoretical Physics, inventor and discoverer.
*[[Leonard Euler]]: Many discoveries in mathematics. "He is our teacher in all things."

==F==
*[[Daniel Gabriel Fahrenheit]]: Physicist and engineer who invented the [[alcohol thermometer]] (1709), the [[mercury thermometer]] (1714), and [[Fahrenheit|a temperature scale now named after him]].
*[[Hermann Emil Fischer]]: Discoveries in chemistry.
*[[Joseph von Fraunhofer]]: Discovery of the dark absorption lines known as Fraunhofer lines in the Sun's spectrum, and for making excellent optical glass and achromatic telescope objectives.
*[[Nikolaus Friedreich]]: Discovery of Friedreich-Auerbach disease (together with Leopold Auerbach) among other things.
[[File:Johannes Gutenberg.jpg|right|thumb|175px|[[Johannes Gutenberg]] in a 16th century copper engraving]]

==G==
*[[Brothers Grimm]]: Academic pioneers of [[linguistics]] and storytelling; [[Grimm's law]].
*[[Carl Friedrich Gauss|Johann Carl Friedrich Gauß]]: Inventor of the first electromagnetic telegraph; discoverer of [[list of topics named after Carl Friedrich Gauss|many things in mathematics and physics]], known as "the Prince of Mathematicians".
*[[Walter Gerlach]]: Physicist who co-discovered spin quantization in a magnetic field, the [[Stern-Gerlach effect]].
*[[Heinrich Göbel]]: Inventor of the first light bulb in 1854.
*[[Kurt Gödel]]: Important discoveries in math and logic, such as the [[incompleteness theorem]]s
*[[Peter Carl Goldmark]]: Engineer who was instrumental in developing the long-playing (LP) microgroove 33-1/3 rpm vinyl phonograph disc.
*[[Heinz Guderian]]: The father of modern mechanized warfare.
*[[Otto von Guericke]]: Discovered vacuum pump.
*[[Johannes Gutenberg]]: Inventor of the technology of printing with movable type.

==H==
[[Image:Bundesarchiv Bild 183-46019-0001, Otto Hahn.jpg|right|thumb|175px|[[Otto Hahn]] the first man to split the atom.]]
*[[Fritz Haber]]:German chemist and Nobel laureate who pioneered synthetic ammonia and chemical warfare.
*[[Otto Hahn]]: German chemist and Nobel laureate who pioneered the fields of radioactivity and radiochemistry. "The father of nuclear chemistry" and the "founder of the atomic age". Discovered nuclear fission.
*[[Henry J. Heinz]]: Tomato ketchup and fifty six other things.
*[[Werner Heisenberg]]: Theoretical physicist who made foundational contributions to quantum mechanics. Discovered a particle's position and velocity cannot be known at the same time. Discovered atomic nuclei are made of protons and neutrons.
*[[Rudolf Hell]]: Inventor of the first fax machine ([[Hellschreiber]]).
*[[Hellmann's and Best Foods|Richard Hellmann]]: Hellmann's (Blue Ribbon) Mayonnaise, 1905.
[[Image:Felix Hoffman.jpg|thumb|right|175px|[[Felix Hoffmann]]]]
*[[Hermann von Helmholtz]]: Discovered the principal of the conservation of energy.
*[[Peter Henlein]]: Inventor of the watch.
*[[Friedrich Wilhelm Herschel]]: Discovered infrared radiation among other things.
*[[Heinrich Hertz]]: Physicist, Discoverer of electromagnetic/radio waves.
*[[Victor Francis Hess]]: Discovered [[Cosmic rays]]. Also won the [[Nobel Prize]].
*[[David Hilbert]]: influental Mathematician, discovered and developed a broad range of fundamental ideas in math.
*[[Albert Hoffman]]: Discovered the chemical properties of [[chitin]] and [[lysergic acid diethylamide]].
*[[Felix Hoffmann]]: Aspirin (Bayer), 1897.
*[[Gottlob Honold]]: Inventor of the spark plug and the modern internal combustion engine, as well as headlights.
*[[Christian Huelsmeyer]]: Inventor of Radar.
*[[Alexander von Humboldt]]: Naturalist and explorer. His quantitative work on botanical geography was foundational to the field of biogeography.
*[[Wilhelm von Humboldt]]: Originator of the [[linguistic relativity]] hypothesis.

==I==
*[[Otmar Issing]]: Economist who invented the "two pillar" decision algorithm now used by the [[European Central Bank|ECB]].

==J==
*[[Hugo Junkers]]: Pioneer of all-metal aircraft construction with the [[Junkers J 1]] (1915–16).

==K==
[[File:Robert Koch berlin.jpg|thumb|175px|right|Monument to Robert Koch on his name square in [[Berlin]]]]
*[[Johannes Kepler]]: Discovered the laws of planetary motion.
*[[Erhard Kietz]]: Pioneer discoverer of video technology.
*[[Gustav Kirchoff]]: Discovery of the principles upon which [[spectrum analysis]] is founded.
*[[Martin Heinrich Klaproth]]: Discovered the element [[Uranium]].
*[[Robert Koch]]: Physician, discoverer, inventor and Nobel Prize winner. He became famous for isolating Bacillus anthracis (1877), the Tuberculosis bacillus (1882) and the Vibrio cholera (1883) and for his development of Koch's postulates.
*[[Arthur Korn]]: Inventor involved in development of the fax machine, specifically the transmission of photographs or telephotography, known as the Bildetelegraph.
*[[Julius H. Kroehl]]: Inventor and engineer, who built the first functioning submarine of the world.
*[[Alfred Krupp]]: Pioneer in metal casting and metal working process and procedures.

==L==
*[[Eugen Langen]]: Entrepreneur, engineer and inventor, involved in the development of the petrol engine and the Wuppertal monorail.
*[[Max von Laue]]: Discoveries regarding the diffraction of X-rays in crystals.
*[[Gottfried Wilhelm Leibniz]]: Philosopher known for discovering the mathematical field of calculus and coherently laying down its basic operations in 1684.
*[[Justus von Liebig]]: Discoveries in chemistry.
*[[Otto Lilienthal]]: Father of Aviation and first successful aviator of humanity. Main discovery was the properties and shape of the [[wing]].

==M==
[[Image:Marx3.jpg|right|thumb|175px|[[Karl Marx]]]]
*[[Ernst Mach]]: Discovered many effects of high speed projectiles; the [[Mach number]] is dedicated to his memory.
*[[Georg Hans Madelung]]: Academic and aeronautical engineer; a participant in the development of the [[Junkers F.13]].
*[[Karl Marx]]: Political economist and philosopher, who discovered the mechanics of capitalism and whose ideas shape the world we now live in.
*[[Wilhelm Maybach]]: Together with Gottlieb Daimler the first gasoline-powered motorcycle, power-engined boat and later, 1902, the Mercedes car model.
*[[Ottomar von Mayenburg]]: Inventor of "Chlorodont", the first commercial brand of toothpaste.
*[[Lise Meitner]]: Nuclear physicist, who, together with [[Otto Hahn]], explained theoretically nuclear fission.
*[[Gregor Mendel]]: Discoveries in genetics.
*[[Johannes Peter Müller]]: Discoveries in physiology.


==N==
*[[Walther Nernst]]: Inventor of the [[Nernst lamp]] and Nobel laureate.
[[Image:Max Planck (Nobel 1918).jpg|right|175px|thumb|[[Max Planck]]]]
==O==
*[[Hermann Oberth]]: Pioneer of rocket science and discoverer of the [[Oberth effect]].
*[[Hans von Ohain|Hans Joachim Pabst von Ohain]]: The modern jet engine in 1933, patented in 1936.
*[[Nikolaus Otto|Nikolaus August Otto]]: Inventor of the first internal-combustion engine to efficiently burn fuel directly in a piston chamber.

==P==

*[[Max Planck]]: Physicist, Scientist. He is considered to be the founder of the quantum theory, and one of the most important physicists of the twentieth century.
*[[Robert Pohl|Robert Wichard Pohl]]: In 1938, together with Rudolf Hilsch, built first functioning solid-state amplifier using salt as the semiconductor.

==Q==

{{Empty section|date=July 2010}}
==R==
*[[Johann Philipp Reis]]: Inventor of the first phone transmitter in 1861, he also invented the term ''Telephone''.
*[[Ralf Reski]]: Moss [[Bioreactor]] (1998)
[[Image:Paul Julius Reuter 1869.jpg|thumb|right|175px|Paul Reuter aged 53 years (1869) by [[Rudolf Lehmann (artist)|Rudolf Lehmann]]]]
*[[Paul Reuter|Paul Julius Freiherr von Reuter]]: Communications pioneer.
*[[Wilhelm Conrad Röntgen]]: Physicist and discoverer of x-rays/Röntgen rays (8 November 1895), this earned him the first Nobel Prize in Physics in 1901.

==S==
*[[Hans Sauer]]: Inventor of miniature high power relays; 309 patents worldwide in relay conceptions.
*[[Heinrich Schliemann]]: Father of [[archaeology]], among other things he discovered Homeric [[Troy]].
*[[Bernhard Schmidt]]: Discovered major improvements to the telescope.
*[[Paul Schmidt (inventor)]]: Developed since 1928 his idea of a new drive, the "pulsating incineration", also used in the V1-Rocket (engine was called "Argus-Schmidtrohr"); [[pulsejet]] was a development by Schmidt.
*[[Theodor Schwann]]: Discovery of properties of cells in animals.
*[[Johann Lukas Schönlein]]: Professor of medicine, he discovered among other things the parasitic cause of ringworm or favus (Achorion Schönleinii).
*[[Marx Schwab]]: Silversmith, invented coining with the screw press around 1550.
*[[Werner Sell]]: (Georg Robert Werner Sell) invented airplane kitchens, fitted kitchens and the prefabricated house among other things.
*[[Ernst Werner von Siemens]]: Dynamo, pointer telegraph that used a needle to point to the right letter, first electric elevator, trolleybus.
*[[Otto Stern]]: Nobel laureate; contributed to the discovery of spin quantization in the [[Stern-Gerlach experiment]] with [[Walther Gerlach]] in 1922.
*[[Eduard Suess]]: Discoveries in geology.

==T==
*[[Louis Tuchscherer]]: Inventor and mechanical engineer.

==U==
*[[Dietrich Uhlhorn|Dietrich "Diedrich" Uhlhorn]]: Engineer, mechanic and inventor, who invented the first mechanical tachometer (1817), between 1817 and 1830 inventor of the Presse Monétaire (level coin press known as Uhlhorn Press) which bears his name.

[[Image:Anna Berthe Roentgen.gif|thumb|''Hand mit Ringen'': print of Wilhelm Röntgen's first "medical" x-ray, of his wife's hand, taken on 22 December 1895 and presented to [[Ludwig Zehnder|Professor Ludwig Zehnder]] of the Physik Institut, University of Freiburg, on 1 January 1896]]

==V==
*[[Abraham Vater]]: Ampulla of Vater.
*[[Tri-Ergon|Hans Vogt]]: Invented sound-on-film (idea 1905) together with Jo Engl and Joseph Massolle, first sound-on-film for the public on 17 September 1922 in Filmtheater ''Alhambra '', Berlin, Germany.
*[[Woldemar Voigt]]: Physicist, who taught at the Georg August University of Göttingen. He worked on crystal physics, thermodynamics and electro-optics. He discovered the Voigt effect in 1898.
*[[Woldemar Voigt (aerospace engineer)]]: Chief designer at [[Messerschmitt|Messerschmitt's]] Oberammergau offices.

==W==
*[[Felix Wankel]]: Inventor of the Rotary Motor.
*[[Max Weber]]: Discovered the mass effects of capitalism and modernity.
*[[Wilhelm Eduard Weber]]: Inventor of the first electromagnetic telegraph together with Carl Friedrich Gauss.
*[[Gustav Weißkopf]]: Aviation pioneer - Worlds First Motorized Flight: August 14, 1901.
*[[Clemens Alexander Winkler]]: Chemist who discovered the element germanium in 1886.
*[[Friedrich Wöhler]]: The first to synthesis [[urea]].

==X==

{{Empty section|date=July 2010}}

==Y==

{{Empty section|date=July 2010}}

[[File:Zuse Z1-2.jpg|thumb|right|Zuse Z1 replica in the [[German Museum of Technology (Berlin)|German Museum of Technology]] in Berlin]]

==Z==
* [[Carl Zeiss]]: Pioneered glass casting and allied procedures and processes for high quality optics.
* [[Ferdinand Graf von Zeppelin]] (1838–1917): Inventor of the airship named after him. Start of the airship LZ1 in 1900.
*[[Karl Zimmer]]: Discovered the effects of [[ionizing radiation]] on [[DNA]].
*[[Konrad Zuse]]: Inventor of the first computing machine/computer, and the first [[high-level programming language]].

== Notes ==
'''When adding an inventor or a discoverer to the main section, please check first to see if it is already in the list .''' If it isn't, you might also check to see if the article exists (by entering the title in the Search box and pressing Go), as some editors may have forgotten to add their articles on German inventors to this list. '''When you add an inventor or a discoverer to this list, please add it in proper alphabetical order within the appropriate section.'''

Please include the year of invention or discovery and list key informations of the invention, as well as the references for it.

After an inventor article has been created, the link on this page will be blue. Please move these titles into the main (existing article) section after creating the show article.

Please be sure that the inventor or discoverer is German or of German heritage. For example, [[Wolfgang Amadeus Mozart]] is often classified as Austrian, but lived in an Era, where Austria was part of the [[Holy Roman Empire of the German Nation]], his nationality is therefore German.

==See also==
* [[German inventions and discoveries]]
* [[List of German Americans]]

==External links==
* [http://germanoriginality.com/madein/inventions.php "Made in Germany"]
* [http://german.about.com/library/blerfindung.htm German Inventions - Discoveries]
* [http://www.zoomwhales.com/inventors/germany.shtml A Sampling of German Inventors and Inventions]

{{Inventions}}

[[Category:German inventions]]
[[Category:History of Germany]]
[[Category:Lists of inventors|German]]

[[ar:ملحق:قائمة مخترعين ومكتشفين ألمان]]

Liste deutscher Erfinder und Entdecker

2011-01-26T20:29:22Z

Paul August: /* C */ Under the normal meaning I don't think that Cantor qualifies as an inventor -- otherwise, all mathematicians would as well

__NOTOC__
This is a list of '''German inventors and discoverers'''. The following list comprises people from Germany or German-speaking Europe, also of people of predominantly German heritage, in alphabetical order of the surname. The main section includes existing articles, indicated by blue links, and possibly non-existing, indicated by red links. '''Please read the [[#Notes|Notes]] section below for more information BEFORE adding an inventor or discoverer.'''

----
{| align=center
! bgcolor=#dddddd align=left | '''Existing''': 
| bgcolor=#dddddd | [[#A|A]]
| bgcolor=#dddddd | [[#B|B]]
| bgcolor=#dddddd | [[#C|C]]
| bgcolor=#dddddd | [[#D|D]]
| bgcolor=#dddddd | [[#E|E]]
| bgcolor=#dddddd | [[#F|F]]
| bgcolor=#dddddd | [[#G|G]]
| bgcolor=#dddddd | [[#H|H]]
| bgcolor=#dddddd |  [[#I|I]] 
| bgcolor=#dddddd |  [[#J|J]] 
| bgcolor=#dddddd | [[#K|K]]
| bgcolor=#dddddd | [[#L|L]]
| bgcolor=#dddddd | [[#M|M]]
| bgcolor=#dddddd | [[#N|N]]
| bgcolor=#dddddd | [[#O|O]]
| bgcolor=#dddddd | [[#P|P]]
| bgcolor=#dddddd | [[#Q|Q]]
| bgcolor=#dddddd | [[#R|R]]
| bgcolor=#dddddd | [[#S|S]]
| bgcolor=#dddddd | [[#T|T]]
| bgcolor=#dddddd | [[#U|U]]
| bgcolor=#dddddd | [[#V|V]]
| bgcolor=#dddddd | [[#W|W]]
| bgcolor=#dddddd | [[#X|X]]
| bgcolor=#dddddd | [[#Y|Y]]
| bgcolor=#dddddd | [[#Z|Z]]
|-
|-
| colspan=9 align=center bgcolor=#bbbbbb | [[#Notes|Notes]]
| colspan=9 align=center bgcolor=#bbbbbb | [[#See also|See also]]
| colspan=10 align=center bgcolor=#bbbbbb | [[#External links|External links]]
|}
----

[[Image:Bundesarchiv Bild 183-K0917-501, Prof. Manfred v. Ardenne.jpg|thumb| right|175px| Ardenne in 1933]]
==A==
*[[Ernst Abbe]]: Invented the first [[refractometer]], and many other devices.
*[[Robert Adler]]: Invented a better television remote control.
*[[Alois Alzheimer]]: Psychiatrist who discovered [[Alzheimer]]´s disease, a degeneration of the brain in old age.
*[[Manfred von Ardenne]]: Self taught researcher, applied physicist and inventor. Inventor of television among other things. 600 patents in fields including electron microscopy, medical technology, nuclear technology, plasma physics, and radio and television technology.
*[[Leopold Auerbach]]: Discovery of Plexus myentericus Auerbachi, or Auerbach's plexus.

==B==
*[[Karl Ernst von Baer]]: Discovered mammal ovum.
*[[Oscar Barnack]]: The father of the first mass marketed 35mm camera and [[Leica]].
*[[Wilhelm Bauer]]: Inventor and engineer, who built several hand-powered submarines.
*[[Hans Beck]]: Inventor of the toy [[Playmobil]].
*[[Martin Behaim]]: Inventor of the first globe of the world ([[Erdapfel]]).
*[[Emil Adolf von Behring]]: Discoverer of diphtheria antitoxin. He was awarded histories first Nobel Prize in Physiology or Medicine in 1901.
*[[Melitta Bentz]]: Inventor of the coffee filter, 1908.
*[[Karl Benz]]: Father and inventor of the gasoline-powered automobile, 1885, and pioneering founder of automobile manufacturing.
[[Image:S-IC engines and Von Braun.jpg|right|thumb|200px|[[Wernher von Braun]]]]
*[[Johann Elert Bode]] Discovered the Titus-Bode Law
*[[Carl Bosch]]: Chemist and Nobel laureate, discovered the processes of industrial high pressure chemistry.
*[[Robert Bosch]]: He invented, engineered and launched various innovations for the motor vehicle.
*[[Karlheinz Brandenburg]]: Inventor and audio engineer; father of audio compression format MPEG Audio Layer 3, more commonly known as MP3.
*[[Wernher von Braun]]: The preeminent rocket engineer of the 20th century. Developed the [[V2 rocket]] for Germany. Built [[Saturn V rocket]] in USA which put the man on the moon.
*[[Carl Friedrich Bückling]]: Inventor of the first German steam-powered machine.
*[[Robert Bunsen]]: Chemist who developed the [[Bunsen burner]], and with [[Gustav Kirchhoff]] he discovered caesium (1860) and rubidium (1861).
*[[Wilhelm Busch]]: Caricaturist, painter and poet; father of comics.

==C==

[[Image:gottliebdaimler1.jpg|thumb|right|175px|Gottlieb Daimler]]
*[[Carl von Clausewitz]]: The father of modern military theory.

==D==
*[[Gottlieb Daimler]]: He invented the first high-speed petrol engine and the first four-wheel automobile, also the first gasoline-powered motorcycle.
*[[Adolf Dassler|Adolf "Adi" Dassler]]: Sport sneakers with and without spikes. ([[Addidas]]..)
*[[Rudolf Dassler]]: First sport shoes with screw-in shoe spikes, 1949. ([[Puma AG|Puma]]..)
*[[Johann Friedrich Dieffenbach]]: Pioneer of skin transplantation and cosmetic surgery.
*[[Rudolf Diesel]]: Inventor of the [[diesel engine]].
*[[Christian Doppler]]: Discovered the [[Doppler effect]].
*[[Walter Robert Dornberger]]: Co-inventor of the [[V2 rocket]].
*[[Karl Drais]]: Inventor of the bicycle and typewriter (1821) among other things.
*[[Peter Ferdinand Drucker]]: Invented the science of modern management.

==E==
[[Image:Einstein1921 by F Schmutzer 4.jpg|thumb|right|175px|[[Albert Einstein]] in [[List of Nobel Laureates in Physics|1921]], the year he was awarded the [[Nobel Prize in Physics]]]]
*[[Paul Ehrlich]]: Scientist in the fields of hematology, immunology, and chemotherapy, and Nobel laureate. Developed an effective treatment against syphilis.
*[[Albert Einstein]]: Godfather of Theoretical Physics, inventor and discoverer.
*[[Leonard Euler]]: Many discoveries in mathematics. "He is our teacher in all things."

==F==
*[[Daniel Gabriel Fahrenheit]]: Physicist and engineer who invented the [[alcohol thermometer]] (1709), the [[mercury thermometer]] (1714), and [[Fahrenheit|a temperature scale now named after him]].
*[[Hermann Emil Fischer]]: Discoveries in chemistry.
*[[Joseph von Fraunhofer]]: Discovery of the dark absorption lines known as Fraunhofer lines in the Sun's spectrum, and for making excellent optical glass and achromatic telescope objectives.
*[[Nikolaus Friedreich]]: Discovery of Friedreich-Auerbach disease (together with Leopold Auerbach) among other things.
[[File:Johannes Gutenberg.jpg|right|thumb|175px|[[Johannes Gutenberg]] in a 16th century copper engraving]]

==G==
*[[Brothers Grimm]]: Academic pioneers of [[linguistics]] and storytelling; [[Grimm's law]].
*[[Carl Friedrich Gauss|Johann Carl Friedrich Gauß]]: Inventor of the first electromagnetic telegraph; discoverer of [[list of topics named after Carl Friedrich Gauss|many things in mathematics and physics]], known as "the Prince of Mathematicians".
*[[Walter Gerlach]]: Physicist who co-discovered spin quantization in a magnetic field, the [[Stern-Gerlach effect]].
*[[Heinrich Göbel]]: Inventor of the first light bulb in 1854.
*[[Kurt Gödel]]: Important discoveries in math and logic, such as the [[incompleteness theorem]]s
*[[Peter Carl Goldmark]]: Engineer who was instrumental in developing the long-playing (LP) microgroove 33-1/3 rpm vinyl phonograph disc.
*[[Heinz Guderian]]: The father of modern mechanized warfare.
*[[Otto von Guericke]]: Discovered vacuum pump.
*[[Johannes Gutenberg]]: Inventor of the technology of printing with movable type.

==H==
[[Image:Bundesarchiv Bild 183-46019-0001, Otto Hahn.jpg|right|thumb|175px|[[Otto Hahn]] the first man to split the atom.]]
*[[Fritz Haber]]:German chemist and Nobel laureate who pioneered synthetic ammonia and chemical warfare.
*[[Otto Hahn]]: German chemist and Nobel laureate who pioneered the fields of radioactivity and radiochemistry. "The father of nuclear chemistry" and the "founder of the atomic age". Discovered nuclear fission.
*[[Henry J. Heinz]]: Tomato ketchup and fifty six other things.
*[[Werner Heisenberg]]: Theoretical physicist who made foundational contributions to quantum mechanics. Discovered a particle's position and velocity cannot be known at the same time. Discovered atomic nuclei are made of protons and neutrons.
*[[Rudolf Hell]]: Inventor of the first fax machine ([[Hellschreiber]]).
*[[Hellmann's and Best Foods|Richard Hellmann]]: Hellmann's (Blue Ribbon) Mayonnaise, 1905.
[[Image:Felix Hoffman.jpg|thumb|right|175px|[[Felix Hoffmann]]]]
*[[Hermann von Helmholtz]]: Discovered the principal of the conservation of energy.
*[[Peter Henlein]]: Inventor of the watch.
*[[Friedrich Wilhelm Herschel]]: Discovered infrared radiation among other things.
*[[Heinrich Hertz]]: Physicist, Discoverer of electromagnetic/radio waves.
*[[Victor Francis Hess]]: Discovered [[Cosmic rays]]. Also won the [[Nobel Prize]].
*[[David Hilbert]]: influental Mathematician, discovered and developed a broad range of fundamental ideas in math.
*[[Albert Hoffman]]: Discovered the chemical properties of [[chitin]] and [[lysergic acid diethylamide]].
*[[Felix Hoffmann]]: Aspirin (Bayer), 1897.
*[[Gottlob Honold]]: Inventor of the spark plug and the modern internal combustion engine, as well as headlights.
*[[Christian Huelsmeyer]]: Inventor of Radar.
*[[Alexander von Humboldt]]: Naturalist and explorer. His quantitative work on botanical geography was foundational to the field of biogeography.
*[[Wilhelm von Humboldt]]: Originator of the [[linguistic relativity]] hypothesis.

==I==
*[[Otmar Issing]]: Economist who invented the "two pillar" decision algorithm now used by the [[European Central Bank|ECB]].

==J==
*[[Hugo Junkers]]: Pioneer of all-metal aircraft construction with the [[Junkers J 1]] (1915–16).

==K==
[[File:Robert Koch berlin.jpg|thumb|175px|right|Monument to Robert Koch on his name square in [[Berlin]]]]
*[[Johannes Kepler]]: Discovered the laws of planetary motion.
*[[Erhard Kietz]]: Pioneer discoverer of video technology.
*[[Gustav Kirchoff]]: Discovery of the principles upon which [[spectrum analysis]] is founded.
*[[Martin Heinrich Klaproth]]: Discovered the element [[Uranium]].
*[[Robert Koch]]: Physician, discoverer, inventor and Nobel Prize winner. He became famous for isolating Bacillus anthracis (1877), the Tuberculosis bacillus (1882) and the Vibrio cholera (1883) and for his development of Koch's postulates.
*[[Arthur Korn]]: Inventor involved in development of the fax machine, specifically the transmission of photographs or telephotography, known as the Bildetelegraph.
*[[Julius H. Kroehl]]: Inventor and engineer, who built the first functioning submarine of the world.
*[[Alfred Krupp]]: Pioneer in metal casting and metal working process and procedures.

==L==
*[[Eugen Langen]]: Entrepreneur, engineer and inventor, involved in the development of the petrol engine and the Wuppertal monorail.
*[[Max von Laue]]: Discoveries regarding the diffraction of X-rays in crystals.
*[[Gottfried Wilhelm Leibniz]]: Philosopher known for discovering the mathematical field of calculus and coherently laying down its basic operations in 1684.
*[[Justus von Liebig]]: Discoveries in chemistry.
*[[Otto Lilienthal]]: Father of Aviation and first successful aviator of humanity. Main discovery was the properties and shape of the [[wing]].

==M==
[[Image:Marx3.jpg|right|thumb|175px|[[Karl Marx]]]]
*[[Ernst Mach]]: Discovered many effects of high speed projectiles; the [[Mach number]] is dedicated to his memory.
*[[Georg Hans Madelung]]: Academic and aeronautical engineer; a participant in the development of the [[Junkers F.13]].
*[[Karl Marx]]: Political economist and philosopher, who discovered the mechanics of capitalism and whose ideas shape the world we now live in.
*[[Wilhelm Maybach]]: Together with Gottlieb Daimler the first gasoline-powered motorcycle, power-engined boat and later, 1902, the Mercedes car model.
*[[Ottomar von Mayenburg]]: Inventor of "Chlorodont", the first commercial brand of toothpaste.
*[[Lise Meitner]]: Nuclear physicist, who, together with [[Otto Hahn]], explained theoretically nuclear fission.
*[[Gregor Mendel]]: Discoveries in genetics.
*[[Johannes Peter Müller]]: Discoveries in physiology.


==N==
*[[Walther Nernst]]: Inventor of the [[Nernst lamp]] and Nobel laureate.
[[Image:Max Planck (Nobel 1918).jpg|right|175px|thumb|[[Max Planck]]]]
==O==
*[[Hermann Oberth]]: Pioneer of rocket science and discoverer of the [[Oberth effect]].
*[[Hans von Ohain|Hans Joachim Pabst von Ohain]]: The modern jet engine in 1933, patented in 1936.
*[[Nikolaus Otto|Nikolaus August Otto]]: Inventor of the first internal-combustion engine to efficiently burn fuel directly in a piston chamber.

==P==

*[[Max Planck]]: Physicist, Scientist. He is considered to be the founder of the quantum theory, and one of the most important physicists of the twentieth century.
*[[Robert Pohl|Robert Wichard Pohl]]: In 1938, together with Rudolf Hilsch, built first functioning solid-state amplifier using salt as the semiconductor.

==Q==

{{Empty section|date=July 2010}}
==R==
*[[Johann Philipp Reis]]: Inventor of the first phone transmitter in 1861, he also invented the term ''Telephone''.
*[[Ralf Reski]]: Moss [[Bioreactor]] (1998)
[[Image:Paul Julius Reuter 1869.jpg|thumb|right|175px|Paul Reuter aged 53 years (1869) by [[Rudolf Lehmann (artist)|Rudolf Lehmann]]]]
*[[Paul Reuter|Paul Julius Freiherr von Reuter]]: Communications pioneer.
*[[Wilhelm Conrad Röntgen]]: Physicist and discoverer of x-rays/Röntgen rays (8 November 1895), this earned him the first Nobel Prize in Physics in 1901.

==S==
*[[Hans Sauer]]: Inventor of miniature high power relays; 309 patents worldwide in relay conceptions.
*[[Heinrich Schliemann]]: Father of [[archaeology]], among other things he discovered Homeric [[Troy]].
*[[Bernhard Schmidt]]: Discovered major improvements to the telescope.
*[[Paul Schmidt (inventor)]]: Developed since 1928 his idea of a new drive, the "pulsating incineration", also used in the V1-Rocket (engine was called "Argus-Schmidtrohr"); [[pulsejet]] was a development by Schmidt.
*[[Theodor Schwann]]: Discovery of properties of cells in animals.
*[[Johann Lukas Schönlein]]: Professor of medicine, he discovered among other things the parasitic cause of ringworm or favus (Achorion Schönleinii).
*[[Marx Schwab]]: Silversmith, invented coining with the screw press around 1550.
*[[Werner Sell]]: (Georg Robert Werner Sell) invented airplane kitchens, fitted kitchens and the prefabricated house among other things.
*[[Ernst Werner von Siemens]]: Dynamo, pointer telegraph that used a needle to point to the right letter, first electric elevator, trolleybus.
*[[Otto Stern]]: Nobel laureate; contributed to the discovery of spin quantization in the [[Stern-Gerlach experiment]] with [[Walther Gerlach]] in 1922.
*[[Eduard Suess]]: Discoveries in geology.

==T==
*[[Louis Tuchscherer]]: Inventor and mechanical engineer.

==U==
*[[Dietrich Uhlhorn|Dietrich "Diedrich" Uhlhorn]]: Engineer, mechanic and inventor, who invented the first mechanical tachometer (1817), between 1817 and 1830 inventor of the Presse Monétaire (level coin press known as Uhlhorn Press) which bears his name.

[[Image:Anna Berthe Roentgen.gif|thumb|''Hand mit Ringen'': print of Wilhelm Röntgen's first "medical" x-ray, of his wife's hand, taken on 22 December 1895 and presented to [[Ludwig Zehnder|Professor Ludwig Zehnder]] of the Physik Institut, University of Freiburg, on 1 January 1896]]

==V==
*[[Abraham Vater]]: Ampulla of Vater.
*[[Tri-Ergon|Hans Vogt]]: Invented sound-on-film (idea 1905) together with Jo Engl and Joseph Massolle, first sound-on-film for the public on 17 September 1922 in Filmtheater ''Alhambra '', Berlin, Germany.
*[[Woldemar Voigt]]: Physicist, who taught at the Georg August University of Göttingen. He worked on crystal physics, thermodynamics and electro-optics. He discovered the Voigt effect in 1898.
*[[Woldemar Voigt (aerospace engineer)]]: Chief designer at [[Messerschmitt|Messerschmitt's]] Oberammergau offices.

==W==
*[[Felix Wankel]]: Inventor of the Rotary Motor.
*[[Max Weber]]: Discovered the mass effects of capitalism and modernity.
*[[Wilhelm Eduard Weber]]: Inventor of the first electromagnetic telegraph together with Carl Friedrich Gauss.
*[[Gustav Weißkopf]]: Aviation pioneer - Worlds First Motorized Flight: August 14, 1901.
*[[Clemens Alexander Winkler]]: Chemist who discovered the element germanium in 1886.
*[[Friedrich Wöhler]]: The first to synthesis [[urea]].

==X==

{{Empty section|date=July 2010}}

==Y==

{{Empty section|date=July 2010}}

[[File:Zuse Z1-2.jpg|thumb|right|Zuse Z1 replica in the [[German Museum of Technology (Berlin)|German Museum of Technology]] in Berlin]]

==Z==
* [[Carl Zeiss]]: Pioneered glass casting and allied procedures and processes for high quality optics.
* [[Ferdinand Graf von Zeppelin]] (1838–1917): Inventor of the airship named after him. Start of the airship LZ1 in 1900.
*[[Karl Zimmer]]: Discovered the effects of [[ionizing radiation]] on [[DNA]].
*[[Konrad Zuse]]: Inventor of the first computing machine/computer, and the first [[high-level programming language]].

== Notes ==
'''When adding an inventor or a discoverer to the main section, please check first to see if it is already in the list .''' If it isn't, you might also check to see if the article exists (by entering the title in the Search box and pressing Go), as some editors may have forgotten to add their articles on German inventors to this list. '''When you add an inventor or a discoverer to this list, please add it in proper alphabetical order within the appropriate section.'''

Please include the year of invention or discovery and list key informations of the invention, as well as the references for it.

After an inventor article has been created, the link on this page will be blue. Please move these titles into the main (existing article) section after creating the show article.

Please be sure that the inventor or discoverer is German or of German heritage. For example, [[Wolfgang Amadeus Mozart]] is often classified as Austrian, but lived in an Era, where Austria was part of the [[Holy Roman Empire of the German Nation]], his nationality is therefore German.

==See also==
* [[German inventions and discoveries]]
* [[List of German Americans]]

==External links==
* [http://germanoriginality.com/madein/inventions.php "Made in Germany"]
* [http://german.about.com/library/blerfindung.htm German Inventions - Discoveries]
* [http://www.zoomwhales.com/inventors/germany.shtml A Sampling of German Inventors and Inventions]

{{Inventions}}

[[Category:German inventions]]
[[Category:History of Germany]]
[[Category:Lists of inventors|German]]

[[ar:ملحق:قائمة مخترعين ومكتشفين ألمان]]

0,999…

2010-10-13T23:50:45Z

Paul August: Reverted edits by 174.20.87.25 (talk) to last version by DASHBotAV

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', '''0.9̇''' or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
\end{align}
</math>

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p. 170</ref>

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

===Dedekind cuts===
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form
:<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p. 399</ref>
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
:<math>\begin{align}\tfrac{a}{b}<1\end{align},</math>
which implies
:<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp. 388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

:<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1},</math>

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111...2 follows directly from Berlekamp's Rule.</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + ''x'' = 1"
has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp. 9–11)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |url=http://jstor.org/stable/2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |url=http://jstor.org/stable/2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |url=http://jstor.org/stable/2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |url=http://jstor.org/stable/2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |url=http://jstor.org/stable/2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |url=http://jstor.org/stable/2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |url=http://jstor.org/stable/2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |url=http://jstor.org/stable/2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0665548087}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Almops

2010-08-19T19:32:07Z

Paul August: sp

{{orphan|date=March 2010}}

'''Almops''' ([[Ancient Greek|Gr.]] '''{{polytonic|Ἄλμωψς}}''') was in [[Greek mythology]] a [[Giants (Greek mythology)|giant]], and son of the god [[Poseidon]] and the half-nymph [[Helle (mythology)|Helle]].<ref name="DGRBM">{{cite encyclopedia | last = Schmitz | first = Leonhard | authorlink = Leonhard Schmitz | title = Almops | editor = [[William Smith (lexicographer)|William Smith]] | encyclopedia = [[Dictionary of Greek and Roman Biography and Mythology]] | volume = 1 | pages = 132 | publisher = [[Little, Brown and Company]] | location = Boston | year = 1867 | url = http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=moa;cc=moa;idno=acl3129.0001.001;q1=demosthenes;size=l;frm=frameset;seq=147}}</ref> He was the brother of [[Paeon (son of Poseidon)|Paeon]] (called "Edonus" in some accounts).<ref>{{cite book | last = Bell | first = Robert E. | authorlink = | coauthors = | title = Women of Classical Mythology | publisher = [[ABC-CLIO]] | year = 1991 | location = | pages = 230 | url = http://www.google.com/books?id=1KIYAAAAIAAJ | isbn = 0-8743-6581-3}}</ref> With the others of his kind, the [[Gigantes]], he [[Gigantomachy|waged war]] on [[Zeus]] and the gods of Olympus.

It is from Almops that the now-obsolete name for the region of [[Almopia]] and its inhabitants, the Almopes, in [[Macedonia (Greece)|Macedonia]], [[Greece]], were believed to have derived their name.<ref>[[Stephanus of Byzantium]], ''s.v.'' {{polytonic|Ἀλμωπία}}</ref>

==References==
{{reflist}}

{{SmithDGRBM}}

[[Category:Offspring of Poseidon]]
[[Category:Greek mythology]]
[[Category:Greek mythological giants]]
[[Category:Mythology of Macedonia (region)]]

[[el:Άλμωψ]]

Almops

2010-08-19T19:27:04Z

Paul August: dab link

{{orphan|date=March 2010}}

'''Almops''' ([[Ancient Greek|Gr.]] '''{{polytonic|Ἄλμωψς}}''') was in [[Greek mythology]] a [[Giants (Greek mythology)|giant]], and son of the god [[Poseidon]] and the half-nymph [[Helle (mythology)|Helle]].<ref name="DGRBM">{{cite encyclopedia | last = Schmitz | first = Leonhard | authorlink = Leonhard Schmitz | title = Almops | editor = [[William Smith (lexicographer)|William Smith]] | encyclopedia = [[Dictionary of Greek and Roman Biography and Mythology]] | volume = 1 | pages = 132 | publisher = [[Little, Brown and Company]] | location = Boston | year = 1867 | url = http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=moa;cc=moa;idno=acl3129.0001.001;q1=demosthenes;size=l;frm=frameset;seq=147}}</ref> He was the brother of [[Paeon (son of Posidon)|Paeon]] (called "Edonus" in some accounts).<ref>{{cite book | last = Bell | first = Robert E. | authorlink = | coauthors = | title = Women of Classical Mythology | publisher = [[ABC-CLIO]] | year = 1991 | location = | pages = 230 | url = http://www.google.com/books?id=1KIYAAAAIAAJ | isbn = 0-8743-6581-3}}</ref> With the others of his kind, the [[Gigantes]], he [[Gigantomachy|waged war]] on [[Zeus]] and the gods of Olympus.

It is from Almops that the now-obsolete name for the region of [[Almopia]] and its inhabitants, the Almopes, in [[Macedonia (Greece)|Macedonia]], [[Greece]], were believed to have derived their name.<ref>[[Stephanus of Byzantium]], ''s.v.'' {{polytonic|Ἀλμωπία}}</ref>

==References==
{{reflist}}

{{SmithDGRBM}}

[[Category:Offspring of Poseidon]]
[[Category:Greek mythology]]
[[Category:Greek mythological giants]]
[[Category:Mythology of Macedonia (region)]]

[[el:Άλμωψ]]

Almops

2010-08-19T11:58:35Z

Paul August: better link

{{orphan|date=March 2010}}

'''Almops''' ([[Ancient Greek|Gr.]] '''{{polytonic|Ἄλμωψς}}''') was in [[Greek mythology]] a [[Giants (Greek mythology)|giant]], and son of the god [[Poseidon]] and the half-nymph [[Helle (mythology)|Helle]].<ref name="DGRBM">{{cite encyclopedia | last = Schmitz | first = Leonhard | authorlink = Leonhard Schmitz | title = Almops | editor = [[William Smith (lexicographer)|William Smith]] | encyclopedia = [[Dictionary of Greek and Roman Biography and Mythology]] | volume = 1 | pages = 132 | publisher = [[Little, Brown and Company]] | location = Boston | year = 1867 | url = http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=moa;cc=moa;idno=acl3129.0001.001;q1=demosthenes;size=l;frm=frameset;seq=147}}</ref> He was the brother of [[Paean (disambiguation)|Paeon]] (called "Edonus" in some accounts).<ref>{{cite book | last = Bell | first = Robert E. | authorlink = | coauthors = | title = Women of Classical Mythology | publisher = [[ABC-CLIO]] | year = 1991 | location = | pages = 230 | url = http://www.google.com/books?id=1KIYAAAAIAAJ | isbn = 0-8743-6581-3}}</ref> With the others of his kind, the [[Gigantes]], he [[Gigantomachy|waged war]] on [[Zeus]] and the gods of Olympus.

It is from Almops that the now-obsolete name for the region of [[Almopia]] and its inhabitants, the Almopes, in [[Macedonia (Greece)|Macedonia]], [[Greece]], were believed to have derived their name.<ref>[[Stephanus of Byzantium]], ''s.v.'' {{polytonic|Ἀλμωπία}}</ref>

==References==
{{reflist}}

{{SmithDGRBM}}

[[Category:Offspring of Poseidon]]
[[Category:Greek mythology]]
[[Category:Greek mythological giants]]
[[Category:Mythology of Macedonia (region)]]

[[el:Άλμωψ]]

0,999…

2010-06-30T01:16:36Z

Paul August: Reverted edits by 79.173.233.184 (talk) to last version by The Nut

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999…'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999…. The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999…=1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students. Some reject it due to their intuitions that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999… eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111… equals 0.999… and 9 × {{frac|1|9}} equals 1, so 0.999… = 1:

:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
\end{align}
</math>

Another form of this proof multiplies {{frac|1|3}} = 0.3… by 3.

===Digit manipulation{{anchor|Digit manipulation}}===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number. To see this, consider that in subtracting 0.999… from 9.999…, each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999… = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999… = 1 but 0.333… < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p.186</ref> William Byers argues that a student who agrees that 0.999… = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp.39-41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman pp.396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

===Dedekind cuts===
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typical the definitions allow <math>\{\ x:x<1\}</math> to be a cut but not <math>\{x: x \le 1\}</math> (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the reverse [[factorial number system]] (using bases 2,3,4,… for positions ''after'' the decimal point), 1 = 1.000… = 0.1234….

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999…/1.000… pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow '9' in each position while forbidding an infinite succession of '9's). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1….So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each number a sequence of digits,
:0.d1d2d3…;…d∞−1d∞d∞+1…,
indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333…;…000…" and "0.999…;…000…" do not correspond to any number.

At the same time, the hyperreal number ''u''''H''=0.999…;…999000…, with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1=0.9,}} {{nowrap|1=''u''2=0.99,}} {{nowrap|1=''u''3=0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999…":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, …)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999….<ref>Stewart 2009, p.175; the full discussion of 0.999… is spread through pp.172-175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999… < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999… less than 1? |journal=[[The Montana Mathematics Enthusiast]] |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (p 27-31) as infinite decimals with 0.999…=1 as part of the definition.
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
*{{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
*{{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
*{{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999… and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
*{{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
*{{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*:This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999…}} falling short of 1 by an infinitesimal {{nowrap|0.000…1.}}
*{{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
*{{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
*{{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
*{{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1-3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
*{{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
*{{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
*{{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
*{{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
*{{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999…] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999… ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Kronos (Band)

2010-04-03T17:42:48Z

Paul August: ce

{{Infobox musical artist 
| Name = Kronos
| Img =
| Img_capt =
| Img_size =
| Landscape =
| Background = group_or_band
| Alias =
| Origin = [[France]]
| Genre = [[Death metal]]
| Years_active = since 1994
| Label = [[Xtreem Music]]
| Associated_acts =
| URL = [http://kronosbrutaldeath.free.fr/ Kronos' Website]
| Current_members = Grams Tom Richard Mike
| Past_members =Nicolas Temmar Marrot Jeremy Kristof
}}

'''Kronos''' is a [[death metal]] band from France. Their name is a reference to the titan [[Cronus]], the father of [[Zeus]] in [[greek mythology]].

==Lyrical subjects==
The lyrics of their songs are mainly about myths, legends and historical subjects from the ancient [[civilization]]s of [[Ancient Rome|Rome]], [[Ancient Greece|Greece]] and [[Ancient Egypt|Egypt]] or from [[Scandinavia]] or even from old kingdoms belonging to [[barbarians]].

== History ==
Kronos was formed in 1994 in [[Thaon-les-Vosges]] ([[France]]) by Grams (14 years old), Marot (17 years old), Jeremy (17 years old) and Mike (13 years old). The band played at this point some Heavy-thrash-metal. Two years later Tems (19 years old) replaced Marot (guitar) and Kristof (18 years old) joins the band for the vocals. Kronos started then to play Death Metal music and recorded its first demo called ''Outrance''.
In 1999, Tom (20 years old) replaces Jéremy and the band is now definitely oriented towards '[[Brutal Death Metal]]'. The following year the recording of a new demo ''Split Promo 2000'' containing 4 titles is finalized.

The long awaited first album ''[[Titan's Awakening]]'' was released in 2001. This first edition of this album was produced by the band itself. The band has then been acclaimed by critics and got its first contract with the label [[Warpath Records]] (former ''Shockwave''). A new edition of ''[[Titan's Awakening]]'' is released at the end of October the same year with a new cover designed by Deather ([[Angel Corpse]], [[Gurkkhas]], [[Vital Remains]]...) as well as a new graphical booklet.
Richard (26 years old) replaces Tems and plays the lead guitar from 2003. The band signs a contract for two upcoming albums with the new Spanish label [[Xtreem music]] managed by [[Dave Rotten]] from the band [[Avulsed]].
The second album was released in 2004 and is named ''[[Colossal Titan Strife]]''. The band toured Europe in 2005 promoting the album.
The third album was released in April 2007 and is named ''[[The Hellenic Terror]]''.

== Members ==
=== Current members ===
* ''Grams'' (guitar)
* ''Richard'' (guitar)
* ''Tom'' (bass guitar and backup vocals)
* ''Mike'' (drums)

=== Previous members ===
* ''Tems'' (1996-2003) (guitar)
* ''Jérémy'' (1994-1999) (bass guitar)
* ''Marot'' (1994-1996) (guitar)
* ''Kristof'' (1994-2009) (vocals)

==Discography==
===Studio releases===
* ''[[Titan's Awakening]]'' (2001)
* ''[[Colossal Titan Strife]]'' (2004)
* ''[[The Hellenic Terror]]'' (2007)

=== Demos ===
* ''[[Outrance]]'' (tape) (1994)
* ''[[Split Promo 2000]]'' with [[None Divine]] (2000)

== External links ==
* [http://kronosbrutaldeath.free.fr/ Kronos official homepage]
* [http://www.myspace.com/kronostitan myspace page]
* [http://www.xtreemmusic.com/ Xtreem music]
* {{musicbrainz artist|id=4b0af58d-b181-4ee9-b986-85631c382c02|name=Kronos}}
* [http://www.metal-archives.com/band.php?id=2825 Kronos] at [[Encyclopaedia Metallum]]
* [http://halfmanhalfmachine.net/Mike_K_main.html Kronos's drummer on halfmanhalfmachine]
{{commonscat|Kronos (Band)|Kronos}}

{{DEFAULTSORT:Kronos}}
[[Category:French death metal musical groups]]
[[Category:French heavy metal musical groups]]
[[Category:Musical groups established in 1994]]

[[fr:Kronos (groupe)]]

Benutzer:JonskiC/Kleinste-Quadrate-Schätzung

2010-03-14T04:10:48Z

Paul August: /* Least squares, regression analysis and statistics */ fix link

The method of '''least squares''' is a standard approach to the approximate solution of [[overdetermined system]]s, i.e. sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every single equation.

The most important application is in [[curve fitting|data fitting]]. The best fit in the least-squares sense minimizes the sum of squared [[errors and residuals in statistics|residuals]], a residual being the difference between an observed value and the value provided by a model.

Least squares problems fall into two categories, [[linear least squares]] and [[nonlinear least squares]], depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical [[regression analysis]]; it has a closed form solution. The non-linear problem has no closed solution and is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, thus the core calculation is similar in both cases.

The least-squares method was first described by [[Carl Friedrich Gauss]] around 1794.<ref name=brertscher>{{cite book|author = Bretscher, Otto|title = Linear Algebra With Applications, 3rd ed.|publisher = Prentice Hall|year = 1995|location = Upper Saddle River NJ}}</ref> Least squares corresponds to the [[maximum likelihood]] criterion if the experimental errors have a [[normal distribution]] and can also be derived as a [[method of moments (statistics)|method of moments]] estimator.

The following discussion is mostly presented in terms of [[linear]] functions but the use of least-squares is valid and practical for more general families of functions. For example, the [[Fourier series]] approximation of degree ''n'' is optimal in the least-squares sense, amongst all approximations in terms of [[trigonometric polynomial|trigonometric polynomials]] of degree ''n''. Also, by iteratively applying local quadratic approximation to the likelihood (through the [[Fisher information]]), the least-squares method may be used to fit a [[generalized linear model]].

[[Image:Linear least squares2.png|right|thumb|The result of fitting a set of data points with a quadratic function.]]

==History==
===Context===
The method of least squares grew out of the fields of [[astronomy]] and [[geodesy]] as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the [[Age of Exploration]]. The accurate description of the behavior of celestial bodies was key to enabling ships to sail in open seas where before sailors had relied on land sightings to determine the positions of their ships.

The method was the culmination of several advances that took place during the course of the eighteenth century<ref name=stigler>{{cite book
| author = Stigler, Stephen M.
| title = The History of Statistics: The Measurement of Uncertainty Before 1900
| publisher = Belknap Press of Harvard University Press
| year = 1986
| location = Cambridge, MA
}}</ref>:

*The combination of different observations taken under the ''same'' conditions contrary to simply trying one's best to observe and record a single observation accurately. This approach was notably used by [[Tobias Mayer]] while studying the [[libration]]s of the moon.
*The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, perhaps first expressed by [[Roger Cotes]].
*The combination of different observations taken under ''different'' conditions as notably performed by [[Roger Joseph Boscovich]] in his work on the shape of the earth and [[Pierre-Simon Laplace]] in his work in explaining the differences in motion of [[Jupiter]] and [[Saturn]].
*The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved, developed by Laplace in his Method of Situation.

===The method itself===
[[Image:Stamp Carl Friedrich Gauß.jpg|thumb|240px|[[Carl Friedrich Gauss]]]]
[[Carl Friedrich Gauss]] is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen. [[Adrien-Marie Legendre|Legendre]] was the first to publish the method, however.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid [[Ceres (asteroid)|Ceres]]. On January 1, 1801, the Italian astronomer [[Giuseppe Piazzi]] discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, it was desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated [[Kepler's laws of planetary motion|Kepler's nonlinear equations]] of planetary motion. The only predictions that successfully allowed Hungarian astronomer [[Franz Xaver von Zach]] to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium''.
In 1829, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. This result is known as the [[Gauss–Markov theorem]].

The idea of least-squares analysis was also independently formulated by the Frenchman [[Adrien-Marie Legendre]] in 1805 and the American [[Robert Adrain]] in 1808.

==Problem statement==
The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of ''n'' points (data pairs) <math>(x_i,y_i)\!</math>, ''i'' = 1, ..., ''n'', where <math>x_i\!</math> is an [[independent variable]] and <math>y_i\!</math> is a [[dependent variable]] whose value is found by observation. The model function has the form <math>f(x,\boldsymbol \beta)</math>, where the ''m'' adjustable parameters are held in the vector <math>\boldsymbol \beta</math>. The parameter values for which the model "best" fits the data need be found. The least squares method finds its optimum when the sum, ''S'', of squared residuals
:<math>S=\sum_{i=1}^{n}{r_i}^2</math>
is a minimum. A [[errors and residuals in statistics|residual]] is defined as the difference between the value of the dependent variable and the model value

:<math>r_i= y_i - f(x_i, \beta).</math>

An example of a model is that of the straight line. Denoting the intercept as <math>\beta_0</math> and the slope as <math>\beta_1</math>, the model function is given by <math>f(x,\boldsymbol \beta)=\beta_0+\beta_1 x</math>. See [[linear least squares#Motivational_example|linear least squares]] for a fully worked out example of this model.

A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, ''x'' and ''z'', say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point. 

==Solving the least squares problem==

The [[Maxima and minima|minimum]] of the sum of squares is found by setting the [[gradient]] to zero. Since the model contains ''m'' parameters there are ''m'' gradient equations.

:<math>\frac{\partial S}{\partial \beta_j}=2\sum_i r_i\frac{\partial r_i}{\partial \beta_j}=0,\ j=1,\ldots,m</math>

and since <math>r_i=y_i-f(x_i,\boldsymbol \beta)\,</math> the gradient equations become

:<math>-2\sum_i \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} r_i=0,\ j=1,\ldots,m</math>

The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.

=== Linear least squares ===
{{main|Linear least squares}}
A regression model is a linear one when the model comprises a [[linear combination]] of the parameters, i.e.

:<math> f(x_i, \beta) = \sum_{j = 1}^{m} \beta_j \phi_j(x_{i})</math>

where the coefficients, <math>\phi_{j}</math>, are functions of <math> x_{i} </math>.

Letting

:<math> X_{ij}= \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j}= \phi_j(x_{i}) . \, </math>

we can then see that in that case the least square estimate (or estimator, in the context of a random sample), <math> \boldsymbol \beta</math> is given by

:<math> \boldsymbol{\hat\beta} =( X ^TX)^{-1}X^{T}\boldsymbol y </math>

For a derivation of this estimate see [[Linear least squares]].

=== Non-linear least squares ===
{{main|Non-linear least squares}}
There is no closed-form solution to a non-linear least squares problem. Instead, numerical algorithms are used to find the value of the parameters <math>\beta</math> which minimize the objective. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation.
:<math>{\beta_j}^{k+1}={\beta_j}^k+\Delta \beta_j</math>
''k'' is an iteration number and the vector of increments, <math>\Delta \beta_j\,</math> is known as the shift vector. In some commonly used algorithms, at each iteration the model may be linearized by approximation to a first-order [[Taylor series]] expansion about <math> \boldsymbol \beta^k\!</math>

:<math>
\begin{align}
f(x_i,\boldsymbol \beta) & = f^k(x_i,\boldsymbol \beta) +\sum_j \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} \left(\beta_j-{\beta_j}^k \right) \\
& = f^k(x_i,\boldsymbol \beta) +\sum_j J_{ij} \Delta\beta_j.
\end{align}
</math>

The [[Jacobian matrix and determinant|Jacobian]], '''J''', is a function of constants, the independent variable ''and'' the parameters, so it changes from one iteration to the next. The residuals are given by

:<math>r_i=y_i- f^k(x_i,\boldsymbol \beta)- \sum_{j=1}^{m} J_{ij}\Delta\beta_j=\Delta y_i- \sum_{j=1}^{m} J_{ij}\Delta\beta_j</math>.

To minimize the sum of squares of <math>r_i</math>, the gradient equation is set to zero and solved for <math> \Delta \beta_j\!</math>

:<math>-2\sum_{i=1}^{n}J_{ij} \left( \Delta y_i-\sum_{j=1}^{m} J_{ij}\Delta \beta_j \right)=0</math>

which, on rearrangement, become ''m'' simultaneous linear equations, the '''normal equations'''.

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} J_{ij}J_{ik}\Delta \beta_k=\sum_{i=1}^{n} J_{ij}\Delta y_i \qquad (j=1,\ldots,m)\,</math>

The normal equations are written in matrix notation as

:<math>\mathbf{\left(J^TJ\right)\Delta \boldsymbol \beta=J^T\Delta y}.\,</math>


These are the defining equations of the [[Gauss–Newton algorithm]].

=== Differences between linear and non-linear least squares ===
* The model function, ''f'', in LLSQ (linear least squares) is a linear combination of parameters of the form <math>f = X_{i1}\beta_1 + X_{i2}\beta_2 +\cdots</math> The model may represent a straight line, a parabola or any other polynomial-type function. In NLLSQ (non-linear least squares) the parameters appear as functions, such as <math>\beta^2, e^{\beta x}</math> and so forth. If the derivatives <math>\partial f /\partial \beta_j</math> are either constant or depend only on the values of the independent variable, the model is linear in the parameters. Otherwise the model is non-linear.
*Many solution algorithms for NLLSQ require initial values for the parameters, LLSQ does not.
*Many solution algorithms for NLLSQ require that the Jacobian be calculated. Analytical expressions for the partial derivatives can be complicated. If analytical expressions are impossible to obtain the partial derivatives must be calculated by numerical approximation.
*In NLLSQ non-convergence (failure of the algorithm to find a minimum) is a common phenomenon whereas the LLSQ is globally concave so non-convergence is not an issue.
*NLLSQ is usually an iterative process. The iterative process has to be terminated when a convergence criterion is satisfied. LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the [[Gauss–Seidel]] method.
*In LLSQ the solution is unique, but in NLLSQ there may be multiple minima in the sum of squares.
*Under the condition that the errors are uncorrelated with the predictor variables, LLSQ yields unbiased estimates, but even under that condition NLLSQ estimates are generally biased.
These differences must be considered whenever the solution to a non-linear least squares problem is being sought.

==Least squares, regression analysis and statistics==
The methods of least squares and [[regression analysis]] are conceptually different. However, the method of least squares is often used to generate estimators and other statistics in regression analysis.

Consider a simple example drawn from physics. A spring should obey [[Hooke's law]] which states that the extension of a spring is proportional to the force, ''F'', applied to it.
:<math>f(F_i,k)=kF_i\!</math>
constitutes the model, where ''F'' is the independent variable. To estimate the [[force constant]], ''k'', a series of ''n'' measurements with different forces will produce a set of data, <math>(F_i, y_i), i=1,n\!</math>, where ''yi'' is a measured spring extension. Each experimental observation will contain some error. If we denote this error <math>\varepsilon</math>, we may specify an empirical model for our observations,

: <math> y_i = kF_i + \varepsilon_i. \, </math>

There are many methods we might use to estimate the unknown parameter ''k''. Noting that the ''n'' equations in the ''m'' variables in our data comprise an [[overdetermined system]] with one unknown and ''n'' equations, we may choose to estimate ''k'' using least squares. The sum of squares to be minimized is

:<math> S = \sum_{i=1}^{n} \left(y_i - kF_i\right)^2. </math>

The least squares estimate of the force constant, ''k'', is given by

:<math>\hat k=\frac{\sum_i F_i y_i}{\sum_i {F_i}^2}.</math>

Here it is assumed that application of the force '''''causes''''' the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.

In regression analysis the researcher specifies an empirical model. For example, a very common model is the straight line model which is used to test if there is a linear relationship between dependent and independent variable. If a linear relationship is found to exist, the variables are said to be [[correlated]]. However, correlation does not prove causation, as both variables may be correlated with other, hidden, variables, or the dependent variable may "reverse" cause the independent variables, or the variables may be otherwise spuriously correlated. For example, suppose there is a correlation between deaths by drowning and the volume of ice cream sales at a particular beach. Yet, both the number of people going swimming and the volume of ice cream sales increase as the weather gets hotter, and presumably the number of deaths by drowning is correlated with the number of people going swimming. Perhaps an increase in swimmers causes both the other variables to increase.

In order to make statistical tests on the results it is necessary to make assumptions about the nature of the experimental errors. A common (but not necessary) assumption is that the errors belong to a [[Normal distribution]]. The [[central limit theorem]] supports the idea that this is a good assumption in many cases.
* The [[Gauss–Markov theorem]]. In a linear model in which the errors have [[expectation]] zero conditional on the independent variables, are [[uncorrelated]] and have equal [[variance]]s, the best linear [[unbiased]] estimator of any linear combination of the observations, is its least-squares estimator. "Best" means that the least squares estimators of the parameters have minimum variance. The assumption of equal variance is valid when the errors all belong to the same distribution.
*In a linear model, if the errors belong to a [[Normal distribution]] the least squares estimators are also the [[maximum likelihood estimator]]s.

However, if the errors are not normally distributed, a [[central limit theorem]] often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large. For this reason, given the important property that the error is mean independent of the independent variables, the distribution of the error term is not an important issue in regression analysis. Specifically, it is not typically important whether the error term follows a normal distribution.

In a least squares calculation with unit weights, or in linear regression, the variance on the ''j''th parameter,
denoted <math>\text{var}(\hat{\beta}_j)</math>, is usually estimated with

:<math>\text{var}(\hat{\beta}_j)= \sigma^2\left( \left[X^TX\right]^{-1}\right)_{jj} \approx \frac{S}{n-m}\left( \left[X^TX\right]^{-1}\right)_{jj},</math>
where the true residual variance σ2 is replaced by an estimate based on the minimised value of the sum of squares objective function ''S''.

[[Confidence limits]] can be found if the [[probability distribution]] of the parameters is known, or an asymptotic approximation is made, or assumed. Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed. The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed. Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.

==Weighted least squares==
:''See also: [[Weighted mean]]''
The expressions given above are based on the implicit assumption that the errors are uncorrelated with each other and with the independent variables and have equal variance. The [[Gauss–Markov theorem]] shows that, when this is so, <math>\hat\boldsymbol\beta</math> is a [[best linear unbiased estimator]] (BLUE). If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted. [[Alexander Aitken|Aitken]] showed that when a weighted sum of squared residuals is minimized, <math>\hat\boldsymbol\beta</math> is BLUE if each weight is equal to the reciprocal of the variance of the measurement.
:<math> S = \sum_{i=1}^{n} W_{ii}{r_i}^2,\qquad W_{ii}=\frac{1}{{\sigma_i}^2} </math>
The gradient equations for this sum of squares are

:<math>-2\sum_i W_{ii}\frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} r_i=0,\qquad j=1,\ldots,n</math>

which, in a linear least squares system give the modified normal equations

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} X_{ij}W_{ii}X_{ik}\hat \beta_k=\sum_{i=1}^{n} X_{ij}W_{ii}y_i, \qquad j=1,\ldots,m\,</math>

or

:<math>\mathbf{\left(X^TWX\right)\hat \boldsymbol \beta=X^TWy}.</math>

When the observational errors are uncorrelated the weight matrix, '''W''', is diagonal. If the errors are correlated, the resulting estimator is BLUE if the weight matrix is equal to the inverse of the [[variance-covariance matrix]] of the observations.

When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as <math>w_{ii}=\sqrt W_{ii}</math>. The normal equations can then be written as

:<math>\mathbf{\left(X'^TX'\right)\hat \boldsymbol \beta=X'^Ty'}\,</math>

where

: <math>\mathbf{X'}=\mathbf{wX}, \mathbf{y'}=\mathbf{wy}.\,</math>

For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.

:<math>\mathbf{\left(J^TWJ\right)\boldsymbol \Delta\beta=J^TW \boldsymbol\Delta y}.\,</math>

Note that for empirical tests, the appropriate '''W''' is not known for sure and must be
estimated. For this [[Feasible Generalized Least Squares]] (FGLS) techniques may be used.

===Principal components===
The first principal component about the mean of a set of points is equivalent to the linear least squares solution. One of the most computationally efficient ways to solve a linear least squares problem is to use the [[Principal_component_analysis#Computing_principal_components_with_expectation_maximization|EM technique]] to compute the first principal component about the mean of the data. This algorithm can be trivially modified to compute a weighted least squares solution as well.

===LASSO method===
In some contexts a [[Regularization (machine learning)|regularized]] version of the least squares solution may be preferable. The ''LASSO'' algorithm, for example, finds a least-squares solution with the constraint that <math>|\beta|_1</math>, the [[L1-norm|L1-norm]] of the parameter vector, is no greater than a given value. Equivalently, it may solve an unconstrained minimization of the least-squares penalty with <math>\alpha|\beta|_1</math> added, where <math>\alpha</math> is a constant (this is the [[Lagrange multipliers|Lagrangian]] form of the constrained problem.) This problem may be solved using [[quadratic programming]] or more general [[convex optimization]] methods, as well as by specific algorithms such as the [[least angle regression]] algorithm. The L1-regularized formulation is useful in some contexts due to its tendency to prefer solutions with fewer nonzero parameter values, effectively reducing the number of variables upon which the given solution is dependent <ref> Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Royal. Statist. Soc B., Vol. 58, No. 1, pages 267–288</ref>. For this reason, the LASSO and its variants are fundamental to the field of [[compressed sensing]].

==See also==
* [[Ordinary least squares]], aka OLS
* [[L2 norm|''L''2 norm]]
* [[Least absolute deviation]]
* [[Measurement uncertainty]]
* [[Root mean square]]
* [[Squared deviations]]
* [[Iteratively re-weighted least squares]]
* [[Total least squares]], aka orthogonal regression
* [[Levenberg–Marquardt algorithm]]
* [[Regression analysis]]
* [[Partial least squares regression]]
* [[Best linear unbiased prediction]] (BLUP)

==Notes==
<references />

==References==
*Å. Björck, ''Numerical Methods for Least Squares Problems'', SIAM, 1996 [http://www.ec-securehost.com/SIAM/ot51.html].
*C.R. Rao, H. Toutenburg, A. Fieger, C. Heumann, T. Nittner and S. Scheid, ''Linear Models: Least Squares and Alternatives'', Springer Series in Statistics, 1999.
*T. Kariya, H. Kurata, ''Generalized Least Squares'', Wiley, 2004.
*J. Wolberg, ''Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments'', Springer, 2005.

==External links==
* [http://www.personal.psu.edu/faculty/j/h/jhm/f90/lectures/lsq2.html Derivation of quadratic least squares]
* [http://www2.uta.edu/infosys/baker/STATISTICS/Keller7/Keller%20PP%20slides-7/Chapter17.ppt Power Point Statistics Book] -- Excellent slides providing an introductory regression example (University of Texas at Arlington)

{{Least Squares and Regression Analysis}}

[[Category:Applied mathematics]]
[[Category:Mathematical optimization]]
[[Category:Statistical methods]]
[[Category:Regression analysis]]
[[Category:Single equation methods (econometrics)]]
[[Category:Mathematical and quantitative methods (economics)]]

[[af:Kleinste-kwadratemetode]]
[[ca:Mínims quadrats ordinaris]]
[[cs:Metoda nejmenších čtverců]]
[[de:Methode der kleinsten Quadrate]]
[[es:Mínimos cuadrados]]
[[eu:Karratu txikienen erregresio]]
[[fa:کمترین مربعات]]
[[fr:Méthode des moindres carrés]]
[[gl:Mínimos cadrados]]
[[ko:최소제곱법]]
[[hi:न्यूनतम वर्ग की विधि]]
[[it:Metodo dei minimi quadrati]]
[[he:שיטת הריבועים הפחותים]]
[[la:Methodus quadratorum minimorum]]
[[hu:Legkisebb négyzetek módszere]]
[[nl:Kleinste-kwadratenmethode]]
[[ja:最小二乗法]]
[[pl:Metoda najmniejszych kwadratów]]
[[pt:Método dos mínimos quadrados]]
[[ru:Метод наименьших квадратов]]
[[su:Kuadrat leutik]]
[[fi:Pienimmän neliösumman menetelmä]]
[[sv:Minstakvadratmetoden]]
[[tr:En küçük kareler yöntemi]]
[[uk:Метод найменших квадратів]]
[[ur:لکیری اقل مربعات]]
[[vi:Bình phương tối thiểu]]
[[zh:最小二乘法]]

Almops

2010-03-07T00:43:11Z

Paul August: fix link

'''Almops''' ([[Ancient Greek|Gr.]] {{polytonic|Ἄλμωψς}}) was in [[Greek mythology]] a [[Giant (mythology)|giant]], and son of the god [[Poseidon]] and the half-nymph [[Helle (mythology)|Helle]].<ref name="DGRBM">{{cite encyclopedia | last = Schmitz | first = Leonhard | authorlink = Leonhard Schmitz | title = Almops | editor = [[William Smith (lexicographer)|William Smith]] | encyclopedia = [[Dictionary of Greek and Roman Biography and Mythology]] | volume = 1 | pages = 132 | publisher = [[Little, Brown and Company]] | location = Boston | year = 1867 | url = http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=moa;cc=moa;idno=acl3129.0001.001;q1=demosthenes;size=l;frm=frameset;seq=147}}</ref> He was the brother of [[Paean (disambiguation)|Paeon]] (called "Edonus" in some accounts).<ref>{{cite book | last = Bell | first = Robert E. | authorlink = | coauthors = | title = Women of Classical Mythology | publisher = [[ABC-CLIO]] | date = 1991 | location = | pages = 230 | url = http://www.google.com/books?id=1KIYAAAAIAAJ | isbn = 0-8743-6581-3}}</ref> With the others of his kind, the [[Gigantes]], he [[Gigantomachy|waged war]] on [[Zeus]] and the gods of Olympus.

It is from Almops that the now-obsolete name for the region of [[Almopia]] and its inhabitants, the Almopes, in [[Macedonia (Greece)|Macedonia]], [[Greece]], were believed to have derived their name.<ref>[[Stephanus of Byzantium]], ''s.v.'' {{polytonic|Ἀλμωπία}}</ref>

==References==
{{reflist}}

{{SmithDGRBM}}

[[Category:Offspring of Poseidon]]
[[Category:Greek mythology]]
[[Category:Greek mythological giants]]
[[Category:Mythology of Macedonia (region)]]

[[el:Άλμωψ]]
hes also a faggot

Lateinische Übersetzungen im Hochmittelalter

2010-02-01T17:00:02Z

Paul August: /* Translators in Italy */ link

[[File:De Ludiciis Natiuitatum Albohali Nuremberg 1546.jpg|thumb|[[Albohali]]'s ''De Iudiciis Natiuitatum'' was translated into Latin by [[Plato of Tivoli]] in 1136, and again by [[John of Seville]] in 1153.<ref>[http://books.google.com/books?id=7CP7fYghBFQC&pg=PA875 Houtsma, p.875]</ref> Here is the [[Nuremberg]] edition of [[John of Seville]]'s translation, 1546.]]
The [[Renaissance of the 12th century]] saw a major search by [[Europe]]an scholars for new learning, which led them to the areas of Europe that once been under [[Muslim]] rule and still had substantial [[Arabic Language|Arabic]]-speaking populations, but that had recently been reconquered by [[Christians]]. This meant central [[Spain]] and [[Sicily]], both of which had come under Christian rule in the eleventh century. The combination of a substantial numbers of Arabic-speaking scholars and Christian rulers made these areas intellectually attractive yet culturally and politically accessible to [[Latin]] scholars. A typical story is that of [[Gerard of Cremona]] (c. 1114-87), who is said to have made his way to Toledo, well after its reconquest by Christians in 1085, because he<ref>C. Burnett, "Arabic-Latin Translation Program in Toledo", p. 255.</ref>
{{blockquote|arrived at a knowledge of each part of [philosophy] according to the study of the Latins, nevertheless, because of his love for the ''[[Almagest]]'', which he did not find at all amongst the Latins, he made his way to [[Toledo, Spain|Toledo]], where seeing an abundance of books in Arabic on every subject, and pitying the poverty he had experienced among the Latins concerning these subjects, out of his desire to translate he thoroughly learnt the Arabic language....}}

Unlike the interest in the literature and history of [[classical antiquity]] during the [[Renaissance]], 12th century translators sought new [[Islamic science|scientific]], [[Early Islamic philosophy|philosophical]] and, to a lesser extent, religious texts. The latter concern was reflected in a renewed interest in translations of the [[Greek language|Greek]] [[Church Fathers]] into [[Latin]], a concern with translating [[Jew]]ish teachings from [[Hebrew language|Hebrew]], and most significantly, an interest in the [[Qur'an]] and other [[Islam]]ic religious texts.<ref>M.-T. d'Alverny, "Translations and Translators," pp. 426-33</ref> In addition, some [[Arabic literature]] was also translated into Latin.<ref name=Irwin/>

==Translators in Italy==
Just before the burst of translations in the 12th century, [[Constantine the African]], a [[Christian]] from [[Carthage]] who studied medicine in [[History of Arab Egypt|Egypt]] and ultimately became a monk at the monastery of [[Monte Cassino]] in [[Italy]], translated [[Islamic medicine|medical works]] from Arabic. Constantine's many translations included [[Ali ibn Abbas al-Majusi]]'s medical encyclopedia ''[[Liber pantegni|The Complete Book of the Medical Art]]'' (as ''Liber pantegni''),<ref name=Bieber>Jerome B. Bieber. [http://inst.santafe.cc.fl.us/~jbieber/HS/trans2.htm Medieval Translation Table 2: Arabic Sources], [[Santa Fe Community College (Florida)|Santa Fe Community College]].</ref>
the ancient medicine of [[Hippocrates]] and [[Galen]] as adapted by [[Islamic medicine|Arabic physicians]],<ref>M.-T. d'Alverny, "Translations and Translators," pp. 422-6</ref>
and the ''Isagoge ad Tegni Galeni''<ref name=Danielle-981/> by [[Hunayn ibn Ishaq]] (Johannitius) and his nephew Hubaysh ibn al-Hasan.<ref>D. Campbell, ''Arabian Medicine and Its Influence on the Middle Ages'', p. 4-5.</ref>
Other medical works he translated include [[Isaac Israeli ben Solomon]]'s ''Liber febribus, Liber de dietis universalibus et particularibus'' and ''Liber de urinis''; Ishaq ibn Imran's [[Islamic psychology|psychological]] work ''al-Maqala fi al-Malikhukiya'' as ''De melancolia''; and [[Ibn Al-Jazzar]]'s ''[[De Gradibus]], Viaticum, Liber de stomacho, De elephantiasi, De coitu'' and ''De oblivione''.<ref name=Danielle-981>{{citation|last=Jacquart|first=Danielle|contribution=The Influence of Arabic Medicine in the Medieval West|page=981}} in {{Harv|Morelon|Rashed|1996|pp=963-84}}</ref>

[[Sicily]] had been part of the Byzantine Empire until 878, was under [[Emirate of Sicily|Muslim control]] from 878-1060, and came under Norman control between 1060 and 1090. As a consequence the Norman [[Kingdom of Sicily]] maintained a trilingual bureaucracy, which made it an ideal place for translations. Sicily also maintained relations with the [[Greek East]], which allowed for exchange of ideas and manuscripts.<ref>C. H. Haskins, ''Studies in Mediaeval Science,'' pp 155-7</ref>

[[File:Ibn Butlan Receuil de Sante Rhenanie 2nd half 15th century.jpg|thumb|[[Ibn Butlan]]'s ''[[Tacuinum sanitatis]]'', [[Rhineland]], 2nd half of 15th century.]]
A copy of [[Ptolemy]]'s ''[[Almagest]]'' was brought back to Sicily by [[Henry Aristippus]], as a gift from the Emperor to [[William I of Sicily|King William I]]. Aristippus, himself, translated [[Plato]]'s ''[[Meno]]'' and ''[[Phaedo]]'' into Latin, but it was left to an anonymous student at Salerno to travel to Sicily and translate the ''Almagest'', as well as several works by [[Euclid]] from Greek to Latin.<ref>M.-T. d'Alverny, "Translations and Translators," pp. 433-4</ref> Although the Sicilians generally translated directly from the Greek, when Greek texts were not available, they would translate from Arabic. [[Admiral Eugene of Sicily]] translated Ptolemy's ''[[Ptolemy#Other works|Optics]]'' into Latin, drawing on his knowledge of all three languages in the task.<ref>M.-T. d'Alverny, "Translations and Translators," p. 435</ref> Accursius of [[Province of Pistoia|Pistoja]]'s translations included the works of [[Galen]] and [[Hunayn ibn Ishaq]].<ref name=Campbell-3>D. Campbell, ''Arabian Medicine and Its Influence on the Middle Ages'', p. 3.</ref> Gerard de [[Sabbioneta|Sabloneta]] translated [[Avicenna]]'s ''[[The Canon of Medicine]]'' and [[al-Razi]]'s ''Almansor''. [[Fibonacci]] presented the first complete European account of the [[Hindu-Arabic numeral system]] from [[Arabic numerals|Arabic sources]] in his ''[[Liber Abaci]]'' (1202).<ref name=Bieber/> The ''Aphorismi'' by [[Masawaiyh]] (Mesue) was translated by an anonymous translator in late 11th or early 12th century Italy.<ref name=Danielle>{{citation|last=Jacquart|first=Danielle|contribution=The Influence of Arabic Medicine in the Medieval West|page=982}} in {{Harv|Morelon|Rashed|1996|pp=963-84}}</ref>

[[James of Venice]], who probably spent some years in Constantinople, translated Aristotle's ''Posterior Analytics'' from Greek into Latin in the mid-twelfth century,<ref>L.D. Reynolds and Nigel G. Wilson, ''Scribes and Scholars,'' Oxford, 1974, p. 106.</ref> thus making the complete Aristotelian logical corpus, the Organon, available in Latin for the first time.

In 13th century [[Padua]], Bonacosa translated [[Averroes]]' medical work ''Kitab al-Kulliyyat'' as ''Colliget'',<ref name=Danielle-983/> and [[John of Capua]] translated the ''Kitab al-Taysir'' by [[Ibn Zuhr]] (Avenzoar) as ''Theisir''. In 13th century [[Sicily]], [[Faraj ben Salem]] translated [[Rhazes]]' ''al-Hawi'' as ''Continens'' as well as [[Ibn Butlan]]'s ''[[Tacuinum sanitatis]]''. Also in 13th century Italy, Simon of [[Genoa]] and Abraham Tortuensis translated [[Abu al-Qasim al-Zahrawi|Abulcasis]]' ''[[Al-Tasrif]]'' as ''Liber servitoris'', Alcoati's ''Congregatio sive liber de oculis'', and the ''Liber de simplicibus medicinis'' by a [[Serapion the Younger|pseudo-Serapion]]<ref name=Danielle-984/>

==Translators on the Spanish frontier==
[[File:Dioscorides De Materia Medica Spain 12th 13th century.jpg|thumb|[[Dioscorides]] ''[[Materia medica|De Materia Medica]]'' in [[Arabic]], [[Spain]], 12th-13th century.]]
As early as the end of the tenth century, European scholars travelled to Spain to study. Most notable among these was [[Gerbert of Aurillac]] (later Pope Sylvester II) who [[Islamic mathematics|studied mathematics]] in the region of the [[Spanish March]] around [[Barcelona]]. Translations, however, did not begin in Spain until after 1085 when Toledo was reconquered by Christians.<ref>C. H. Haskins, ''Studies in Mediaeval Science'', pp. 8-10</ref> The early translators in Spain focused heavily on [[Islamic science|scientific works]], especially [[Islamic mathematics|mathematics]] and [[Islamic astronomy|astronomy]], with a second area of interest including the [[Qur'an]] and other [[Islam]]ic texts.<ref>M.-T. d'Alverny, "Translations and Translators," pp. 429-30, 451-2</ref> Spanish collections included many scholarly works written in Arabic, so translators worked almost exclusively from Arabic, rather than Greek texts, often in cooperation with a local speaker of Arabic.<ref>[[Charles Homer Haskins|C. H. Haskins]], ''Renaissance of the Twelfth Century,'' p. 288</ref>

One of the more important translation projects was sponsored by [[Peter the Venerable]], the [[Abbot of Cluny|abbot]] of [[Cluny Abbey|Cluny]]. In 1142 he called upon [[Robert of Ketton]] and [[Herman of Carinthia]], [[Peter of Poitiers (translator)|Peter of Poitiers]], and a [[Muslim]] known only as "Mohammed" to produce the first Latin translation of the Qur'an (the ''[[Lex Mahumet pseudoprophete]]'').<ref>M.-T. d'Alverny, "Translations and Translators," p. 429</ref>

Translations were produced throughout Spain and [[Provence]]. [[Plato Tiburtinus|Plato of Tivoli]] worked in [[Catalonia]], Herman of Carinthia in Northern Spain and across the [[Pyrenees]] in [[Languedoc]], [[Hugh of Santalla]] in [[Aragon]], Robert of Ketton in [[Navarre]] and [[Robert of Chester]] in [[Segovia]].<ref>M.-T. d'Alverny, "Translations and Translators," pp. 444-8</ref> The most important center of translation was the great cathedral library of [[Toledo, Spain|Toledo]].

Plato of Tivoli's translations into Latin include [[Muhammad ibn Jābir al-Harrānī al-Battānī|al-Battani]]'s astronomical and [[trigonometry|trigonometrical]] work ''De motu stellarum'', [[Abraham bar Hiyya]]'s ''Liber embadorum'', [[Theodosius of Bithynia]]'s ''Spherica'', and [[Archimedes]]' ''[[Measurement of a Circle]]''. Robert of Chester's translations into Latin included [[Muhammad ibn Mūsā al-Khwārizmī|al-Khwarizmi]]'s ''[[The Compendious Book on Calculation by Completion and Balancing|Algebra]]'' and astronomical tables (also containing trigonometric tables).<ref name=Katz/> Abraham of [[Tortosa]]'s translations include Ibn Sarabi's ([[Serapion the Younger|Serapion Junior]]) ''De Simplicibus'' and [[Abu al-Qasim|Abulcasis]]' ''[[Al-Tasrif]]'' as ''Liber Servitoris''.<ref name=Campbell-3/> In 1126, [[Muhammad al-Fazari]]'s ''Great Sindhind'' (based on the [[Sanskrit]] works of ''[[Surya Siddhanta]]'' and [[Brahmagupta]]'s ''[[Brahmasphutasiddhanta]]'') was translated into Latin.<ref>G. G. Joseph, ''The Crest of the Peacock'', p. 306.</ref>

In addition to philosophical and scientific literature, the Jewish writer [[Petrus Alphonsi]] translated a collection of 33 tales from [[Arabic literature]] into [[Latin]]. Some of the tales he drew on were from the ''[[Panchatantra]]'' and ''[[One Thousand and One Nights|Arabian Nights]]'', such as the story cycle of "[[Sinbad the Sailor]]".<ref name=Irwin>{{citation|title=The Arabian Nights: A Companion|first=Robert|last=Irwin|publisher=[[I.B. Tauris|Tauris Parke Paperbacks]]|year=2003|isbn=1860649831|page=93}}</ref>

===The "Toledo School"===
[[Image:Las Siete Partidas.jpg|250px|thumb|King [[Alfonso X]] (the Wise)]]
One of the sponsors of translations in Spain was Archbishop [[Raymond de Sauvetât|Raymond of Toledo]], (1125–52), to whom [[John of Seville]] dedicated a translation in appreciation. Starting from this fragmentary evidence, nineteenth-century historians proposed that Raymond had established a formal translation school, but no specific evidence for such a school has emerged and its existence is now doubted. Many of the translators worked outside Toledo and those who did work in Toledo, worked after Raymond's episcopacy.<ref>M.-T. d'Alverny, "Translations and Translators," pp. 444-7</ref>

Toledo, however, was a center of multilingual culture, with a large population of Arabic speaking Christians ([[Mozarabs]]) and had prior importance as a center of learning. This tradition of scholarship, and the books that embodied it, survived the conquest of the city by [[Alfonso VI of Castile|King Alfonso VI]] in 1085. A further factor was that Toledo's early bishops and clergy came from France, where Arabic was not widely known. Consequently the cathedral became a center of translations, which were on a scale and importance that "has no match in the history of western culture".<ref>C. Burnett, "Arabic-Latin Translation Program in Toledo", pp. 249-51, 270.</ref>

Among the early translators at Toledo were an Avendauth (who some have identified with [[Abraham ibn Daud]]), who translated [[Avicenna]]'s encyclopedia, the ''[[The Book of Healing|Kitāb al-Shifa]]'' (''The Book of Healing''), in cooperation with [[Domingo Gundisalvo]], Archdeacon of Cuéllar.<ref>M.-T. d'Alverny, "Translations and Translators," pp. 444-6, 451</ref>

[[File:Al Razi Receuil de traite de medecine translated by Gerard de Cremone Second half of 13th century.jpg|thumb|[[Al-Razi]]'s ''Recueil des traités de médecine'' translated by [[Gerard of Cremona]], second half of 13th century.]]
[[Image:Al-RaziInGerardusCremonensis1250.JPG|thumb|Depiction of the Arab physician [[Al-Razi]], in Gerard of Cremona's "Recueil des traités de medecine" 1250-1260.]]
The most productive of the Toledo translators was [[Gerard of Cremona]],<ref>C. H. Haskins, ''Renaissance of the Twelfth Century,'' p. 287. "more of Arabic science passed into Western Europe at the hands of Gerard of Cremona than in any other way."</ref> who translated 87 books,<ref>For a list of Gerard of Cremona's translations see: Edward Grant (1974) ''A Source Book in Medieval Science'', (Cambridge: Harvard Univ. Pr.), pp. 35-8 or Charles Burnett, "The Coherence of the Arabic-Latin Translation Program in Toledo in the Twelfth Century," ''Science in Context'', 14 (2001): at 249-288, at pp. 275-281.</ref> including [[Ptolemy]]'s ''[[Almagest]]'', many of the works of [[Aristotle]], including his [[Posterior Analytics]], [[Physics (Aristotle)|Physics]], [[On the Heavens|On the Heavens and the World]], [[On Generation and Corruption]], and [[Meteorology (Aristotle)|Meteorology]], [[Muhammad ibn Mūsā al-Khwārizmī|al-Khwarizmi]]'s ''[[The Compendious Book on Calculation by Completion and Balancing|On Algebra and Almucabala]]'', [[Archimedes]]' ''[[On the Measurement of the Circle]]'', [[Aristotle]], [[Euclid]]'s ''[[Euclid's Elements|Elements of Geometry]]'', [[Jabir ibn Aflah]]'s ''Elementa astronomica'',<ref name=Katz/> [[Al-Kindi]]'s ''On Optics'', [[Ahmad ibn Muhammad ibn Kathīr al-Farghānī|al-Farghani]]'s ''On Elements of Astronomy on the Celestial Motions'', [[al-Farabi]]'s ''On the Classification of the Sciences'', the [[Alchemy (Islam)|chemical]] and [[Islamic medicine|medical]] works of [[al-Razi]] (Rhazes),<ref name=Bieber/> the works of [[Thabit ibn Qurra]] and [[Hunayn ibn Ishaq]],<ref>D. Campbell, ''Arabian Medicine and Its Influence on the Middle Ages'', p. 6.</ref> and the works of [[al-Zarkali]], [[Jabir ibn Aflah]], the [[Banū Mūsā|Banu Musa]], [[Abū Kāmil Shujā ibn Aslam|Abu Kamil]], [[Abu al-Qasim]], and [[Ibn al-Haytham]] (including the ''[[Book of Optics]]'').<ref name=Zaimeche>Salah Zaimeche (2003). [http://www.muslimheritage.com/uploads/Main%20-%20Aspects%20of%20the%20Islamic%20Influence1.pdf Aspects of the Islamic Influence on Science and Learning in the Christian West], p. 10. Foundation for Science Technology and Civilisation.</ref> The medical works he translated include [[Ali ibn Ridwan|Haly Abenrudian]]'s ''Expositio ad Tegni Galeni''; the ''Practica, Brevarium medicine'' by [[Yahya ibn Sarafyun|Yuhanna ibn Sarabiyun]] (Serapion); [[Al-Kindi|Alkindus]]' ''[[De Gradibus]]''; [[Muhammad ibn Zakarīya Rāzi|Rhazes]]' ''Liber ad Almansorem, Liber divisionum, Introductio in medicinam, De egritudinibus iuncturarum, Antidotarium'' and ''Practica puerorum''; [[Isaac Israeli ben Solomon]]'s ''De elementis'' and ''De definitionibus'';<ref name=Danielle/> [[Abu al-Qasim al-Zahrawi|Abulcasis]]' ''[[Al-Tasrif]]'' as ''Chirurgia''; [[Avicenna]]'s ''[[The Canon of Medicine]]'' as ''Liber Canonis''; and the ''Liber de medicamentis simplicus'' by Ibn Wafid ([[Abenguefit]]).<ref name=Danielle-983>{{citation|last=Jacquart|first=Danielle|contribution=The Influence of Arabic Medicine in the Medieval West|page=983}} in {{Harv|Morelon|Rashed|1996|pp=963-84}}</ref>

At the close of the twelfth and the beginning of the thirteenth centuries, [[Mark of Toledo]] translated the [[Qur'an]] (once again) and various [[Islamic medicine|medical works]].<ref>M.-T. d'Alverny, "Translations and Translators," pp. 429, 455</ref> He also translated [[Hunayn ibn Ishaq]]'s medical work ''Liber isagogarum''.<ref name=Danielle-983/>

===Later translators===
[[Michael Scot]] (c. 1175-1232)<ref>William P. D. Wightman (1953) ''The Growth of Scientific Ideas'', p.332. New Haven: Yale University Press. ISBN 1135460426.</ref> translated the works of [[Nur Ed-Din Al Betrugi|al-Betrugi]] (Alpetragius) in 1217,<ref name=Bieber/> [[al-Bitruji]]'s ''On the Motions of the Heavens'', and [[Averroes]]' influential commentaries on the scientific works of [[Aristotle]].<ref>[http://www.bautz.de/bbkl/m/michael_sco.shtml ''Biographisch-Bibliographisches Kirchenlexicon'']</ref>

King [[Alfonso X of Castile]] (reigned 1252–84) continued to promote translations, as well as the production of original scholarly works.

David the Jew (c. 1228-1245) translated the works of [[al-Razi]] (Rhazes) into Latin. [[Arnaldus de Villa Nova]]'s (1235–1313) translations include the works of [[Galen]] and [[Avicenna]]<ref>D. Campbell, ''Arabian Medicine and Its Influence on the Middle Ages'', p. 5.</ref> (including his ''Maqala fi Ahkam al-adwiya al-qalbiya'' as ''De viribus cordis''), the ''De medicinis simplicibus'' by Abu al-Salt (Albuzali),<ref name=Danielle-983/> and [[Qusta ibn Luqa|Costa ben Luca]]'s ''De physicis ligaturis''.<ref name=Danielle/>

In 13th century [[Portugal]], [[Giles of Santarem]] translated [[Rhazes]]' ''De secretis medicine, Aphorismi Rasis'' and [[Masawaiyh|Mesue]]'s ''De secretis medicine''. In [[Murcia]], Rufin of [[Alexandria]] translated the ''Liber questionum medicinalium discentium in medicina'' by [[Hunayn ibn Ishaq]] (Hunen), and Dominicus Marrochinus translated the ''Epistola de cognitione infirmatum oculorum'' by [[Ali Ibn Isa]] (Jesu Haly).<ref name=Danielle-983/> In 14th century [[Lleida|Lerida]], John Jacobi translated Alcoati's medical work ''Liber de la figura del uyl'' into [[Catalan language|Catalan]] and then Latin.<ref name=Danielle-984/>

[[William of Moerbeke|Willem van Moerbeke]], known in the English speaking world as William of [[Moerbeke]] (c. 1215–1286) was a prolific medieval translator of philosophical, medical, and scientific texts from Greek into Latin. At the request of Aquinas, so it is assumed—the source document is not clear—he undertook a complete translation of the works of [[Aristotle]] or, for some portions, a revision of existing translations. He was the first translator of the ''[[Politics]]'' (c. 1260) from Greek into Latin. The reason for the request was that the many copies of Aristotle in Latin then in circulation had originated in [[Spain]] (see [[Gerard of Cremona]]). These earlier translations were assumed to have been influenced by the rationalist [[Averroes]], who was suspected of being a source of philosophical and theological errors found in the earlier translations of Aristotle. Moerbeke's translations have had a long history; they were already standard classics by the 14th century, when [[Henricus Hervodius]] put his finger on their enduring value: they were literal (''de verbo in verbo''), faithful to the spirit of Aristotle and ''without elegance.'' For several of William's translations, the Greek texts have since disappeared: without him the works would be lost. William also translated mathematical treatises by [[Hero of Alexandria]] and [[Archimedes]]. Especially important was his translation of the ''Theological Elements'' of [[Proclus]] (made in 1268), because the ''Theological Elements'' is one of the fundamental sources of the revived [[Neo-Platonic]] philosophical currents of the 13th century. The [[Vatican Library|Vatican]] collection holds William's own copy of the translation he made of the greatest [[Hellenistic]] mathematician, [[Archimedes]], with commentaries of [[Eutocius]], which was made in 1269 at the papal court in Viterbo. William consulted two of the best Greek manuscripts of Archimedes, both of which have since disappeared.

==Other European translators==
[[Adelard of Bath]]'s (fl. 1116-1142) translations into Latin included al-Khwarizmi's astronomical and trigonometrical work ''Astronomical tables'' and his [[arithmetic]]al work ''Liber ysagogarum Alchorismi'', the ''Introduction to Astrology'' of [[Abu Mashar|Abū Ma'shar]], as well as Euclid's ''Elements''.<ref>Charles Burnett, ed. ''Adelard of Bath, Conversations with His Nephew,'' (Cambridge: Cambridge University Press, 1999), p. xi.</ref> Adelard associated with other scholars in Western England such as [[Petrus Alphonsi|Peter Alfonsi]] and [[Walcher of Malvern]] who translated and developed the astronomical concepts brought from Spain.<ref>M.-T. d'Alverny, "Translations and Translators," pp. 440-3</ref> [[Abū Kāmil Shujā ibn Aslam|Abu Kamil]]'s ''Algebra'' was also translated into Latin during this period, but the translator of the work is unknown.<ref name=Katz>V. J. Katz, ''A History of Mathematics: An Introduction'', p. 291.</ref>

[[Alfred of Sareshel]]'s (c. 1200-1227) translations include the works of [[Nicolaus of Damascus]] and [[Hunayn ibn Ishaq]]. Antonius Frachentius Vicentinus' translations include the works of [[Avicenna|Ibn Sina]] (Avicenna). Armenguad's translations include the works of Avicenna, [[Averroes]], Hunayn ibn Ishaq, and [[Maimonides]]. Berengarius of [[Valentia]] translated the works of [[Abu al-Qasim]] (Abulcasis). Drogon (Azagont) translated the works of [[al-Kindi]]. Farragut (Faradj ben Salam) translated the works of Hunayn ibn Ishaq, Ibn Zezla (Byngezla), [[Masawaiyh]] (Mesue), and [[al-Razi]] (Rhazes). Andreas Alphagus Bellnensis' translations include the works of Avicenna, Averroes, [[Serapion]], al-Qifti, and Albe'thar.<ref>D. Campbell, ''Arabian Medicine and Its Influence on the Middle Ages'', p. 4.</ref>

In 13th century [[Montpellier]], Profatius and Bernardus Honofredi translated the ''Kitab alaghdiya'' by [[Ibn Zuhr]] (Avenzoar) as ''De regimine sanitatis''; and Armengaudus Blasius translated the ''al-Urjuza fi al-tibb'', a work combining the medical writings of [[Avicenna]] and [[Averroes]], as ''Cantica cum commento''.<ref name=Danielle-984>{{citation|last=Jacquart|first=Danielle|contribution=The Influence of Arabic Medicine in the Medieval West|page=984}} in {{Harv|Morelon|Rashed|1996|pp=963-84}}</ref>

Other texts translated during this period include the [[Alchemy and chemistry in Islam|alchemical works]] of [[Geber|Jabir ibn Hayyan]] (Geber), whose treatises became standard texts for European [[Alchemy|alchemists]]. These include the ''Kitab al-Kimya'' (titled ''Book of the Composition of Alchemy'' in Europe), translated by [[Robert of Chester]] (1144); the ''Kitab al-Sab'een'' translated by [[Gerard of Cremona]] (before 1187), and the ''Book of the Kingdom'', ''Book of the Balances'' and ''Book of Eastern Mercury'' translated by [[Marcelin Berthelot]]. Another work translated during this period was ''De Proprietatibus Elementorum'', an [[Islamic science|Arabic work]] on [[Islamic geography|geology]] written by a [[pseudo-Aristotle]].<ref name=Bieber/> A pseudo-[[Masawaiyh|Mesue]]'s ''De consolatione medicanarum simplicum, Antidotarium, Grabadin'' was also translated into Latin by an anonymous translator.<ref name=Danielle-983/>

==See also==
* [[Renaissance of the 12th century]]
* [[Islamic contributions to Medieval Europe]]
* [[Lex Mahumet pseudoprophete]]
* [[List of translators]]
** [[Mark of Toledo]]

==Notes==
{{reflist}}

==References==
* Burnett, Charles. "The Coherence of the Arabic-Latin Translation Program in Toledo in the Twelfth Century," ''Science in Context'', 14 (2001): 249-288.
* Campbell, Donald (2001). ''Arabian Medicine and Its Influence on the Middle Ages''. [[Routledge]]. (Reprint of the London, 1926 edition). ISBN 0415231884.
* d'Alverny, Marie-Thérèse. "Translations and Translators", in Robert L. Benson and Giles Constable, eds., ''Renaissance and Renewal in the Twelfth Century'', p. 421-462. Cambridge: Harvard Univ. Pr., 1982.
* Haskins, Charles Homer. ''The Renaissance of the Twelfth Century''. Cambridge: Harvard Univ. Pr., 1927. See especially chapter 9, "The Translators from Greek and Arabic".
* Haskins, Charles Homer. ''Studies in the History of Mediaeval Science.'' New York: Frederick Ungar Publishing, 1967 (reprint of the Cambridge, Mass., 1927 ed.) Most of the book deals with the translations of Arabic and Greek scientific literature.
* Joseph, George G. (2000). ''The Crest of the Peacock''. [[Princeton University Press]]. ISBN 0691006598.
* Katz, Victor J. (1998). ''A History of Mathematics: An Introduction''. [[Addison Wesley]]. ISBN 0321016181.
*{{Citation
|last1=Morelon
|first1=Régis
|last2=Rashed
|first2=Roshdi
|year=1996
|title=[[Encyclopedia of the History of Arabic Science]]
|publisher=[[Routledge]]
|isbn=0415124107
}}

==External sources==
*[http://www.muslimheritage.com/topics/default.cfm?ArticleID=344 The Impact of Translations of Muslim Sciences on the West]
*[http://libro.uca.edu/alfonso10/emperor5.htm Norman Roth, "Jewish Collaborators in Alfonso's Scientific Work,"] in Robert I. Burns, ed., ''Emperor of Culture: Alfonso X the Learned of Castile and His Thirteenth-Century Renaissance Culture''

[[Category:Translations]]

[[fr:Traductions latines du XIIe siècle]]

Papst (Titel)

2010-02-01T16:22:40Z

Paul August: /* Nicea to East-West Schism (325–1054) */ better link I think

{{otheruses1|the head of the Catholic Church}}
{{pp-semi-protected|small=yes}}
{{Infobox Bishopric
|border = catholic
|font_color = black
|bishopric = [[Rome]]
|coatofarms = Coat of arms of the Holy See.svg
|image = Pope Benedictus XVI january,20 2006 (2).JPG
|incumbent = [[Benedict XVI]] Elected: 19 April 2005
|province = [[List of Roman Catholic dioceses (structured view)#Ecclesiastical Province of Rome|Ecclesiastical Province of Rome]]
|diocese = [[Diocese of Rome|Holy See]]
|cathedral = [[Basilica of St. John Lateran]]
|first_bishop = [[Saint Peter]]
|date = 53 AD
|website = [http://www.vatican.va/holy_father/benedict_xvi/index.htm/ Benedict XVI]
}}
[[File:Clemens I.jpg|thumb|250px|[[Pope Clement I]], one of the 1st Century Bishops of Rome, considered successors to [[Saint Peter]] as leaders of the Catholic Church. Monastic mural from [[Ohrid]], Macedonia.]]
The '''pope''' (from [[Latin]]: "papa" or "father" from [[Greek language|Greek]] {{polytonic|πάππας}}, ''pappas'')<ref>[http://artfl.uchicago.edu/cgi-bin/philologic/getobject.pl?c.51:9:99.lsj Liddell and Scott]</ref> is the [[Bishop]] of [[diocese of Rome|Rome]] and as such, is leader of the worldwide [[Catholic Church]] (that is, both the [[Latin Rite]] and the [[Eastern Catholic Churches]] in full communion with the Roman Pontiff). The current<ref>The [[Annuario Pontificio]] does not assign numbers to its list of popes because of [[canon law|canonical]] obscurities regarding the legitimacy of some members of the list.</ref> office-holder is [[Pope Benedict XVI]], who was elected in [[Papal conclave, 2005|papal conclave]] on 19 April 2005.

The office of the pope is called the ''Papacy'', and his ecclesiastical jurisdiction the "[[Holy See]]" (''Sancta Sedes'' in Latin) or "[[Apostolic See]]" (the latter on the basis that both [[Saint Peter|St. Peter]] and [[Paul the Apostle|St. Paul]] were [[martyr]]ed at [[Rome]]). The pope is also [[head of state]] of [[Vatican City]], a sovereign [[city-state]] entirely [[Enclave and exclave|enclaved]] by Rome.

Early popes helped to spread [[Christianity]] and resolve doctrinal disputes.<ref name = "World History" /> After the conversion of the rulers of the Roman Empire (the conversion of the populace was already advanced even before the [[Edict of Milan]], 313), the Roman emperors became the popes' secular allies until, with the loss of the emperors' power in the west, [[Pope Stephen II]] was forced in the 8th century to appeal to the [[Franks]] for help,<ref name="AF:CC">[[Will Durant|Durant, Will]]. The Age of Faith. New York: Simon and Schuster. 1972. Chapter XXI: Christianity in Conflict 529-1085. p. 517–551</ref> beginning a period of close interaction with the rulers of the west. For centuries, the forged [[Donation of Constantine]] also provided the basis for the papacy's claim of political supremacy over the entire former [[Western Roman Empire]]. In [[medieval]] times, popes played powerful roles in Western Europe, often struggling with monarchs for power over wide-ranging affairs of church and state,<ref name = "World History" /> crowning emperors ([[Charlemagne]] was the first emperor crowned by a pope) and regulating disputes among secular rulers.<ref>Such as regulating the [[colonization]] of the [[New World]]. See [[Treaty of Tordesillas]] and [[Inter caetera]].</ref>

Gradually forced to give up secular power, popes now focus almost exclusively on spiritual matters.<ref name = "World History" /> Over the centuries, popes' claims of spiritual authority have been ever more clearly expressed, culminating in the proclamation of the [[Dogma (Roman Catholic)|dogma]] of [[papal infallibility]] for rare occasions when the pope speaks ''[[ex cathedra]]'' (literally "from the chair (of Peter)") to issue a solemn definition of [[dogma|faith]] or [[morals]].<ref name = "World History" /> The first (after the proclamation) and so far the last such occasion was in 1950, with the definition of the dogma of the [[Assumption of Mary]].
== History ==
{{Roman Catholicism}}
{{main|History of the Papacy}}

Catholics recognize the Pope as a successor to [[Saint Peter]], whom, according to the [[Bible]], Jesus named as the "shepherd" and "rock" of the Church.<ref>{{cite web |url=http://www.vatican.va/archive/catechism/p123a9p4.htm |title=Catechism of the Catholic Church |accessdate=2008-08-02 |publisher=Vatican Library }}, 880-884</ref><ref>"St. Peter, ''[[The Catholic Encyclopedia]]''</ref> Peter never bore the ''title'' of "Pope", which came into use much later, but Catholics recognize him as the first Pope,<ref>Wilken, p. 281, quote: "Some (Christian communities) had been founded by Peter, the disciple Jesus designated as the founder of his church. ... Once the position was institutionalized, historians looked back and recognized Peter as the first pope of the Christian church in Rome"</ref> while official declarations of the Church only speak of the Popes as holding within the college of the Bishops a position analogous to that held by Peter within the college of the Apostles, of which the college of the Bishops, a distinct entity, is the successor.<ref>[http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19641121_lumen-gentium_en.html Second Vatican Council, ''Lumen Gentium'', 22]</ref><ref>[http://www.vatican.va/holy_father/john_paul_ii/audiences/alpha/data/aud19921007en.html [[Pope John Paul II]], Talk on 7 October 1992</ref><ref>[http://books.google.com/books?id=j8-GHiYUSX8C&printsec=frontcover&dq=Dulles+catholicity#v=onepage&q=episcopal%20college&f=false [[Avery Dulles]], ''The Catholicity of the Church'', Oxford University Press, 1987, ISBN 0198266952,] page 140</ref>

The study of the New Testament offers no uncontested proof that Jesus established the papacy nor even that he established Peter as the first bishop of Rome.<ref>{{cite book |first=John |last=O'Grady |title=The Roman Catholic church: its origins and nature |page=143}}</ref> The Catholic Church teaches that Jesus personally appointed Peter as leader of the Church and in its dogmatic constitution ''[[Lumen Gentium]]'' makes a clear distinction between apostles and bishops, presenting the latter as the successors of the former, with the Pope as successor of Peter in that he is head of the bishops as Peter was head of the apostles.<ref>[http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19641121_lumen-gentium_en.html ''Lumen gentium'', 22]</ref> Some historians have argued that the notion that Peter was the first bishop of Rome and founded the Christian church there can be traced back no earlier than the third century.<ref name="O'Grady 146">{{cite book |first=John |last=O'Grady |title=The Roman Catholic church: its origins and nature |page=146}}</ref> The writings of the [[Church Father]] [[Irenaeus]] who wrote around 180 AD indicate a belief that Peter "founded and organised" the Church at Rome.<ref>{{cite book |first=J. |last=Stevenson |title=A New Eusebius| page=114}}</ref> However, Irenaeus was not the first to write of Peter's presence in the early Roman Church. [[Clement of Rome]] wrote in a letter to the Corinthians, ''c.'' 96<ref name="fn_2">{{cite web|url=http://www.newadvent.org/fathers/1010.htm|title=Letter of Clement to the Corinthians}}</ref> about the awesome persecution of Christians in Rome as the “struggles in our time” and presented to the Corinthians its heroes, “first, the greatest and most just columns, the “good apostles” Peter and Paul.<ref name="Gröber, 510">Gröber, 510</ref> [[St. Ignatius of Antioch]] wrote shortly after Clement and in his letter from the city of Smyrna to the Romans he said he would not command them as Peter and Paul did.<ref name="fn_3">{{cite web |url=http://www.crossroadsinitiative.com/library_article/244/Letter_of_Ignatius_of_Antioch_to_the_Romans.html |title=Letter of Ignatius of Antioch to the Romans}}</ref> Given this and other evidence, many scholars conclude that Peter was indeed martyred in Rome under Nero.<ref>"[M]any scholars ... accept Rome as the location of the martyrdom and the reign of Nero as the time." Daniel William O’Connor, "Saint Peter the Apostle." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 25 Nov. 2009 [http://www.britannica.com/EBchecked/topic/453832/Saint-Peter-the-Apostle].</ref>

Various Christian communities would have had a group of presbyter-bishops functioning as leaders of the local church. Eventually this evolved into a monarchical episcopacy in certain cities.<ref name="O'Grady 140">{{cite book |first=John |last=O'Grady |title=The Roman Catholic church: its origins and nature |page=140}}</ref> Some historians would argue that it is possible that the monarchical episcopacy probably developed in other churches in the Christian world before it took shape in Rome. For example, it has been conjectured that Antioch may have been one of the first Christian communities to have adopted such a structure.<ref name="O'Grady 140"/> Indeed, in Rome there were many who claimed to be the rightful bishop though again Irenaeus stressed the validity of one line of bishops from the time of St. Peter up to his contemporary [[Pope Victor I]] and listed them.<ref>{{cite book |first=J. |last=Stevenson |title=A New Eusebius| page=114-115}}</ref> Some writers claim that the emergence of a single bishop in Rome probably did not arise until the middle of the second century. In their view, Linus, Cletus and Clement were possibly prominent presbyter-bishops but not necessarily monarchical bishops.<ref name="O'Grady 146"/> Though this would not necessarily affect their authority as Popes in terms of Catholic Theology.

The see of Rome was early accorded prominence in issues related to matters of the universal church.<ref>"From an historical perspective, there is no conclusive documentary evidence from the first century or the early decades of the second of the exercise of, or even the claim to, a primacy of the Roman bishop or to a connection with Peter, although documents from this period accord the church at Rome some kind of pre‑eminence" ([http://www.goarch.org/ourfaith/ourfaith8523 Emmanuel Clapsis, Papal Primacy,] extract from ''Orthodoxy in Conversation'' (2000), p. 110]); and "The see of Rome, whose prominence was associated with the deaths of Peter and Paul, became the principle centre in matters concerning the universal Church" (Clapsis, p. 102). The same writer quotes with approval the words of [[Pope Benedict XVI|Joseph Ratzinger]]: "In Phanar, on 25 July 1976, when [[Patriarch Athenagoras I of Constantinople|Patriarch Athenegoras]] addressed [[Pope Paul VI|the visiting pope]] as Peter's successor, the first in honour among us, and the presider over charity, this great church leader was expressing the essential content of the declarations of the primacy of the first millennium" (Clapsis, p. 113).</ref>

=== Early Christianity (''c.'' 30 – 325) ===

It seems that at first the terms 'episcopos' and 'presbyter' were used interchangeably.<ref>Oxford Dictionary of the Christian Church, 1997 edition revised 2005, page 211: "It seems that at first the terms 'episcopos' and 'presbyter' were used interchangeably ..."</ref> The general consensus among scholars has been that, at the turn of the first and second centuries, local congregations were led by bishops and presbyters whose offices were overlapping or indistinguishable.<ref>Cambridge History of Christianity, volume 1, 2006, "The general consensus among scholars has been that, at the turn of the first and second centuries, local congregations were led by bishops and presbyters whose offices were overlapping or indistinguishable."</ref> There was probably no single 'monarchical' bishop in Rome before the middle of the second century ... and likely later."<ref>Cambridge History of Christianity, volume 1, 2006, page 418: "Probably there was no single 'monarchical' bishop in Rome before the middle of the second century ... and likely later."</ref>

In the early Christian era, Rome and a few other cities had claims on the leadership of worldwide ("Catholic") church. [[James the Just]], known as "the brother of the Lord", served as head of the Jerusalem church, which is still honored as the "Mother Church" in Orthodox tradition. Alexandria had been a center of Jewish learning and became a center of Christian learning. Rome had a large congregation early in the apostolic period whom Paul the Apostle addressed in his [[Epistle to the Romans]], and Paul himself was martyred there.

During the first century of the Christian Church (''ca.'' 30–130), the Roman capital became recognized as a Christian center of exceptional importance. [[Pope Clement I]] at the end of the 1st century wrote an epistle to the Church in Corinth, Greece, intervening in a major dispute, and apologising for not having taken action earlier.<ref>Chadwick, Henry, ''Oxford History of Christianity'', OUP, quote: "Towards the latter part of the 1st century, Rome's presiding cleric named Clement wrote on behalf of his church to remonstrate with the Corinthian Christians who had ejected clergy without either financial or charismatic endowment in favour of a fresh lot; Clement apologized not for intervening but for not having acted sooner. Moreover, during the second century the Roman community's leadership was evident in its generous alms to poorer churches. About 165 they erected monuments to their martyred apostles, to Peter in a necropolis on the Vatican Hill, to Paul on the road to Ostia, at the traditional sites of their burial. Roman bishops were already conscious of being custodians of the authentic tradition of true interpretation of the apostolic writings. In the conflict with Gnosticism Rome played a decisive role, and likewise in the deep division in Asia Minor created by the claims of the Montanist prophets.."</ref> However there are only a few other references of that time to recognition of the [[Primacy of the Roman Pontiff|authoritative primacy]] of the [[Holy See|Roman See]] outside of Rome. In the [http://www.vatican.va/roman_curia/pontifical_councils/chrstuni/ch_orthodox_docs/rc_pc_chrstuni_doc_20071013_documento-ravenna_en.html Ravenna Document] of 13 October 2007, theologians chosen by the Roman Catholic and the Eastern Orthodox Churches stated: "41. Both sides agree ... that Rome, as the Church that 'presides in love' according to the phrase of St [[Ignatius of Antioch]] ([http://www.crossroadsinitiative.com/library_article/244/Letter_of_Ignatius_of_Antioch_to_the_Romans.html ''To the Romans'',] Prologue), occupied the first place in the ''taxis'', and that the [[bishop]] of Rome was therefore the ''protos'' among the patriarchs. They disagree, however, on the interpretation of the historical evidence from this era regarding the prerogatives of the Bishop of Rome as ''protos'', a matter that was already understood in different ways in the first millennium." In addition, in the last years of the first century AD the Church in Rome [[Epistles of Clement|intervened]] in the affairs of the Christian Church in [[Corinth]] to help solve their internal disputes.

Later in the second century AD, there were further manifestations of Roman authority over other churches. In 189 AD, assertion of the primacy of the Church of Rome may be indicated in [[Irenaeus of Lyons]]'s ''[[On the Detection and Overthrow of the So-Called Gnosis|Against Heresies]]'' (3:3:2): "With [the Church of Rome], because of its superior origin, all the churches must agree... and it is in her that the faithful everywhere have maintained the apostolic tradition." And in 195 AD, [[Pope Victor I]], in what is seen as an exercise of Roman authority over other churches, excommunicated the [[Quartodecimans]] for observing Easter on the 14th of Nisan, the date of the Jewish [[Passover]], a tradition handed down by [[John the Evangelist|St. John the Evangelist]] (see [[Easter controversy]]). Celebration of Easter on a Sunday, as insisted on by the Pope, is the system that has prevailed (see [[computus]]).

Early popes helped spread Christianity and resolve doctrinal disputes.<ref name = "World History"/>

=== Nicea to East-West Schism (325–1054) ===
During these seven centuries, the church unified by Emperor Constantine within his empire effectively split first, after the 451 [[Council of Chalcedon]], into [[Chalcedonian Christianity]] and [[Oriental Orthodoxy]], and then, after the 1054 [[East-West Schism]], into a [[Greek East and Latin West]]. In the West, the pope became independent of the Emperor in the East, and became a major force in politics there.

==== Imperial capitals: Rome and Constantinople ====
With the conversion of Roman Emperor [[Constantine I|Constantine]] to Christianity and the [[First Council of Nicaea|Council of Nicea]], the Christian religion received imperial backing.

At the time of the Council (325), Rome was still seen as the capital of the empire, although the emperor rarely lived there. With the establishment of a new fixed capital in [[Constantinople]] (330), there arose a new centre, which soon grew in prominence, rivalling those in Rome, Alexandria and Antioch, which previously had been the most important centres of Christianity.

Of these, Rome claimed the chief place, as illustrated by [[Pope Leo I|Pope Leo the Great]]'s statement, in about 446, that "the care of the universal Church should converge towards Peter's one seat, and nothing anywhere should be separated from its Head",<ref>[http://www.newadvent.org/fathers/3604014.htm Letter XIV]</ref> clearly articulating the extension of papal authority as doctrine, and promulgating his right to exercise "the full range of apostolic powers that Jesus had first bestowed on the apostle Peter".

The early [[ecumenical council]]s, particularly the [[First Council of Constantinople]] (381), affirmed the importance of the Bishop of Rome's position, though all the councils in the Church's early history took place in cities in the East, and the Pope did not personally attend the council in 381. It was at the ecumenical [[Council of Chalcedon]] in 451 that Leo I (through his emissaries) stated that he was "speaking with the voice of Peter". At this same council, the Bishop of Constantinople was given "equal privileges" to those of the Bishop of Rome, because "Constantinople is the New Rome". Pope Leo rejected this decree on the ground that it contravened the sixth canon of Nicaea and infringed the rights of Alexandria and Antioch.<ref>[http://www.1911encyclopedia.org/Council_of_Chalcedon Council of Chalcedon (1911 Encyclopaedia Britannica)]</ref>

==== Medieval development ====
[[File:Gregorythegreat.jpg|thumb|left|Gregory the Great (''c'' 540–604) who established medieval themes in the Church, in a painting by [[Carlo Saraceni]], circa 1610, Rome.]]
After the fall of Rome, the pope served as a source of authority and continuity. [[Gregory the Great]] (''c'' 540–604) administered the church with stern reform.<ref name="AF:CC"/> From an ancient senatorial family, Gregory worked with the stern judgment and discipline typical of ancient Roman rule.<ref name="AF:CC"/> Theologically, he represents the shift from the classical to the medieval outlook, his popular writings full of dramatic miracles, potent relics, demons, angels, ghosts, and the approaching end of the world.<ref name="AF:CC"/>

Gregory's successors were mostly dominated by the exarch or the Eastern emperor.<ref name="AF:CC"/> These humiliations, the weakening of the Empire in the face of Muslim expansion, and the inability of the Emperor to protect the papal estates made Pope Stephen II turn from the Emperor.<ref name="AF:CC"/> Seeking protection against the Lombards and getting no help from Emperor Constantine V, the pope appealed to the Franks to protect his lands.<ref name="AF:CC"/> Pepin the Short subdued the Lombards and donated Italian land to the Papacy.<ref name="AF:CC"/> When Leo III crowned Charlemagne (800), he established the precedent that no man would be emperor without anointment by a pope.<ref name="AF:CC"/>

Around 850, a forger, probably from among the French opposers of [[Hincmar]], [[Archbishop]] of [[Reims]]<ref name=ODCC:fd>"False Decretals." Cross, F. L., ed. The Oxford Dictionary of the Christian Church. New York: Oxford University Press. 2005</ref> made a collection of church legislation that contained forgeries as well as genuine documents.<ref name=ODCC:fd/><ref name=EB:fd>[http://www.britannica.com/EBchecked/topic/200996/False-Decretals Encyclopaedia Britannica: ''False Decretals'']</ref> At first some attacked it as false, but it was taken as genuine throughout the rest of the [[Middle Ages]]<ref name=ODCC:fd/> It is now known as the [[False Decretals]]. It was part of a series of falsifications of past legislation by a party in the Carolingian Empire whose principal aim was to free the church and the bishops from interference by the state and the [[Metropolitan archbishop#Roman Catholic|metropolitans]] respectively,<ref name=ODCC:fd/><ref name=EB:fd/> and who were concerned for papal supremacy as guaranteeing those rights.<ref name=ODCC:fd/> The author, a French cleric calling himself Isidore Mercator, created false documents purportedly by early church popes, demonstrating that supremacy of the papacy dated back to the church's oldest traditions.<ref name="AF:CC"/> The decretals include the ''[[Donation of Constantine]]'', in which [[Constantine I|Constantine]] grants [[Pope Sylvester I]] secular authority over all Western Europe.<ref name="AFself">[[Will Durant|Durant, Will]]. The Age of Faith. New York: Simon and Schuster. 1972. p. 525–526</ref> Thanks to this forgery in the collection, the decretals became one of the most persuasive forgeries in the history of the West. It supported Papal policies for centuries.<ref name="AF:CC"/>

Pope Nicholas I (858–867) asserted that the pope should have suzerain authority over all Christians, even royalty, in matters of faith and morals.<ref name="AF:CC"/> Only Photius, bishop of Constantinople, dared gainsay him.<ref name="AF:CC"/> He sternly defended morality and justice in a decadent age.<ref name="AF:CC"/> After his death, the authority of the papacy was acknowledged more widely than ever before.<ref name="AF:CC"/>

The low point of the Papacy was 867–1049.<ref name = "AF"/> The Papacy came under the control of vying political factions.<ref name="AF"/> Popes were variously imprisoned, starved, killed, and deposed by force.<ref name="AF"/> The family of a certain papal official made and unmade popes for fifty years.<ref name="AF"/> The official's great-grandson, Pope John XII, held orgies of debauchery in the Lateran palace.<ref name="AF"/> Emperor Otto I of Germany had John accused in an ecclesiastical court, which deposed him and elected a layman as Pope Leo VIII.<ref name="AF"/> John mutilated the Imperial representatives in Rome and had himself reinstated as Pope.<ref name="AF"/> Conflict between the Emperor and the papacy continued, and eventually dukes in league with the emperor were buying bishops and popes almost openly.<ref name="AF"/>

In 1049, Leo IX became pope, at last a pope with the character to face the papacy's problems.<ref name="AF"/> He traveled to the major cities of Europe to deal with the church's moral problems firsthand, notably the sale of church offices or services (simony) and clerical marriage and concubinage.<ref name="AF"/> With his long journey, he restored the prestige of the Papacy in the north.<ref name="AF"/>

=== East–West Schism to Reformation (1054–1517) ===
[[File:Grand schisme 1378-1417.png|thumb|260px|Historical map of the Western Schism: red is support for Avignon, blue for Rome]]
The East and West churches split definitively in 1054. This split was caused more by political events than by slight diversities of creed.<ref name = "AF"/> Popes had galled the emperors by siding with the king of the Franks, crowning a rival Roman emperor, appropriating the exarchate of Ravenna, and driving into Greek Italy.<ref name="AF">[[Will Durant|Durant, Will]]. The Age of Faith. New York: Simon and Schuster. 1972</ref>

In the [[Middle Ages]], popes struggled with monarchs over power.<ref name = "World History"/>

From 1309 to 1377, the pope resided not in Rome but in Avignon (see [[Avignon Papacy]]). The Avignon Papacy was notorious for greed and corruption.<ref name="R:RCC">[[Will Durant|Durant, Will]]. The Reformation. New York: Simon and Schuster. 1957. "Chapter I. The Roman Catholic Church." 1300-1517. p. 3–25</ref> During this period, the pope was effectively an ally of France, alienating France's enemies, such as England.<ref name="R:EWCGR">[[Will Durant|Durant, Will]]. The Reformation. New York: Simon and Schuster. 1957. "Chapter II. England: Wyclif, Chaucer, and the Great Revolt." 1308-1400. p. 26–57</ref>

The pope was understood to have the power to draw on the "treasury" of merit built up by the saints and by Christ, so that he could grant indulgences, reducing one's time in [[purgatory]].<ref name="R:RCC"/> The concept that a monetary fine or donation accompanied contrition, confession, and prayer eventually gave way to the common understanding that indulgences depended on a simple monetary contribution.<ref name="R:RCC"/> Popes condemned misunderstandings and abuses but were too pressed for income to exercise effective control over indulgences.<ref name="R:RCC"/>

Popes also contended with the cardinals, who sometimes attempted to assert the authority of councils over the pope's. Conciliar theory holds that the supreme authority of the church lies with a General Council, not with the pope.<ref name="ReferenceA">"Conciliar theory." Cross, F. L., ed. The Oxford dictionary of the Christian church. New York: Oxford University Press. 2005</ref> Its foundations were laid early in the 13th century, and it culminated in the 15th century.<ref name="ReferenceA"/> The failure of the conciliar theory to win general acceptance after the 15th century is taken as a factor in the Protestant Reformation.<ref name="ReferenceA"/>

Various antipopes challenged papal authority, especially during the [[Western Schism]] (1378–1417). In this schism, the papacy had returned to Rome from Avignon, but an antipope was installed in Avignon, as if to extend the papacy there.

The Eastern Church continued to decline with the Eastern Roman (Byzantine) Empire, undercutting Constantinople's claim to equality with Rome. Twice an Eastern Emperor tried to force the Eastern Church to reunify with the West. Papal claims of superiority were a sticking point in reunification, which failed in any event. In the 15th century, the Ottoman Turks captured Constantinople.

=== Reformation to present (1517 to today) ===
[[File:Council Trent.jpg|frame|As part of the Catholic Reformation, [[Pope Paul III]] (1534–1549) initiated the [[Council of Trent]] (1545–1563), which established the triumph of the Papacy over those who sought to reconcile with Protestants or oppose Papal claims.]]
Protestant Reformers criticized the Papacy as corrupt and characterized the pope as the antichrist.

Popes instituted the [[Catholic Reformation]]<ref name = "World History"/> (1560–1648), which addressed challenges of the [[Protestant Reformation]] and instituted internal reforms. Pope Paul III (1534–1549) initiated the [[Council of Trent]] (1545–1563), which established the triumph of the Papacy over rulers who sought to reconcile with Protestants and against French and Spanish bishops opposed to Papal claims.<ref>"Counter-Reformation." Cross, F. L., ed. The Oxford dictionary of the Christian church. New York: Oxford University Press. 2005</ref>

Gradually forced to give up secular power, popes focused on spiritual issues.<ref name = "World History"/>

The pope's claims of spiritual authority have been ever more clearly expressed since the first centuries. In 1870, the [[First Vatican Council]] proclaimed the [[dogma]] of [[papal infallibility]] for those rare occasions the pope speaks ''[[ex cathedra]]'' (literally "from the chair (of Peter)") when issuing a solemn definition of [[dogma|faith]] or [[morals]].<ref name="World History">Wetterau, Bruce. World history. New York: Henry Holt and company. 1994.</ref>

Later in 1870, Victor Emmanuel II [[Capture of Rome|seized Rome]] from the pope's control and substantially completed the unification of Italy.<ref name="World History"/> The Papal States that the pope lost had been used to support papal independence.<ref name="World History"/>

In 1929, the [[Lateran Treaty]] between Italy and Pope Pius XI established the Vatican guaranteed papal independence from secular rule.<ref name="World History"/>

In 1950, the pope defined the [[Assumption of Mary]] as dogma, the only time that a pope has spoken ex cathedra since papal infallibility was explicitly declared.

The [[Petrine Doctrine]] is still controversial as an issue of doctrine that continues to divide the eastern and western churches as well as separating Protestants from Rome.

== Saint Peter and the origin of the office ==
{{seealso|Primacy of Simon Peter}}
The [[dogma]]s and traditions of the [[Roman Catholic Church]] teach that the institution of the papacy was first mandated by interpretations of several Biblical passages, mainly Matthew 16:13–19:<ref group="nb" name="ex01">See also Isaiah 22:20–22, John 21:15–17, Luke 12:41, and Luke 22:31–32.</ref>

{{cquote|"When Jesus came into the coasts of [[Caesarea]] Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am? ... And Simon Peter answered and said, Thou art the Christ, the Son of the living God. And Jesus answered and said unto him, Blessed art thou, Simon Bar-jona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven. And I also say to you that you are Peter, and upon this rock I will build my church, and the gates of the netherworld will not prevail against it. I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven."}}

Catholics believe that this passage shows Jesus establishing his church on the shoulders of Simon son of John (Peter). In the past, some authorities have held that that the "rock" Jesus referred to was Jesus himself or was Peter's faith.<ref>Daniel William O'Connor. "Saint Peter the Apostle." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 27 Nov. 2009 [http://www.britannica.com/EBchecked/topic/453832/Saint-Peter-the-Apostle].</ref> The general scholarly consensus is that this account is authentic, and almost all current scholars agree with the straightforward interpretation that the "rock" Jesus refers to in this passage is Peter.<ref>Such is "the consensus of the great majority of scholars today." "Saint Peter the Apostle." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 27 Nov. 2009 [http://www.britannica.com/EBchecked/topic/453832/Saint-Peter-the-Apostle].</ref>

The reference to the "keys of the kingdom of heaven" here is the basis for the symbolic keys often found in Catholic papal symbolism, such as in the Vatican Coat of Arms (see below).{{citation needed|date=August 2009}}

== Election, death and abdication ==
=== Election ===
{{main|Papal conclave}}
[[File:Christ Handing the Keys to St. Peter by Pietro Perugino.jpg|thumb|300px|right|''The Giving of the Keys to Saint Peter'' painted by [[Pietro Perugino]] (1492)]]
The pope was originally chosen by those senior [[clergy]]men resident in and near Rome. In 1059 the electorate was restricted to the [[Cardinal (Catholicism)|Cardinals]] of the Holy Roman Church, and the individual votes of all Cardinal Electors were made equal in 1179. [[Pope Urban VI]], elected 1378, was the last pope who was not already a cardinal at the time of his election. [[Canon law]] requires that if a layman or non-bishop is elected, he receives episcopal consecration from the [[Dean of the College of Cardinals]] before assuming the Pontificate. Under present canon law, the pope is elected by the cardinal electors, comprising those cardinals who are under the age of 80.

The [[Second Council of Lyons]] was convened on 7 May 1274, to regulate the election of the pope. This Council decreed that the cardinal electors must meet within ten days of the pope's death, and that they must remain in seclusion until a pope has been elected; this was prompted by the three-year ''[[Sede Vacante]]'' following the death of [[Pope Clement IV]] in 1268. By the mid-sixteenth century, the electoral process had more or less evolved into its present form, allowing for alteration in the time between the death of the pope and the meeting of the cardinal electors.

Traditionally, the vote was conducted by [[acclamation]], by selection (by committee), or by plenary vote. Acclamation was the simplest procedure, consisting entirely of a voice vote, and was last used in 1621. [[Pope John Paul II]] abolished vote by acclamation and by selection by committee, and henceforth all Popes will be elected by full vote of the [[College of Cardinals|Sacred College of Cardinals]] by [[ballot]].

[[File:Konklave Konzilsgebaude Konstanz.jpg|thumb|left|The conclave in [[Konstanz]] where [[Pope Martin V]] was elected]]
[[File:Habemus Papam 1415.jpg|thumb|left|The formal declaration of "[[Habemus Papam]]" after the election of Pope Martin V]]
The election of the pope almost always takes place in the [[Sistine Chapel]], in a sequestered meeting called a "[[Papal conclave|conclave]]" (so called because the cardinal electors are theoretically locked in, ''cum clave'', until they elect a new pope). Three cardinals are chosen by lot to collect the votes of absent cardinal electors (by reason of illness), three are chosen by lot to count the votes, and three are chosen by lot to review the count of the votes. The ballots are distributed and each cardinal elector writes the name of his choice on it and pledges aloud that he is voting for "one whom under God I think ought to be elected" before folding and depositing his vote on a plate atop a large chalice placed on the altar (in the 2005 conclave, a special urn was used for this purpose instead of a chalice and plate). The plate is then used to drop the ballot into the chalice, making it difficult for any elector to insert multiple ballots. Before being read, the number of ballots are counted while still folded; if the total number of ballots does not match the number of electors, the ballots are burned unopened and a new vote is held. Otherwise, each ballot is read aloud by the presiding Cardinal, who pierces the ballot with a needle and thread, stringing all the ballots together and tying the ends of the thread to ensure accuracy and honesty. Balloting continues until a Pope is elected by a two-thirds majority.<ref>With the promulgation of ''[[Universi Dominici Gregis]]'' in 1996, a simple majority after a deadlock of twelve days was allowed, but this was revoked by Pope [[Benedict XVI]] by ''[[motu proprio]]'' in 2007.</ref>

One of the most famous aspects of the papal election process is the means by which the results of a ballot are announced to the world. Once the ballots are counted and bound together, they are burned in a special stove erected in the Sistine Chapel, with the smoke escaping through a small chimney visible from [[St. Peter's Square]]. The ballots from an unsuccessful vote are burned along with a chemical compound in order to produce black smoke, or ''fumata nera''. (Traditionally, wet straw was used to produce the black smoke, but this was not completely reliable. The chemical compound is more reliable.) When a vote is successful, the ballots are burned alone, sending white smoke (''fumata bianca'') through the chimney and announcing to the world the election of a new pope. At the end of the conclave that elected [[Pope Benedict XVI]], church bells were also rung to signal that a new pope had been chosen.

The Dean of the College of Cardinals then asks the cardinal who has been successfully-elected two solemn questions. First he asks, "Do you freely accept your election?" If he replies with the word ''"Accepto"'', his reign as Pope begins at that instant, ''not'' at the inauguration ceremony several days afterward. The Dean then asks, "By what name shall you be called?" The new pope then announces the [[regnal name]] he has chosen for himself. (If the Dean himself is elected pope, the Vice Dean performs this duty).

The new pope is led through the "Door of Tears" to a dressing room in which three sets of white papal vestments (''immantatio'') await: small, medium, and large. Donning the appropriate vestments and reemerging into the Sistine Chapel, the new pope is given the "[[Ring of the Fisherman|Fisherman's Ring]]" by the [[Camerlengo of the Holy Roman Church]], whom he first either reconfirms or reappoints. The pope then assumes a place of honor as the rest of the cardinals wait in turn to offer their first "obedience" (''adoratio'') and to receive his blessing.

The senior [[Cardinal Deacon]] then announces from a balcony over St. Peter's Square the following [[Habemus Papam|proclamation]]: ''Annuntio vobis gaudium magnum! Habemus Papam!'' ("I announce to you a great joy! We have a pope!"). He then announces the new pope's Christian name along with the new name he has adopted as his regnal name.

Until 1978 the pope's election was followed in a few days by the [[Papal Coronation]]. A procession with great pomp and circumstance formed from the [[Sistine Chapel]] to [[St. Peter's Basilica]], with the newly elected pope borne in the ''[[sedia gestatoria]]''. There, after a solemn [[Papal Mass]], the new pope was crowned with the ''[[Papal Tiara|triregnum]]'' (papal tiara) and he gave for the first time as pope the famous blessing ''[[Urbi et Orbi]]'' ("to the City [Rome] and to the World"). Another renowned part of the coronation was the lighting of a bundle of [[flax]] at the top of a gilded pole, which would flare brightly for a moment and then promptly extinguish, with the admonition ''[[Sic transit gloria mundi]]'' ("Thus passes worldly glory"). A similar sombre warning against papal hubris made on this occasion was the ritual exclamation ''"Annos Petri non videbis"'', reminding the newly crowned Pope that he would not live to see his rule lasting as long as that of St. Peter, who according to tradition headed the church for 35 years and has thus far been the longest reigning Pope in the history of the Catholic Church.

A [[traditionalist Catholic]] belief claims the existence of a [[Papal Oath (Traditionalist Catholic)|Papal Oath]] sworn, at their coronation, by all popes from [[Pope Agatho]] to [[Pope Paul VI]], but which since the abolition of the coronation ceremony is no longer used. There is no reliable authority for this claim.

The [[Latin]] term ''sede vacante'' ("vacant seat") refers to a papal [[interregnum]], the period between the death of a pope and the election of his successor. From this term is derived the term [[sedevacantism]], which designates a category of dissident Catholics who maintain that there is no canonically and legitimately elected Pope, and that there is therefore a ''Sede Vacante''. One of the most common reasons for holding this belief is the idea that the reforms of the [[Second Vatican Council]] and especially the replacement of the [[Tridentine Mass]] with the ''[[Mass of Paul VI]]'' are heretical, and that, per the dogma of papal infallibility, it is impossible for a valid Pope to have done these things. Secevacantists are considered to be schismatics by the mainstream Roman Catholic Church.

For centuries, the papacy was an institution dominated by [[Italians]]. Prior to the election of the Polish cardinal [[Karol Wojtyla]] as Pope John Paul II in 1978, the last non-Italian was [[Pope Adrian VI]] of the Netherlands, elected in 1522. John Paul II was followed by the German-born Benedict XVI, leading some to believe the Italian domination of the papacy to be over.

=== Death ===
[[File:Pope johnpaul funeral.jpeg|thumb|right|250px|Funeral of [[Pope John Paul II]] at the Vatican in 2005, presided over by Cardinal Ratzinger, the future Pope Benedict XVI]]
The current regulations regarding a papal [[interregnum]]—that is, a ''[[sede vacante]]'' ("vacant seat")—were promulgated by John Paul II in his 1996 document ''[[Universi Dominici Gregis]]''. During the "Sede Vacante", the [[College of Cardinals|Sacred College of Cardinals]], composed of the pope's principal advisors and assistants, is collectively responsible for the government of the Church and of the Vatican itself, under the direction of the [[Camerlengo of the Holy Roman Church]]; however, canon law specifically forbids the cardinals from introducing any innovation in the government of the Church during the vacancy of the [[Holy See]]. Any decision that requires the assent of the pope has to wait until the new pope has been elected and accepts office.

In recent centuries it was traditional, when a Pope was judged to have died, for the Cardinal Chamberlain to confirm the death ceremonially by gently tapping the Pope's head thrice with a silver hammer, calling his birth name each time. This custom was not followed at the death of Pope John Paul I<ref>Sullivan, George E. Pope John Paul II: The People's Pope. Boston: Walker & Company, 1984.</ref> and probably was not revived upon the death of Pope John Paul II. The Cardinal Chamberlain then retrieves the [[Ring of the Fisherman]] and cuts it in two in the presence of the Cardinals. The deceased pope's seals are defaced, to keep them from ever being used again, and his personal apartment is sealed.

The body then lies in state for a number of days before being interred in the [[crypt]] of a leading church or cathedral; the popes of the 20th century were all interred in [[St. Peter's Basilica]]. A nine-day period of mourning (''novendialis'') follows the interment of the late Pope.

=== Resignation ===
{{mainarticle|Papal resignation}}
The [[Code of Canon Law]] [http://www.vatican.va/archive/ENG1104/_P16.HTM 332 §2] states, "If it happens that the [[Papal resignation|Roman Pontiff resigns]] his office, it is required for validity that the resignation is made freely and properly manifested but not that it is accepted by anyone." This right has been exercised by, among others, [[Pope Celestine V]] in 1294 and [[Pope Gregory XII]] in 1409, Gregory XII being the last to do so.

It was widely reported in June and July 2002 that Pope John Paul II firmly refuted the speculation of his resignation using Canon 332, in a letter to the Milan daily newspaper ''[[Corriere della Sera]]''. Nevertheless, 332 §2 caused speculation that (1) Pope [[John Paul II]] would have resigned as his health failed, or (2) a properly manifested legal instrument had been prepared which effected his resignation if he could not perform his duties.{{citation needed|date=August 2009}} Pope John Paul II, however, did not resign. He died on 2 April 2005 after a long period of ill-health and was buried on 8 April 2005. After his death, it was reported in [[Testament of Pope John Paul II|his last will and testament]] that he considered abdicating in 2000 as he neared his 80th birthday.{{citation needed|date=August 2009}} That portion of the will, however, is unclear and others interpret it differently.

== Titles ==
{{infobox popestyles
|papal name=The Pope
|dipstyle=[[His Holiness]]
|offstyle=Your Holiness
|relstyle=Holy Father
|deathstyle=NA|}}

=== Official list of titles ===

The official list of titles of the Pope, in the order in which they are given in the [[Annuario Pontificio]], is: Bishop of [[Diocese of Rome|Rome]], [[Vicar of Christ|Vicar of Jesus Christ]], Successor of the [[Prince of the Apostles]], Supreme [[Pontiff#Christianity|Pontiff]] of the Universal Church, [[Primate (religion)|Primate]] of [[Italy]], [[Metropolitan bishop|Archbishop and Metropolitan]] of the Roman [[Ecclesiastical province|Province]], Sovereign of the State of [[Vatican City]], [[Servus Servorum Dei|Servant of the Servants of God]].<ref>Annuario Pontificio, published annually by Libreria Editrice Vaticana, p. 23*. ISBN of the 2009 edition: 978-88-209-8191-4.</ref>

The official list of titles does not include all the titles that are officially used.

=== Pope ===

The best-known title of the Popes, that of "Pope", does not appear in the official list, but is commonly used in the titles of documents, and appears, in abbreviated form, in their signatures. Thus [[Pope Paul VI]] signed as "Paulus PP. VI", the "PP." standing for "''Papa''" ("Pope").

The title "Pope" was from the early third century an honorific designation used for ''any'' bishop in the West.<ref name=ODCC:Pope>Oxford Dictionary of the Christian Church ([[Oxford University Press]] 2005 ISBN 978-0-19-280290-3), article ''Pope''</ref> In the East it was used only for the Bishop of Alexandria.<ref name=ODCC:Pope/> [[Pope Marcellinus]] (d. 304) is the first Bishop of Rome shown in sources to have had the title "Pope" used of him. From the 6th century, the imperial chancery of [[Byzantine Empire|Constantinople]] normally reserved this designation for the Bishop of Rome.<ref name=ODCC:Pope/> From the early sixth century it began to be confined in the West to the Bishop of Rome, a practice that was firmly in place by the eleventh century,<ref name=ODCC:Pope/> when [[Pope Gregory VII]] declared it reserved for the Bishop of Rome.

In [[Eastern Christianity]], where the title "pope" is used also of the Bishop of Alexandria, the Bishop of Rome is often referred to as the "Pope of Rome", regardless of whether the speaker or writer is in communion with Rome or not.

=== Vicar of Peter and Vicar of Christ ===

Early [[Bishop (Catholic Church)|bishops]] occupying the [[See of Rome]] were designated "[[Vicar]] of Peter", indicating that they were successors of [[Saint Peter]], the "Prince of the Apostles" or leader of the apostolic Church. The [[Roman Missal]] uses this title in its prayers for a dead Pope.<ref>[http://www.usccb.org/liturgy/papalnotes.pdf Liturgical Notes and Resource Materials for Use upon the Death of a Pope]</ref>

The designation "Vicar of Christ" was first used of a Pope by the Roman [[Synod]] of 495 with reference to [[Pope Gelasius I]]. But for long after this the stable designation for the Popes was "Vicar of Peter", while "Vicar of Christ" was a title used by the [[Byzantine Empire|Roman Emperors of the East]].<ref name=New>[http://books.google.co.uk/books?id=X5rcnhLnRYMC&dq=%22vicarius+petri%22&q=vicarius+petri#v=snippet&q=vicarius%20petri&f=false New Commentary on the Code of Canon Law: Study Edition By John P. Beal, James A. Coriden, Thomas J. (Thomas Joseph) Green, Thomas J. Green, Canon Law Society of America, p. 432]</ref>

Much earlier, Tertullian (c. 160 – c. 220) used the phrase "Vicar of Christ" of the Holy Spirit with regard to the Spirit's role of maintaining in the Church the teaching given by the [[Twelve Apostles|apostles]]:

<blockquote>
"Grant, then, that all have erred; that the apostle was mistaken in giving his testimony; that the Holy Ghost had no such respect to any one (church) as to lead it into truth, although sent with this view by Christ, ... grant also that He, the Steward of God, the Vicar of Christ neglected His office, permitting the churches for a time to understand differently, (and) to believe differently, what He Himself was preaching by the apostles,— is it likely that so many churches, and they so great, should have gone astray into one and the same faith?"<ref>[http://www.newadvent.org/fathers/0311.htm Prescription Against the Heretics, Chapter 28)]</ref>
</blockquote>

He also referred to the Holy Spirit as the "Vicar of the Lord":
<blockquote>
"For what kind of (supposition) is it, that, while the devil is always operating and adding daily to the ingenuities of iniquity, the work of God should either have ceased, or else have desisted from advancing? whereas the reason why the Lord sent the Paraclete was, that, since human mediocrity was unable to take in all things at once, discipline should, little by little, be directed, and ordained, and carried on to perfection, by that Vicar of the Lord, the Holy Spirit."<ref>[http://www.newadvent.org/fathers/0403.htm Tertullian, On the Veiling of Virgins, Chapter 1)]</ref>
</blockquote>

It was only from the time of [[Pope Innocent III]] (1198–1216) that the title, which has no legal definition or juridical significance, was used stably of the Popes.<ref name=New/> For the Catholic Church, all bishops are vicars of Christ.<ref>[http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19641121_lumen-gentium_en.html Second Vatican Council, Dogmatic Constitution ''Lumen gentium'', 27]</ref>

=== Supreme Pontiff and Pontifex Maximus ===

The term "Supreme Pontiff" (''Summus Pontifex'') or, more completely, "Supreme Pontiff of the Universal Church" (''Summus Pontifex Ecclesiae Universalis'') is one of the official titles of the Pope.

Inscriptions on buildings and coins often use the [[Latin]] title "[[Pontifex Maximus]]", which is not to be confused with "Summus Pontifex". The title "Pontifex Maximus" dates back to the early years of the [[Roman Republic]]. Beginning with [[Julius Caesar]], it was associated with the [[Roman Emperor]]s, until [[Gratian]] (359–383), under the influence of Saint [[Ambrose]], formally renounced the title. It is commonly found in inscriptions on buildings erected in the time of a particular Pope and on coins and medals of his reign, and is usually abbreviated as "Pont. Max." or "P.M." The phrase literally means "Greatest Pontiff", but is often interpreted as "Supreme Pontiff", which is instead a literal translation of "''Summus Pontifex''".

=== Servant of the Servants of God ===

The title "Servant of the Servants of God", although used by Church leaders including [[Augustine of Hippo|St. Augustine]] and [[St. Benedict]], was first used by [[Pope Gregory I|Pope St. Gregory the Great]] in his dispute with the Patriarch of Constantinople after the latter assumed the title "[[Ecumenical Patriarch]]". It was not reserved for the pope until the thirteenth century. The documents of the [[Second Vatican Council]] reinforced the understanding of this title as a reference to the pope's role as a function of collegial authority, in which the Bishop of Rome serves the world's bishops.

=== Patriarch of the West ===

From 1863 until 2005, the ''Annuario Pontificio'' included also the title "[[Patriarch]] of the West". This title was first used by [[Pope Theodore I]] in 642, and was only used occasionally. Indeed, it did not begin to appear in the pontifical yearbook until 1863. On 22 March 2006, the Vatican released a statement explaining this omission on the grounds of expressing a "historical and theological reality" and of "being useful to ecumenical dialogue". The title Patriarch of the West symbolized the pope's special relationship with, and jurisdiction over, the Latin Church—and the omission of the title neither symbolizes in any way a change in this relationship, nor distorts the relationship between the Holy See and the [[Eastern Churches]], as solemnly proclaimed by the Second Vatican Council.<ref>[http://www.vatican.va/roman_curia/pontifical_councils/chrstuni/general-docs/rc_pc_chrstuni_doc_20060322_patriarca-occidente_fr.html Communiqué concernant la suppression du titre «Patriarche d’Occident» dans l'Annuaire pontifical 2006]</ref>

=== Other titles ===

Other titles commonly used are "His Holiness", "Holy Father". In [[Spanish language|Spanish]] and [[Italian language|Italian]], "''Beatísimo/Beatissimo Padre''" (Most Blessed Father) is often used in preference to "''Santísimo/Santissimo Padre''" (Most Holy Father). In the [[Middle Ages|medieval period]], "''Dominus Apostolicus''" ("the [[Saint Peter|Apostolic]] Lord") was also used.

=== Signature ===

As indicated above, a Pope normally signs documents using the title "Papa" in the abbreviated form "PP." and with the numeral, as in "Benedictus PP. XVI" (Pope Benedict XVI). Exceptions are [[papal bull|bulls]] of canonization and decrees of ecumenical councils, which the Pope signs with the formula, "Ego N. Episcopus Ecclesiae catholicae", without the numeral, as in "Ego Paulus Episcopus Ecclesiae catholicae" (I, Paul, Bishop of the catholic/universal Church).<ref>[http://www.1911encyclopedia.org/Curia_Romana Classic Encyclopedia: ''Curia Romana'']</ref> The Pope's signature is followed, in bulls of canonization, by those of all the cardinals resident in Rome, and in decrees of ecumenical councils, by the signatures of the other bishops participating in the council, each signing as Bishop of a particular see.

[[Papal bull]]s are headed ''N. Episcopus [[Servus Servorum Dei]]'' ("Name, Bishop, Servant of the Servants of God"). In general, they are not signed by the Pope, but [[Pope John Paul II]] introduced in the mid-1980s the custom by which the Pope signs not only bulls of canonization but also, using his normal signature, such as "Benedictus PP. XVI", bulls of nomination of bishops.

== Residence and jurisdiction ==

The pope's [[cathedra|official seat]] or [[cathedral]] is the [[Basilica of St. John Lateran]], and his official residence is the [[Palace of the Vatican]]. He also possesses a summer residence at [[Castel Gandolfo]] (situated on the site of the ancient city of [[Alba Longa]]). Until the time of the [[Avignon Papacy]], the residence of the Pope was the [[Lateran Palace]], donated by the [[Roman Emperor]] [[Constantine I of the Roman Empire|Constantine the Great]].

The Pope's ecclesiastical jurisdiction (the [[Holy See]]) is distinct from his secular jurisdiction (Vatican City). It is the Holy See which conducts international relations; for hundreds of years, the papal court (the [[Roman Curia]]) has functioned as the government of the Catholic Church.

The names "Holy See" and "Apostolic See" are in ecclesiastical terminology the [[ordinary jurisdiction]] of the Bishop of Rome (including the Roman Curia); the pope's various honors, powers, and privileges within the Catholic Church and the international community derive from his Episcopate of Rome in lineal succession from the [[Twelve apostles|Apostle]] [[Saint Peter]] (see [[Apostolic Succession]]). Consequently, Rome has traditionally occupied a central position in the Catholic Church, although this is not necessarily so. The pope derives his pontificate from being Bishop of Rome but is not required to live there; according to the Latin formula ''ubi Papa, ibi Curia'', wherever the Pope resides is the central government of the Church, provided that the pope is Bishop of Rome. As such, between 1309 and 1378, the popes lived in [[Avignon]] (see [[Avignon Papacy]]), a period often called the [[Babylonian Captivity]] in allusion to the [[Bible|Biblical]] [[exile]] of [[Israel]].

Though the Pope is the diocesan Bishop of the [[Diocese of Rome]], he delegates most of the day-to-day work of leading the diocese to the [[Cardinal Vicar]], who assures direct episcopal oversight of the diocese's pastoral needs, not in his own name but in that of the Pope. The current Cardinal Vicar is [[Agostino Vallini]], who was appointed to the office in June 2008.

== Regalia and insignia ==

[[File:433px-Pope Pius VII.jpg|thumb|170px|[[Pope Pius VII]], bishop of Rome, next to [[Cardinal (Catholicism)|Cardinal]] Caprara. The Pope wears the [[pallium]], a liturgical [[vestment]] that is used [[heraldry|heraldically]] at the foot of the coat of arms of [[Benedict XVI]].]]
{{main|Papal regalia and insignia}}
* "[[Papal Tiara|Triregnum]]", also called the "tiara" or "triple crown", represents the pope's three functions as "supreme pastor", "supreme teacher" and "supreme priest". Recent popes have not, however, worn the ''triregnum'', though it remains the symbol of the papacy and has not been abolished. In liturgical ceremonies Popes wear an episcopal [[mitre]] (an erect cloth hat).
* [[Pastoral Staff]] topped by a [[crucifix]], a custom established before the 13th century (see [[papal cross]]).
* [[Pallium]], or pall, a circular band of fabric worn around the neck over the [[chasuble]]. It forms a yoke about the neck, breast and shoulders and has two pendants hanging down in front and behind, and is ornamented with six crosses. Previously, the pallium worn by the pope was identical to those he granted to the [[primate (religion)|primates]], but in 2005 Pope Benedict XVI began to use a distinct papal pallium that is larger than the primatial, and was adorned with red crosses instead of black.
* "Keys to the Kingdom of Heaven", the image of two keys, one gold and one silver. The silver key symbolizes the power to bind and loose on Earth, and the gold key the power to bind and loose in Heaven.
* [[Ring of the Fisherman]], a gold ring decorated with a depiction of St. Peter in a boat casting his net, with the name of the reigning Pope around it.
* ''[[Umbraculum]]'' (better known in the Italian form ''ombrellino'') is a canopy or umbrella consisting of alternating red and gold stripes, which used to be carried above the pope in processions.
* ''[[Sedia gestatoria]]'', a mobile throne carried by twelve [[footmen]] (''palafrenieri'') in red uniforms, accompanied by two attendants bearing ''[[flabella]]'' (fans made of white ostrich feathers), and sometimes a large [[baldachin|canopy]], carried by eight attendants. The use of the ''flabella'' was discontinued by [[Pope John Paul I]]. The use of the ''sedia gestatoria'' was discontinued by [[Pope John Paul II]], being replaced by the so-called [[Popemobile]].
[[File:Holysee-arms.svg|thumb|left|The [[coat of arms]] of the Holy See. That of the State of Vatican City is the same except that the positions of the gold and silver keys are interchanged.<ref>[http://www.fotw.net/flags/va).html Vatican City (Holy See) - The Keys and Coat of Arms]</ref>]]
In [[heraldry]], each pope has his own [[Papal Coat of Arms]]. Though unique for each pope, the arms are always surmounted by the aforementioned two keys in [[saltire]] (i.e., crossed over one another so as to form an ''X'') behind the [[Escutcheon (heraldry)|escutcheon]] (shield) (one silver key and one gold key, tied with a red cord), and above them a silver ''triregnum'' with three gold crowns and red ''infulae'' ([[lappet]]s—two strips of fabric hanging from the back of the triregnum which fall over the neck and shoulders when worn). This is [[blazon]]ed: "two keys in saltire or and argent, interlacing in the rings or, beneath a tiara argent, crowned or"). With the recent election of [[Benedict XVI]] in 2005, his personal coat of arms eliminated the papal tiara; a [[mitre]] with three horizontal lines is used in its place, with the pallium, a papal symbol of authority more ancient than the tiara, the use of which is also granted to metropolitan [[archbishops]] as a sign of communion with the See of Rome, was added underneath of the shield. The distinctive feature of the crossed keys behind the shield was maintained. The omission of the tiara in the Pope's personal coat of arms, however, did not mean the total disappearance of it from papal heraldry, since the coat of arms of the Holy See was kept unaltered.

The [[flag]] most frequently associated with the pope is the yellow and white [[flag of Vatican City]], with the arms of the Holy See (blazoned: "Gules, two keys in saltire or and argent, interlacing in the rings or, beneath a tiara argent, crowned or") on the right-hand side (the "fly") in the white half of the flag (the left-hand side—the "hoist"—is yellow). The pope's escucheon does not appear on the flag. This flag was first adopted in 1808, whereas the previous flag had been red and gold, the traditional colors of the papacy. Although Pope Benedict XVI replaced the triregnum with a mitre on his personal coat of arms, it has been retained on the flag.

== Status and authority ==
{{main|Primacy of the Roman Pontiff|Papal infallibility}}
[[File:Kruisheren uden bij paus pius xii Crosiers from Uden Holland with PiusXII.jpg|thumb|350px|left|To maintain contacts with local clergymen and Catholic communities, the popes grant private audiences as well as public ones. Here the [[Canons Regular of the Holy Cross]] from [[Uden]] ([[Netherlands]]) are received by [[Pope Pius XII]].]]
=== First Vatican Council ===
The status and authority of the Pope in the Catholic Church was [[dogma]]tically [[dogmatic definition|defined]] by the [[First Vatican Council]] on 18 July 1870. In its Dogmatic Constitution of the Church of Christ, the Council established the following canons:<ref>The texts of these canons are given in [[Denzinger]], [http://catho.org/9.php?d=byj#dez Latin original;] [http://www.catecheticsonline.com/SourcesofDogma19.php English translation]</ref>

"If anyone says that the blessed Apostle Peter was not established by the Lord Christ as the chief of all the [[twelve apostles|apostles]], and the visible head of the whole militant Church, or, that the same received great honour but did not receive from the same our Lord Jesus Christ directly and immediately the primacy in true and proper jurisdiction: let him be [[anathema]].<ref>Denzinger 3055 (old numbering, 1823)</ref>

If anyone says that it is not from the institution of Christ the Lord Himself, or by divine right that the blessed Peter has perpetual successors in the primacy over the universal Church, or that the Roman Pontiff is not the successor of blessed Peter in the same primacy, let him be anathema.<ref>Denzinger 3058 (old numbering, 1825)</ref>

If anyone thus speaks, that the Roman Pontiff has only the office of inspection or direction, but not the full and supreme power of jurisdiction over the universal Church, not only in things which pertain to faith and morals, but also in those which pertain to the discipline and government of the Church spread over the whole world; or, that he possesses only the more important parts, but not the whole plenitude of this supreme power; or that this power of his is not ordinary and immediate, or over the churches altogether and individually, and over the pastors and the faithful altogether and individually: let him be anathema.<ref> Denzinger 3064 (old numbering, 1831)</ref>

We, adhering faithfully to the tradition received from the beginning of the Christian faith, to the glory of God, our Saviour, the elevation of the Catholic religion and the salvation of Christian peoples, with the approbation of the sacred Council, teach and explain that the dogma has been divinely revealed: that the Roman Pontiff, when he speaks ex cathedra, that is, when carrying out the duty of the pastor and teacher of all Christians by virtue of his supreme apostolic authority he defines a doctrine of faith or morals to be held by the universal Church, through the divine assistance promised him in blessed Peter, operates with that infallibility with which the divine Redeemer wished that His church be instructed in defining doctrine on faith and morals; and so such definitions of the Roman Pontiff from himself, but not from the consensus of the Church, are unalterable. But if anyone presumes to contradict this definition of Ours, which may God forbid: let him be anathema."<ref>Denzinger 3073–3075 (old numbering, 1839–1840</ref>

=== Second Vatican Council ===
In its [[Lumen Gentium|Dogmatic Constitution on the Church]] (1964), the [[Second Vatican Council]] declared:

"Among the principal duties of bishops the preaching of the Gospel occupies an eminent place. For bishops are preachers of the faith, who lead new disciples to Christ, and they are authentic teachers, that is, teachers endowed with the authority of Christ, who preach to the people committed to them the faith they must believe and put into practice, and by the light of the Holy Spirit illustrate that faith. They bring forth from the treasury of Revelation new things and old, making it bear fruit and vigilantly warding off any errors that threaten their flock. Bishops, teaching in communion with the Roman Pontiff, are to be respected by all as witnesses to divine and Catholic truth. In matters of faith and morals, the bishops speak in the name of Christ and the faithful are to accept their teaching and adhere to it with a religious assent. This religious submission of mind and will must be shown in a special way to the authentic magisterium of the Roman Pontiff, even when he is not speaking ex cathedra; that is, it must be shown in such a way that his supreme magisterium is acknowledged with reverence, the judgments made by him are sincerely adhered to, according to his manifest mind and will. His mind and will in the matter may be known either from the character of the documents, from his frequent repetition of the same doctrine, or from his manner of speaking.
[[File:GestatorialChair1.jpg|270px|left|thumb|[[Pope Pius XII]], wearing the traditional 1877 [[Papal Tiara]], is carried through St Peter's Basilica on a [[sedia gestatoria]] circa 1955.]]
... this infallibility with which the Divine Redeemer willed His Church to be endowed in defining doctrine of faith and morals, extends as far as the deposit of Revelation extends, which must be religiously guarded and faithfully expounded. And this is the infallibility which the Roman Pontiff, the head of the college of bishops, enjoys in virtue of his office, when, as the supreme shepherd and teacher of all the faithful, who confirms his brethren in their faith, by a definitive act he proclaims a doctrine of faith or morals. And therefore his definitions, of themselves, and not from the consent of the Church, are justly styled irreformable, since they are pronounced with the assistance of the Holy Spirit, promised to him in blessed Peter, and therefore they need no approval of others, nor do they allow an appeal to any other judgment. For then the Roman Pontiff is not pronouncing judgment as a private person, but as the supreme teacher of the universal Church, in whom the charism of infallibility of the Church itself is individually present, he is expounding or defending a doctrine of Catholic faith. The infallibility promised to the Church resides also in the body of Bishops, when that body exercises the supreme magisterium with the successor of Peter. To these definitions the assent of the Church can never be wanting, on account of the activity of that same Holy Spirit, by which the whole flock of Christ is preserved and progresses in unity of faith."<ref>[http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19641121_lumen-gentium_en.html ''Lumen gentium'', 25]</ref>

== Political role ==
{{main|Politics of the Vatican City}}
{{Infobox sovereignofvatican
|body = Sovereign of the State of the Vatican City
|insignia = Coat of arms of the Vatican City.svg
|insigniasize = 120px
|insigniacaption = Coat of Arms of the Vatican
|image = BentoXVI-28-10052007.jpg
|incumbent = [[Pope Benedict XVI|Benedict XVI]]
|style = [[His Holiness]]
|residence = [[Papal Palace]]
|firstsovereign = [[Pope Pius XI]]
|formation = 11 February 1929
|website = http://www.va
}}
[[File:PapalPolitics2.JPG|270px|left|thumb|''Antichristus'', a woodcut by Lucas Cranach of the pope using the temporal power to grant authority to a generously contributing ruler]]

Though the progressive [[Christianization|Christianisation]] of the [[Roman Empire]] in the fourth century did not confer upon bishops civil authority within the state, the gradual withdrawal of imperial authority during the fifth century left the pope the senior imperial civilian official in Rome, as bishops were increasingly directing civil affairs in other cities of the Western Empire. This status as a secular and civil ruler was vividly displayed by [[Pope Leo I]]'s confrontation with [[Attila]] in 452. The first expansion of papal rule outside of Rome came in 728 with the [[Donation of Sutri]], which in turn was substantially increased in 754, when the [[Frankish people|Frankish]] ruler [[Pippin the Younger]] gave to the pope the land from his conquest of the [[Lombards]]. The pope may have utilized the forged [[Donation of Constantine]] to gain this land, which formed the core of the [[Papal States]]. This document, accepted as genuine until the 1400s, states that [[Constantine I]] placed the entire Western Empire of Rome under papal rule. In 800 [[Pope Leo III]] [[coronation|crowned]] the Frankish ruler [[Charlemagne]] as [[Roman Emperor]], a major step toward establishing what later became known as the [[Holy Roman Empire]]; from that date onward the popes claimed the prerogative to crown the Emperor, though the right fell into disuse after the coronation of [[Charles V, Holy Roman Emperor|Charles V]] in 1530. [[Pope Pius VII]] was present at the coronation of [[Napoleon I]] in 1804, but did not actually perform the crowning. As mentioned above, the pope's sovereignty over the Papal States ended in 1870 with their annexation by [[Italy]].

Popes like [[Pope Alexander VI|Alexander VI]], an ambitious if spectacularly corrupt politician, and [[Pope Julius II]], a formidable general and statesman, were not afraid to use power to achieve their own ends, which included increasing the power of the papacy. This political and temporal authority was demonstrated through the papal role in the Holy Roman Empire (especially prominent during periods of contention with the Emperors, such as during the Pontificates of [[Pope Gregory VII]] and [[Pope Alexander III]]). [[Papal bull]]s, [[Interdict (Roman Catholic Church)|interdict]], and [[excommunication]] (or the threat thereof) have been used many times to increase papal power. The Bull ''[[Laudabiliter]]'' in 1155 authorized [[Henry II of England]] to invade [[Ireland]]. In 1207, [[Innocent III]] placed England under interdict until [[John of England|King John]] made his kingdom a [[fiefdom]] to the Pope, complete with yearly [[tribute]], saying, "we offer and freely yield...to our lord Pope Innocent III and his catholic successors, the whole kingdom of England and the whole kingdom of Ireland with all their rights and appurtenences for the remission of our sins".<ref>Quoted from the [http://www.fordham.edu/halsall/source/innIII-policies.html Medieval Sourcebook]</ref> The Bull ''[[Inter caetera]]'' in 1493 led to the [[Treaty of Tordesillas]] in 1494, which divided the world into areas of [[Spain|Spanish]] and [[Portugal|Portuguese]] rule. The Bull ''[[Regnans in Excelsis]]'' in 1570 excommunicated [[Elizabeth I of England]] and declared that all her subjects were released from all allegiance to her. The Bull ''[[Inter Gravissimas]]'' in 1582 established the [[Gregorian Calendar]].<ref>See [http://tera-3.ul.cs.cmu.edu/cgi-bin/getImage.pl?target=/data/www/NASD/4a7f1db4-5792-415c-be79-266f41eef20a/009/499/PTIFF/00000673.tif&rs=2 selection from ''Concordia Cyclopedia'': Roman Catholic Church, History of]</ref>

== Objections to the papacy ==
[[File:Antichrist1.jpg|thumb|right|230px|''Antichristus'', by [[Lucas Cranach the Elder]], from Luther's 1521 ''Passionary of the Christ and Antichrist''. The Pope is signing and selling [[indulgence]]s.]]

The Pope's claim to authority is either disputed or not recognised at all by other churches. The reasons for these objections differ from denomination to denomination.

===Orthodox, Anglican and Old Catholic churches===
Some Christian churches ([[Assyrian Church of the East]], the [[Oriental Orthodoxy|Oriental Orthodox Church]], the [[Eastern Orthodox Church]], the [[Old Catholic Church]], the [[Anglican Communion]], the [[Independent Catholic Churches]], etc.) accept the doctrine of [[Apostolic Succession]] and, to varying extents, papal claims to a primacy of honour while generally rejecting that the pope is the successor to Peter in any unique sense not true of any other bishop. Primacy is regarded as a consequence of the pope's position as bishop of the original capital city of the [[Roman Empire]], a definition explicitly spelled out in the 28th [[canon law|canon]] of the [[Council of Chalcedon]]. These churches see no foundation to papal claims of ''universal immediate jurisdiction'', or to claims of [[papal infallibility]]. Several of these churches refer to such claims as ''[[ultramontanism]]''.

===Protestant denominations===
{{main|Historicism (Christian eschatology)}}
Many Christian denominations reject the claims of [[Primacy of Simon Peter|Petrine primacy]] of honor, Petrine primacy of jurisdiction, and papal infallibility. These denominations vary from simply not accepting the Pope's claim to authority as legitimate and valid, to believing that the Pope is the [[Antichrist]]<ref>'Therefore on the basis of a renewed study of the pertinent Scriptures we reaffirm the statement of the Lutheran Confessions, that “the Pope is the very Antichrist”' from [http://www.wels.net/cgi-bin/site.pl?2617&collectionID=795&contentID=4441&shortcutID=5297 Statement on the Antichrist], from the [[Wisconsin Evangelical Lutheran Synod]], also [http://www.ianpaisley.org/antichrist.asp The Pope is the Antichrist]</ref> from [http://www.biblegateway.com/passage/?search=1%20John%202:18;&version=9; 1 John 2:18],<ref>[http://www.lcms.org/pages/internal.asp?NavID=579 Brief Statment]</ref> the [[Man of Sin]] from [http://www.biblegateway.com/passage/?search=2%20Thessalonians%202:3-12&version=9 2 Thessalonians 2:3-12],<ref>See Kretzmann's [http://www.kretzmannproject.org/EP_MINOR/2TH_2.htm ''Popular Commentary''], 2 Thessalonians chapter two and [http://www.wlsessays.net/authors/IJ/JeskeThessalonians/JeskeThessalonians.PDF An Exegesis of 2 Thessalonians 2:1-10] by Mark Jeske</ref> and the [[The Beast (Bible)|Beast out of the Earth]] from [http://www.biblegateway.com/passage/?search=Revelation%2013:11-18;&version=9; Revelation 13:11-18].<ref>See See Kretzmann's [http://www.kretzmannproject.org/REV/REV_13.htm ''Popular Commentary''], Revelation Chapter 13</ref> The sweeping rejection includes some denominations of Lutherans: [[Confessional Lutheran]]s hold that the pope is the Antichrist, stating that this article of faith is part of a ''quia'' rather than ''quatenus'' subscription to the [[Book of Concord]]. In 1932, the [[Lutheran Church - Missouri Synod]] (LCMS) adopted ''A Brief Statement of the Doctrinal Position of the Missouri Synod'', which a number of Lutheran church bodies now hold.<ref>The [[Lutheran Churches of the Reformation]][http://www.lcrusa.org/brief_statement.htm], the [[Concordia Lutheran Conference]][http://www.concordialutheranconf.com/clc/doctrine/brief_1932.cfm], the [[Church of the Lutheran Confession]][http://clclutheran.org/library/BriefStatement.html], and the Illinois Lutheran Conference [http://www.illinoislutheranconference.org/our-solid-foundation/doctrinal-position-of-the-ilc.lwp/odyframe.htm] all hold to ''Brief Statement'', which the LCMS adopted in 1932 and places in the [http://www.lcms.org/pages/internal.asp?NavID=579 LCMS.org website]</ref> Statement 43, ''Of the Antichrist'':<ref>Online at[http://www.lcms.org/pages/internal.asp?NavID=579 Of the Antichrist]</ref> [[File:ChristWashingFeet.JPG|thumb|left|250px|''Christus'', by Lucas Cranach. This woodcut of John 13:14–17 is from ''Passionary of the Christ and Antichrist''. Cranach shows Jesus kissing Peter's foot during the footwashing. This stands in contrast to the opposing woodcut, where the Pope demands others kiss his feet.]][[File:PopeKissing Feet.JPG|thumb|right|250px|''Antichristus'', by the Lutheran [[Lucas Cranach the Elder]]. This woodcut of the traditional practice of kissing the Pope's toe is from ''Passionary of the Christ and Antichrist''. The two fingers the Pope is holding up symbolizes his claim to be the Church's substitute for Christ's earthly presence.]]

<blockquote>43. As to the Antichrist we teach that the prophecies of the Holy Scriptures concerning the Antichrist, [http://www.biblegateway.com/passage/?search=2%20Thess.%202:3-12&version=9 2 Thess. 2:3-12];[http://www.biblegateway.com/passage/?search=1%20John%202:18;&version=9; 1 John 2:18], have been fulfilled in the Pope of Rome and his dominion. All the features of the Antichrist as drawn in these prophecies, including the most abominable and horrible ones, for example, that the Antichrist "as God sitteth in the temple of God," [http://www.biblegateway.com/passage/?search=2%20Thess.%202:4;&version=9; 2 Thess. 2:4]; that he anathematizes the very heart of the Gospel of Christ, that is, the doctrine of the forgiveness of sins by grace alone, for Christ's sake alone, through faith alone, without any merit or worthiness in man ([http://www.biblegateway.com/passage/?search=Rom.%203:20-28;&version=9; Rom. 3:20-28]; [http://www.biblegateway.com/passage/?search=Gal.%202:16;&version=9; Gal. 2:16]); that he recognizes only those as members of the Christian Church who bow to his authority; and that, like a deluge, he had inundated the whole Church with his antichristian doctrines till God revealed him through the Reformation—these very features are the outstanding characteristics of the Papacy. (Cf. [http://www.bookofconcord.com/smalcald.html#article4 Smalcald Articles, Triglot, p. 515, Paragraphs 39-41; p. 401, Paragraph 45; M. pp. 336, 258.]) Hence we subscribe to the statement of our Confessions that the Pope is "the very Antichrist." ([http://www.bookofconcord.com/smalcald.html#article4 Smalcald Articles, Triglot, p. 475, Paragraph 10; M., p. 308.])</blockquote>

The claim of temporal power over all secular governments, including territorial claims in Italy, raises objection.<ref>See the [http://books.google.com/books?id=Zr3lGJei6fkC&printsec=frontcover&source=gbs_summary_r#PPA168,M1 Baltimore Catechism] on the temporal power of the pope over governments and Innocent III's [http://www.fordham.edu/halsall/source/innIII-policies.html Letter to the prefect Acerbius and the nobles of Tuscany]. For objection to this, see the [http://www.archive.org/details/concordiacyclope009499mbp Concordia Cyclopedia], p.564 and 750</ref> The papacy's complex relationship with secular states such as the [[Roman Empire|Roman]] and [[Byzantine Empire|Byzantine]] Empires are also objections. Some disapprove of the autocratic character of the papal office.<ref>See Luther, [http://www.bookofconcord.com/smalcald.html#article4 Smalcald Articles, Article four]</ref> In [[Western Christianity]] these objections both contributed to and are products of the [[Protestant Reformation]].

== Antipopes ==
{{main|Antipope|Western Schism}}
Groups sometimes form around [[antipope]]s, who claim the Pontificate without being canonically and properly elected to it.

Traditionally, this term was reserved for claimants with a significant following of cardinals or other clergy. The existence of an antipope is usually due either to doctrinal controversy within the Church ([[heresy]]) or to confusion as to who is the legitimate pope at the time (see schism). Briefly in the 1400s, three separate lines of Popes claimed authenticity (see [[Western Schism|Papal Schism]]). Even Catholics don't all agree whether certain historical figures were Popes or antipopes. Though antipope movements were significant at one time, they are now overwhelmingly minor fringe causes.

== Other popes ==
In the earlier centuries of Christianity, the title "Pope," meaning "father," had been used by all bishops. Some popes used the term and others didn't. Eventually, the title became associated especially with the Bishop of Rome. In a few cases, the term is used for other Christian clerical authorities.

=== In the Roman Catholic Church ===
The "Black Pope" is a name that was popularly, but quite unofficially, given to the [[Superior General of the Society of Jesus]] due to the [[Society of Jesus|Jesuits']] in reference to the importance, within the Church, of the Jesuit order. This name, based on the black colour of his cassock, was used to suggest a parallel between him and the "White Pope" (since the time of [[Pope Pius V]] the Popes dress in white) and the Cardinal Prefect of the [[Congregation for the Evangelization of Peoples]] (formerly called the Sacred Congregation for the Propagation of the Faith), whose red cardinal's cassock gave him the name of the "Red Pope" in view of the authority over all territories that were not considered in some way Catholic. In the present time this cardinal has power over mission territories for Catholicism, essentially the Churches of Africa and Asia,<ref name = "Magister">[http://www.chiesa.espressonline.it/dettaglio.jsp?id=7049&eng=ylink Sandro Magister], Espresso Online.</ref> but in the past his competence extended also to all lands where [[Protestantism|Protestants]] or [[Eastern Christianity]] was dominant. Some remnants of this situation remain, with the result that, for instance, [[New Zealand]] is still in the care of this Congregation.

=== In the Eastern Churches ===
Since the papacy of [[Heraclas]] in the third century, the [[Metropolitan Archbishop|Bishop]] of the [[Coptic Orthodox Church of Alexandria|Alexandria]] in both the [[Coptic Orthodox Church of Alexandria]] and the [[Greek Orthodox Church of Alexandria]] continue to be called "Pope", the former being called "Coptic Pope" or, more properly, "[[Pope of the Coptic Orthodox Church of Alexandria|Pope and Patriarch of All Africa on the Holy Orthodox and Apostolic Throne of Saint Mark the Evangelist and Holy Apostle]]" and the last called "[[Greek Orthodox Patriarch of Alexandria|Pope and Patriarch of Alexandria and All Africa]]".

In the [[Bulgarian Orthodox Church]], [[Russian Orthodox Church]] and [[Serbian Orthodox Church]], it is not unusual for a village priest to be called a "pope" ("поп" ''pop''). However, this should be differentiated from the words used for the head of the Catholic Church (Bulgarian "папа" ''papa'', Russian "папа римский" ''papa rimskiy'').

== Longest-reigning popes ==
{{See also|List of popes by length of reign}}
[[File:Popepiusix.jpg|thumb|[[Pope Pius IX]], excluding Saint Peter, the longest-reigning pope]]

Although the average reign of the pope from the [[Middle Ages]] was a decade, a number of those whose reign lengths can be determined from contemporary historical data are the following:
# [[Pope Pius IX|Pius IX]] (1846–1878): 31 years, 7 months and 23 days (11,560 days).
# [[Pope John Paul II|John Paul II]] (1978–2005): 26 years, 5 months and 18 days (9,665 days).
# [[Pope Leo XIII|Leo XIII]] (1878–1903): 25 years, 5 months and 1 day (9,281 days).
# [[Pope Pius VI|Pius VI]] (1775–1799): 24 years, 6 months and 15 days (8,962 days).
# [[Pope Adrian I|Adrian I]] (772–795): 23 years, 10 months and 25 days (8,729 days).
# [[Pope Pius VII|Pius VII]] (1800–1823): 23 years, 5 months and 7 days (8,560 days).
# [[Pope Alexander III|Alexander III]] (1159–1181): 21 years, 11 months and 24 days (8,029 days).
# [[Pope Sylvester I|St. Sylvester I]] (314–335): 21 years, 11 months and 1 day (8,005 days).
# [[Pope Leo I|St. Leo I]] (440–461): 21 years, 1 month, and 13 days. (7,713 days).
# [[Pope Urban VIII|Urban VIII]] (1623–1644): 20 years, 11 months and 24 days (7,664 days).

[[Saint Peter]] is thought to have reigned for over thirty years (AD 29 – 64?/67?), but the exact length is not reliably known.

== Shortest-reigning popes ==

[[File:urban3355.jpg|thumb|[[Pope Urban VII]], the shortest-reigning pope]]
Conversely, there have been a number of popes whose reign lasted less than a month. In the following list the number of calendar days includes partial days. Thus, for example, if a pope's reign commenced on 1 August and he died on 2 August, this would count as having reigned for two calendar days.
#[[Pope Urban VII|Urban VII]] (15 September–27 September 1590): reigned for 13 calendar days, died before [[consecration]].
#[[Pope Boniface VI|Boniface VI]] (April, 896): reigned for 16 calendar days
#[[Pope Celestine IV|Celestine IV]] (25 October–10 November 1241): reigned for 17 calendar days, died before [[consecration]].
#[[Pope Theodore II|Theodore II]] (December, 897): reigned for 20 calendar days
#[[Pope Sisinnius|Sisinnius]] (15 January–4 February 708): reigned for 21 calendar days
#[[Pope Marcellus II|Marcellus II]] (9 April–1 May 1555): reigned for 22 calendar days
#[[Pope Damasus II|Damasus II]] (17 July–9 August 1048): reigned for 24 calendar days
#[[Pope Pius III|Pius III]] (22 September–18 October 1503): reigned for 27 calendar days
#[[Pope Leo XI|Leo XI]] (1 April–27 April 1605): reigned for 27 calendar days
#[[Pope Benedict V|Benedict V]] (22 May–23 June 964): reigned for 33 calendar days, [[Pope John Paul I|John Paul I]] (26 August–28 September 1978): reigned for 33 calendar days.

Note: [[Pope-elect Stephen|Stephen]] (23 March–26 March 752), died of [[apoplexy]] three days after his election, and before his [[consecration]] as a bishop. He is not recognized as a valid Pope, but was added to the lists of popes in the fifteenth century as ''Stephen II'', causing difficulties in enumerating later Popes named Stephen. He was removed in 1961 from the [[Vatican City|Vatican's]] [[List of Popes|list]] (see "[[Pope-elect Stephen]]" for detailed explanation).

== See also ==
{{col-begin}}
{{col-3}}
* [[African popes]]
* [[Caesaropapism]]
* [[History of the Papacy]]
* [[Investiture Controversy]]
* [[Leaders of Christianity]]
* [[Legends surrounding the Papacy]]
* [[List of canonised popes]]
* [[List of French popes]]
* [[List of German popes]]
* [[List of names of popes]]
{{Col-3}}
* [[List of popes]]
* [[List of popes by length of reign]]
* [[List of popes (graphical)]]
* [[Papal Coronation]]
* [[Papal Inauguration]]
* [[Papal regalia and insignia]]
* [[Papal Slippers]]
* [[Pontiff]]
* [[Prophecy of the Popes]]
* [[Sedevacantism]]
{{col-3}}
{{Christianityportal}}
{{Catholicismportal}}
{{Portalpar | Pope | Holysee-arms.svg | 35}}
{{Col-end}}

== Notes ==
{{reflist|2}}

== References ==
{{refbegin}}
* {{cite book|title=One Faith, One Lord: A Study of Basic Catholic Belief|last=Barry|first=Rev. Msgr. John F|year=2001|[[Nihil obstat]], [[Imprimatur]]|publisher=Gerard F. Baumbach, Ed.D|isbn=0-8215-2207-8|ref=harv}}
* {{cite book|title=A Concise History of the Catholic Church|last=Bokenkotter|first=Thomas|year=2004|publisher=Doubleday|isbn=0385505841|ref=harv}}
* {{cite encyclopedia|last=Chadwick|first=Henry|authorlink=Henry Chadwick (theologian)|editor=John McManners|encyclopedia=The Oxford Illustrated History of Christianity|title=The Early Christian Community|year=1990|publisher=Oxford University Press|isbn=0198229283|ref=harv}}
* {{cite book|title=Saints and Sinners, a History of the Popes|last=Duffy|first=Eamon|authorlink=Eamon Duffy|year=1997|publisher=Yale University Press|isbn=0-3000-7332-1|ref=harv}}
* {{cite book|last=Franzen|first=August|coauthors=John Dolan|title=A History of the Church|publisher=Herder and Herder|year=1969|ref=harv}}
* Hartmann Grisar (1845–1932), ''History of Rome and the Popes in the Middle Ages'', AMS Press; Reprint edition (1912). ISBN 0-404-09370-1
* {{cite book|last=Kelly|first=J. N.|title=Oxford Dictionary of the Popes|year=1986|publisher=Prentice Hall|isbn=9780191909351|ref=harv}}
* {{cite book|title=The Catholic Church: A Short History|last=Kung|first=Hans|authorlink=Hans Kung|year=2003|publisher=Random House|isbn=9780812967623|ref=harv}}
* {{cite book |author=Loomis, Louise Ropes |title=The Book of the Popes (Liber Pontificalis): To the Pontificate of Gregory I |location=[[Evolution Publishing]] |publisher=[[Merchantville, NJ]] |year=2006 |isbn=1-889758-86-8}}. Reprint of an English translation originally published in 1916.
* [[Ludwig von Pastor]], ''History of the Popes from the Close of the Middle Ages; Drawn from the [[Vatican Secret Archives|Secret Archives of the Vatican]] and other original sources'', 40 vols. St. Louis, B. Herder 1898 – ([http://www.worldcatlibraries.org/wcpa/ow/b92040657d7c02f6.html World Cat entry])
* {{cite book|last=Noble|first=Thomas|coauthors=Strauss, Barry|title=Western Civilization|year=2005|publisher=Houghton Mifflin Company|isbn=0618432779|ref=harv}}
* {{cite book|title=A Short History of the Catholic Church|last=Orlandis|first=Jose|authorlink=Jose Orlandis|year=1993|publisher=Scepter Publishers|isbn=1851821252|ref=harv}}
* [[James Joseph Walsh]], [http://books.google.com/books?vid=OCLC22760194&id=B-cQAAAAIAAJ&printsec=titlepage&dq=%22popes+and+science%22 ''The Popes and Science; the History of the Papal Relations to Science During the Middle Ages and Down to Our Own Time''], Fordam University Press, 1908, reprinted 2003, Kessinger Publishing. ISBN 0-7661-3646-9
{{refend}}

== Further reading ==
* Brusher, Joseph H. ''Popes Through The Ages''. Princeton: D. Van Nostland Company, Inc., 1959.
* Chamberlin, E.R. ''The Bad Popes''. 1969. Reprint: Barnes and Noble, 1993. ISBN 9780880291163.
* Dollison, John ''Pope-pourri''. New York: Simon & Schuster, 1994. ISBN 9780671886158.
* Kelly, J.N.D. ''The Oxford Dictionary of Popes''. Oxford: University Press, 1986. ISBN 0-19-213964-9.
* Maxwell-Stuart, P.G. ''Chronicle of the Popes: The Reign-by-Reign Record of the Papacy from St. Peter to the Present; with 308 Illustrations, 105 in Color''. London: Thames and Hudson, 1997. ISBN 0-500-01798-0.
<references group="nb" />

== External links ==
{{Commons|Pope}}
* [http://www.britannica.com/EBchecked/topic/441722/papacy "papacy."] Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 12 Nov. 2009.
* [http://www.vatican.va/holy_father/index.htm The Holy See - The Holy Father]—website for the past and present Holy Fathers (since [[Pope Leo XIII|Leo XIII]])
* [http://www.apostleshipofprayer.org/2008.html The Holy Father's 2008 Prayer Intentions]
* [http://www.newadvent.org/cathen/12260a.htm Catholic Encyclopedia entry]
* [http://kolonisera.rymden.nu/pope/popes.php?l=1 Pope Endurance League - Sortable list of Popes]
* [http://www.wlsessays.net/subjects/R/rsubind.htm#RomanCCPapacy Scholarly articles on the Roman Catholic Papacy from the Wisconsin Lutheran Seminary Library]
* [http://www.documentacatholicaomnia.eu/01_01_Magisterium_Paparum.html Data Base of more than 23,000 documents of the Popes in latin and modern languages]

{{Popes}}
{{Papal symbols and ceremonial}}
{{Vatican City topics}}
{{Catholicism}}
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[[Category:Episcopacy in Roman Catholicism]]
[[Category:Holy See| ]]
[[Category:Popes| ]]
[[Category:Religious leadership roles]]

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Benutzer:JonskiC/Kleinste-Quadrate-Schätzung

2010-01-27T12:56:07Z

Paul August: Reverted edits by 121.52.152.2 (talk) to last version by 128.93.58.17

The method of '''least squares''' is applied to approximate solutions of [[overdetermined system]]s, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly [[regression analysis]].

Least squares may be interpreted as a method of fitting data. The best fit, between modeled data and observed data, in its least-squares sense, is an instance of the model for which the sum of '''squared''' residuals has its '''least''' value, where a [[errors and residuals in statistics|residual]] is the difference between an observed value and the value provided by the model. The method was first described by [[Carl Friedrich Gauss]] around 1794.<ref name=brertscher>{{cite book|author = Bretscher, Otto|title = Linear Algebra With Applications, 3rd ed.|publisher = Prentice Hall|year = 1995|location = Upper Saddle River NJ}}</ref> Least squares corresponds to the [[maximum likelihood]] criterion if the experimental errors have a [[normal distribution]] and can also be derived as a [[method of moments (statistics)|method of moments]] estimator. Regression analysis is available in most [[statistical software]] packages.

The discussion is mostly presented in terms of [[linear]] functions but the use of least-squares is valid and practical for more general families of functions. For example, the [[Fourier series]] approximation of degree <math>n</math> is optimal in the least-squares sense, amongst all approximations in terms of [[trigonometric polynomial|trigonometric polynomials]] of degree <math>n</math>. Also, by iteratively applying local quadratic approximation to the likelihood (through the [[Fisher information]]), the least-squares method may be used to fit a [[generalized linear model]].

[[Image:Linear least squares2.png|right|thumb|The result of fitting a set of data points with a quadratic function.]]

==History==
===Context===
The method of least squares grew out of the fields of [[astronomy]] and [[geodesy]] as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the [[Age of Exploration]]. The accurate description of the behavior of celestial bodies was key to enabling ships to sail in open seas where before sailors had relied on land sightings to determine the positions of their ships.

The method was the culmination of several advances that took place during the course of the eighteenth century<ref name=stigler>{{cite book
| author = Stigler, Stephen M.
| title = The History of Statistics: The Measurement of Uncertainty Before 1900
| publisher = Belknap Press of Harvard University Press
| year = 1986
| location = Cambridge, MA
}}</ref>:

*The combination of different observations taken under the ''same'' conditions contrary to simply trying one's best to observe and record a single observation accurately. This approach was notably used by [[Tobias Mayer]] while studying the [[libration]]s of the moon.
*The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, perhaps first expressed by [[Roger Cotes]].
*The combination of different observations taken under ''different'' conditions as notably performed by [[Roger Joseph Boscovich]] in his work on the shape of the earth and [[Pierre-Simon Laplace]] in his work in explaining the differences in motion of [[Jupiter]] and [[Saturn]].
*The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved, developed by Laplace in his Method of Situation.

===The method itself===
[[Image:Carl Friedrich Gauss.jpg|thumb|120px|[[Carl Friedrich Gauss]]]]
[[Carl Friedrich Gauss]] is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen. [[Adrien-Marie Legendre|Legendre]] was the first to publish the method, however.

An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid [[Ceres (asteroid)|Ceres]]. On January 1, 1801, the Italian astronomer [[Giuseppe Piazzi]] discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, it was desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated [[Kepler's laws of planetary motion|Kepler's nonlinear equations]] of planetary motion. The only predictions that successfully allowed Hungarian astronomer [[Franz Xaver von Zach]] to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium''.
In 1829, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. This result is known as the [[Gauss–Markov theorem]].

The idea of least-squares analysis was also independently formulated by the Frenchman [[Adrien-Marie Legendre]] in 1805 and the American [[Robert Adrain]] in 1808.

==Problem statement==
The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of ''n'' points (data pairs) <math>(x_i,y_i)\!</math>, ''i'' = 1, ..., ''n'', where <math>x_i\!</math> is an [[independent variable]] and <math>y_i\!</math> is a [[dependent variable]] whose value is found by observation. The model function has the form <math>f(x,\boldsymbol \beta)</math>, where the ''m'' adjustable parameters are held in the vector <math>\boldsymbol \beta</math>. The parameter values for which the model "best" fits the data need be found. The least squares method finds its optimum when the sum, ''S'', of squared residuals
:<math>S=\sum_{i=1}^{n}{r_i}^2</math>
is a minimum. A [[errors and residuals in statistics|residual]] is defined as the difference between the value of the dependent variable and the predicted value from the estimated model,

:<math>r_i= y_i - f(x_i, \hat\boldsymbol \beta),\,</math>


An example of a model is that of the straight line. Denoting the intercept as <math>\beta_0</math> and the slope as <math>\beta_1</math>, the model function is given by

:<math>f(x,\boldsymbol \beta)=\beta_0+\beta_1 x.\,</math>

See the [[linear least squares#Motivational_example|example of linear least squares]] for a fully worked out example of this model.

A data point may consist of more than one independent variable. For an example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, ''x'' and ''z'', say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point. 

==Solving the least squares problem==
Least squares problems fall into two categories, linear and non-linear. The linear least squares problem has a closed form solution, but the non-linear problem does not and is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, thus the core calculation is similar in both cases.

The [[Maxima and minima|minimum]] of the sum of squares is found by setting the [[gradient]] to zero. Since the model contains ''m'' parameters there are ''m'' gradient equations.

:<math>\frac{\partial S}{\partial \beta_j}=2\sum_i r_i\frac{\partial r_i}{\partial \beta_j}=0,\ j=1,\ldots,m</math>

and since <math>r_i=y_i-f(x_i,\boldsymbol \beta)\,</math> the gradient equations become

:<math>-2\sum_i \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} r_i=0,\ j=1,\ldots,m</math>

The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.

=== Linear least squares ===
{{main|Linear least squares}}
A regression model is a linear one when the model comprises a [[linear combination]] of the parameters, i.e.

:<math> f(x_i, \beta) = \sum_{j = 1}^{m} \beta_j \phi_j(x_{i})</math>

where the coefficients, <math>\phi_{j}</math>, are functions of <math> x_{i} </math>.

Letting

:<math> X_{ij}= \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j}= \phi_j(x_{i}) . \, </math>

we can then see that in that case the least square estimate (or estimator, in the context of a random sample), <math> \boldsymbol \beta</math> is given by

:<math> \boldsymbol{\hat\beta} =( X ^TX)^{-1}X^{T}\boldsymbol y </math>

For a derivation of this estimate see [[Linear least squares]].

=== Non-linear least squares ===
{{main|Non-linear least squares}}
There is no closed-form solution to a non-linear least squares problem. Instead, numerical algorithms are used to find the value of the parameters <math>\beta</math> which minimize the objective. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation.
:<math>{\beta_j}^{k+1}={\beta_j}^k+\Delta \beta_j</math>
''k'' is an iteration number and the vector of increments, <math>\Delta \beta_j\,</math> is known as the shift vector. In some commonly used algorithms, at each iteration the model may be linearized by approximation to a first-order [[Taylor series]] expansion about <math> \boldsymbol \beta^k\!</math>

:<math>
\begin{align}
f(x_i,\boldsymbol \beta) & = f^k(x_i,\boldsymbol \beta) +\sum_j \frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} \left(\beta_j-{\beta_j}^k \right) \\
& = f^k(x_i,\boldsymbol \beta) +\sum_j J_{ij} \Delta\beta_j.
\end{align}
</math>

The [[Jacobian]], '''J''', is a function of constants, the independent variable ''and'' the parameters, so it changes from one iteration to the next. The residuals are given by

:<math>r_i=y_i- f^k(x_i,\boldsymbol \beta)- \sum_{j=1}^{m} J_{ij}\Delta\beta_j=\Delta y_i- \sum_{j=1}^{m} J_{ij}\Delta\beta_j</math>.

To minimize the sum of squares of <math>r_i</math>, the gradient equation is set to zero and solved for <math> \Delta \beta_j\!</math>

:<math>-2\sum_{i=1}^{n}J_{ij} \left( \Delta y_i-\sum_{j=1}^{m} J_{ij}\Delta \beta_j \right)=0</math>

which, on rearrangement, become ''m'' simultaneous linear equations, the '''normal equations'''.

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} J_{ij}J_{ik}\Delta \beta_k=\sum_{i=1}^{n} J_{ij}\Delta y_i \qquad (j=1,\ldots,m)\,</math>

The normal equations are written in matrix notation as

:<math>\mathbf{\left(J^TJ\right)\Delta \boldsymbol \beta=J^T\Delta y}.\,</math>


These are the defining equations of the [[Gauss–Newton algorithm]].

=== Differences between linear and non-linear least squares ===
* The model function, ''f'', in LLSQ (linear least squares) is a linear combination of parameters of the form <math>f = X_{i1}\beta_1 + X_{i2}\beta_2 +\cdots</math> The model may represent a straight line, a parabola or any other polynomial-type function. In NLLSQ (non-linear least squares) the parameters appear as functions, such as <math>\beta^2, e^{\beta x}</math> and so forth. If the derivatives <math>\partial f /\partial \beta_j</math> are either constant or depend only on the values of the independent variable, the model is linear in the parameters. Otherwise the model is non-linear.
*Many solution algorithms for NLLSQ require initial values for the parameters, LLSQ does not.
*Many solution algorithms for NLLSQ require that the Jacobian be calculated. Analytical expressions for the partial derivatives can be complicated. If analytical expressions are impossible to obtain the partial derivatives must be calculated by numerical approximation.
*In NLLSQ non-convergence (failure of the algorithm to find a minimum) is a common phenomenon whereas the LLSQ is globally concave so non-convergence is not an issue.
*NLLSQ is usually an iterative process. The iterative process has to be terminated when a convergence criterion is satisfied. LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the [[Gauss–Seidel]] method.
*In LLSQ the solution is unique, but in NLLSQ there may be multiple minima in the sum of squares.
*Under the condition that the errors are uncorrelated with the predictor variables, LLSQ yields unbiased estimates, but even under that condition NLLSQ estimates are generally biased.
These differences must be considered whenever the solution to a non-linear least squares problem is being sought.

==Least squares, regression analysis and statistics==
The methods of least squares and [[regression analysis]] are conceptually different. However, the method of least squares is often used to generate estimators and other statistics in regression analysis.

Consider a simple example drawn from physics. A spring should obey [[Hooke's law]] which states that the extension of a spring is proportional to the force, ''F'', applied to it.
:<math>f(F_i,k)=kF_i\!</math>
constitutes the model, where ''F'' is the independent variable. To estimate the [[force constant]], ''k'', a series of ''n'' measurements with different forces will produce a set of data, <math>(F_i, y_i), i=1,n\!</math>, where ''yi'' is a measured spring extension. Each experimental observation will contain some error. If we denote this error <math>\varepsilon</math>, we may specify an empirical model for our observations,

: <math> y_i = kF_i + \varepsilon_i. \, </math>

There are many methods we might use to estimate the unknown parameter ''k''. Noting that the ''n'' equations in the ''m'' variables in our data comprise an [[overdetermined system]] with one unknown and ''n'' equations, we may choose to estimate ''k'' using least squares. The sum of squares to be minimized is

:<math> S = \sum_{i=1}^{n} \left(y_i - kF_i\right)^2. </math>

The least squares estimate of the force constant, ''k'', is given by

:<math>\hat k=\frac{\sum_i F_i y_i}{\sum_i {F_i}^2}.</math>

Here it is assumed that application of the force '''''causes''''' the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.

In regression analysis the researcher specifies an empirical model. For example, a very common model is the straight line model which is used to test if there is a linear relationship between dependent and independent variable. If a linear relationship is found to exist, the variables are said to be [[correlated]]. However, correlation does not prove causation, as both variables may be correlated with other, hidden, variables, or the dependent variable may "reverse" cause the independent variables, or the variables may be otherwise spuriously correlated. For example, suppose there is a correlation between deaths by drowning and the volume of ice cream sales at a particular beach. Yet, both the number of people going swimming and the volume of ice cream sales increase as the weather gets hotter, and presumably the number of deaths by drowning is correlated with the number of people going swimming. Perhaps an increase in swimmers causes both the other variables to increase.

In order to make statistical tests on the results it is necessary to make assumptions about the nature of the experimental errors. A common (but not necessary) assumption is that the errors belong to a [[Normal distribution]]. The [[central limit theorem]] supports the idea that this is a good assumption in many cases.
* The [[Gauss–Markov theorem]]. In a linear model in which the errors have [[expectation]] zero conditional on the independent variables, are [[uncorrelated]] and have equal [[variance]]s, the best linear [[unbiased]] estimator of any linear combination of the observations, is its least-squares estimator. "Best" means that the least squares estimators of the parameters have minimum variance. The assumption of equal variance is valid when the errors all belong to the same distribution.
*In a linear model, if the errors belong to a [[Normal distribution]] the least squares estimators are also the [[linear model#maximum likelihood|maximum likelihood]] estimators.

However, if the errors are not normally distributed, a [[central limit theorem]] often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large. For this reason, given the important property that the error is mean independent of the independent variables, the distribution of the error term is not an important issue in regression analysis. Specifically, it is not typically important whether the error term follows a normal distribution.

In a least squares calculation with unit weights, or in linear regression, the variance on the ''j''th parameter,
denoted <math>\text{var}(\hat{\beta}_j)</math>, is usually estimated with

:<math>\text{var}(\hat{\beta}_j)= \sigma^2\left( \left[X^TX\right]^{-1}\right)_{jj} \approx \frac{S}{n-m}\left( \left[X^TX\right]^{-1}\right)_{jj},</math>
where the true residual variance σ2 is replaced by an estimate based on the minimised value of the sum of squares objective function ''S''.

[[Confidence limits]] can be found if the [[probability distribution]] of the parameters is known, or an asymptotic approximation is made, or assumed. Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed. The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed. Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.

==Weighted least squares==
:''See also: [[Weighted mean]]''
The expressions given above are based on the implicit assumption that the errors are uncorrelated with each other and with the independent variables and have equal variance. The [[Gauss–Markov theorem]] shows that, when this is so, <math>\hat\boldsymbol\beta</math> is a [[best linear unbiased estimator]] (BLUE). If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted. [[Alexander Aitken|Aitken]] showed that when a weighted sum of squared residuals is minimized, <math>\hat\boldsymbol\beta</math> is BLUE if each weight is equal to the reciprocal of the variance of the measurement.
:<math> S = \sum_{i=1}^{n} W_{ii}{r_i}^2,\qquad W_{ii}=\frac{1}{{\sigma_i}^2} </math>
The gradient equations for this sum of squares are

:<math>-2\sum_i W_{ii}\frac{\partial f(x_i,\boldsymbol \beta)}{\partial \beta_j} r_i=0,\qquad j=1,\ldots,n</math>

which, in a linear least squares system give the modified normal equations

:<math>\sum_{i=1}^{n}\sum_{k=1}^{m} X_{ij}W_{ii}X_{ik}\hat \beta_k=\sum_{i=1}^{n} X_{ij}W_{ii}y_i, \qquad j=1,\ldots,m\,</math>

or

:<math>\mathbf{\left(X^TWX\right)\hat \boldsymbol \beta=X^TWy}.</math>

When the observational errors are uncorrelated the weight matrix, '''W''', is diagonal. If the errors are correlated, the resulting estimator is BLUE if the weight matrix is equal to the inverse of the [[variance-covariance matrix]] of the observations.

When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as <math>w_{ii}=\sqrt W_{ii}</math>. The normal equations can then be written as

:<math>\mathbf{\left(X'^TX'\right)\hat \boldsymbol \beta=X'^Ty'}\,</math>

where

: <math>\mathbf{X'}=\mathbf{wX}, \mathbf{y'}=\mathbf{wy}.\,</math>

For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.

:<math>\mathbf{\left(J^TWJ\right)\boldsymbol \Delta\beta=J^TW \boldsymbol\Delta y}.\,</math>

Note that for empirical tests, the appropriate '''W''' is not known for sure and must be
estimated. For this [[Feasible Generalized Least Squares]] (FGLS) techniques may be used.

===Principal components===
The first principal component about the mean of a set of points is equivalent to the linear least squares solution. One of the most computationally efficient ways to solve a linear least squares problem is to use the [[Principal_component_analysis#Computing_Principal_Components_with_Expectation_Maximization|EM technique]] to compute the first principal component about the mean of the data. This algorithm can be trivially modified to compute a weighted least squares solution as well.

===Lasso method===
In some contexts a [[Regularization (machine learning)|regularized]] version of the least squares solution may be preferable. The ''LASSO'' algorithm, for example, finds a least-squares solution with the constraint that <math>|\beta|_1</math>, the [[L1-norm|L1-norm]] of the parameter vector, is no greater than a given value. Equivalently, it may solve an unconstrained minimization of the least-squares penalty with <math>\alpha|\beta|_1</math> added, where <math>\alpha</math> is a constant. (This is the [[Lagrange multipliers|Lagrangian]] form of the constrained problem.) This problem may be solved using [[quadratic programming]] or more general [[convex optimization]] methods, but is most efficiently solved using the results of the [[least angle regression]] algorithm. The L1-regularized formulation is useful in some contexts due to its tendency to prefer solutions with fewer nonzero parameter values, effectively reducing the number of variables upon which the given solution is dependent <ref> Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Royal. Statist. Soc B., Vol. 58, No. 1, pages 267–288</ref>.

==See also==
* [[L2 norm|''L''2 norm]]
* [[Least absolute deviation]]
* [[Measurement uncertainty]]
* [[Root mean square]]
* [[Squared deviations]]
* [[Iteratively re-weighted least squares]]
* [[Total least squares]], aka orthogonal regression
* [[Levenberg–Marquardt algorithm]]
* [[Regression analysis]]
* [[Partial least squares regression]]
* [[Best linear unbiased prediction]] (BLUP)

==Notes==
<references />

==References==
*Å. Björck, ''Numerical Methods for Least Squares Problems'', SIAM, 1996 [http://www.ec-securehost.com/SIAM/ot51.html].
*C.R. Rao, H. Toutenburg, A. Fieger, C. Heumann, T. Nittner and S. Scheid, ''Linear Models: Least Squares and Alternatives'', Springer Series in Statistics, 1999.
*T. Kariya, H. Kurata, ''Generalized Least Squares'', Wiley, 2004.
*J. Wolberg, ''Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments'', Springer, 2005.

==External links==
* [http://www.personal.psu.edu/faculty/j/h/jhm/f90/lectures/lsq2.html Derivation of quadratic least squares]
* [http://www2.uta.edu/infosys/baker/STATISTICS/Keller7/Keller%20PP%20slides-7/Chapter17.ppt Power Point Statistics Book] -- Excellent slides providing an introductory regression example (University of Texas at Arlington)

{{Least Squares and Regression Analysis}}

[[Category:Applied mathematics]]
[[Category:Mathematical optimization]]
[[Category:Statistical methods]]
[[Category:Regression analysis]]
[[Category:Single equation methods (econometrics)]]
[[Category:Mathematical and quantitative methods (economics)]]

[[af:Kleinste-kwadratemetode]]
[[ca:Mínims quadrats ordinaris]]
[[cs:Metoda nejmenších čtverců]]
[[de:Methode der kleinsten Quadrate]]
[[es:Mínimos cuadrados]]
[[eu:Karratu txikienen erregresio]]
[[fa:کمترین مربعات]]
[[fr:Méthode des moindres carrés]]
[[gl:Mínimos cadrados]]
[[hi:न्यूनतम वर्ग की विधि]]
[[it:Metodo dei minimi quadrati]]
[[he:שיטת הריבועים הפחותים]]
[[la:Methodus quadratorum minimorum]]
[[hu:Legkisebb négyzetek módszere]]
[[nl:Kleinste-kwadratenmethode]]
[[ja:最小二乗法]]
[[pl:Metoda najmniejszych kwadratów]]
[[pt:Método dos mínimos quadrados]]
[[ru:Метод наименьших квадратов]]
[[su:Kuadrat leutik]]
[[fi:Pienimmän neliösumman menetelmä]]
[[sv:Minstakvadratmetoden]]
[[tr:En küçük kareler yöntemi]]
[[uk:Метод найменших квадратів]]
[[vi:Bình phương tối thiểu]]
[[zh:最小二乘法]]

0,999…

2010-01-21T01:03:55Z

Paul August: moved Uishshiusfhdui to 0.999...: vandalism

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
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*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:59:42Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by Paul August

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:58:20Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by Paul August

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
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*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
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*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
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*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
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*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
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*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
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*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:56:47Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by Paul August

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:55:31Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by Paul August

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:54:50Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by Paul August

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:53:46Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by Paul August

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
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[[Category:Articles containing proofs]]

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0,999…

2010-01-21T00:50:13Z

Paul August: Reverted edits by Qwertyisbest (talk) to last version by TheDJ

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] equal to [[1 (number)|the number '''one''']]. In other words, the notations ''0.999…'' and ''1'' represent the same number within the real number system. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

The ''0.999...=1'' [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by [[counterintuitive]] behavior of the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates.

Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit of a sequence|limits]]. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = {{frac|1|3}}, 0.111… = {{frac|1|9}}, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × {{frac|1|3}} equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies {{frac|1|9}} = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9''x''. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000… both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==0.999... in alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

Karin Katz and [[Mikhail Katz]] have developed an alternative to the unital evaluation of the symbol "0.999..." The alternative evaluation is
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>\langle\mathbb{N}\rangle</math> is the sequence <math>\langle1,2,3,\ldots\rangle</math> listing all the natural numbers in increasing order, while <math>[\mathbb{N}]</math> is the infinite [[hypernatural]] represented by the sequence, in the [[ultrapower]] construction; see Katz & Katz (2010).

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]

{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
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*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]
*Katz, K.; [[Mikhail Katz|Katz, M.]] (2010) When is .999... less than 1? [[The Montana Mathematics Enthusiast]], Vol. 7, No. 1, pp. 3--30. http://www.math.umt.edu/TMME/vol7no1/

{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Quadratwurzel aus 5

2009-10-18T14:57:40Z

Paul August: remove infobox (please see talk page)

The '''square root of 5''' is the positive [[real number]] that, when multiplied by itself, gives the prime number [[5 (number)|5]]. This number appears in the formula for the [[golden ratio]]. It can be denoted in [[nth root#Working with surds|surd]] form as:

:<math>\sqrt{5}. \, </math>

It is an [[irrational number|irrational]] [[algebraic number]].<ref>Dauben, Joseph W. (June 1983) [[Scientific American]] ''Georg Cantor and the origins of transfinite set theory.'' Volume 248; Page 122.</ref> The first sixty significant digits of its [[decimal expansion]] are:

:2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089... {{OEIS|id=A002163}}

which can be rounded down to 2.236 to within 99.99% accuracy. As of April 1994, its numerical value in decimal had been computed to at least one million digits.<ref>R. Nemiroff and J. Bonnell: ''[http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt5.1mil The first 1 million digits of the square root of 5]''</ref>

== Continued fraction ==

It can be expressed as the [[continued fraction]] [2; 4, 4, 4, 4, 4...] {{OEIS|id=A040002}}. The sequence of [[continued fraction#Best rational approximations|best rational approximations]] is:

:<math>{\color{OliveGreen}\frac{2}{1}}, \frac{7}{3} , {\color{OliveGreen}\frac{9}{4}} , \frac{20}{9} , \frac{29}{13} , {\color{OliveGreen}\frac{38}{17}} , \frac{123}{55} , {\color{OliveGreen}\frac{161}{72}} , \frac{360}{161} , \frac{521}{233} , {\color{OliveGreen}\frac{682}{305}} , \frac{2207}{987} , {\color{OliveGreen}\frac{2889}{1292}}, \dots</math>

[[Convergent (continued fraction)|Convergent]]s of the continued fraction are colored; their numerators are sequence {{OEIS url|id=A001077}}, and their denominators are sequence {{OEIS url|id=A001076}}. The other (non-colored) terms are [[continued fraction#Semiconvergents|semiconvergents]].

==Babylonian method==

When <math>\sqrt{5}</math> is computed with the [[Methods of computing square roots#Babylonian method|Babylonian method]], starting with ''r''0 = 2 and using ''r''''n''+1 = (''r''''n'' + 5/''r''''n'') / 2, the ''n''th approximant ''r''''n'' is equal to the 2''n''-th convergent of the convergent sequence:

:<math>\frac{2}{1} = 2.0,\quad \frac{9}{4} = 2.25,\quad \frac{161}{72} = 2.23611\dots,\quad \frac{51841}{23184} = 2.2360679779 \ldots</math>

== Relation to the golden ratio and Fibonacci numbers==
[[Image:Golden Rectangle Construction.svg|thumb|The {{fraction|√5|2}} diagonal of a half square forms the basis for the geometrical construction of a [[golden rectangle]].]]
This [[golden ratio]] φ is the [[arithmetic mean]] of [[1 (number)|1]] and the square root of 5.<ref>Browne, Malcolm W. (July 30, 1985) [[New York Times]] ''Puzzling Crystals Plunge Scientists into Uncertainty.'' Section: C; Page 1. (Note - this is a widely cited article).</ref> The [[algebra]]ic relationship between the square root of 5, the golden ratio and the [[golden ratio#Conjugate golden ratio|conjugate golden ratio]] (Φ = {{fraction|1|φ}} = φ − 1) are expressed in the following formulae:{{Fact|date=August 2007}}

:<math>\sqrt{5} = \varphi + \Phi = 2\varphi - 1 = 2\Phi + 1</math>

:<math>\varphi = \frac{1 + \sqrt{5}}{2}</math>

:<math>\Phi = \frac{\sqrt{5} - 1}{2}.</math>

(See section below for their geometrical interpretation as decompositions of a root-5 rectangle.)

The square root of 5 then naturally figures in the closed form expression for the [[Fibonacci number]]s, a formula which is usually written in terms of the golden ratio:
:<math>F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}.</math>

The quotient of √5 and φ (or the product of √5 and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the [[Lucas number]]s:<ref>[[Richard K. Guy]]: "The Strong Law of Small Numbers". ''[[American Mathematical Monthly]]'', vol. 95, 1988, pp. 675–712</ref>

:<math>\frac{\sqrt{5}}{\varphi} = \Phi \cdot \sqrt{5} = \frac{5 - \sqrt{5}}{2} = 1.3819660112501051518\dots = [1; 2, 1, 1, 1, 1, 1, 1, 1, \dots]</math>

:<math>\frac{\varphi}{\sqrt{5}} = \frac{1}{\Phi \cdot \sqrt{5}} = \frac{2}{5 - \sqrt{5}} = 0.72360679774997896964\dots = [0; 1, 2, 1, 1, 1, 1, 1, 1, \dots].</math>

The series of convergents to these values feature the series of Fibonacci numbers and the series of [[Lucas number]]s as numerators and denominators, and viceversa, respectively:

:<math>{1, \frac{3}{2}, \frac{4}{3}, \frac{7}{5}, \frac{11}{8}, \frac{18}{13}, \frac{29}{21}, \frac{47}{34}, \frac{76}{55}, \frac{123}{89}}, \dots \dots [1; 2, 1, 1, 1, 1, 1, 1, 1, \dots]</math>

:<math>{1, \frac{2}{3}, \frac{3}{4}, \frac{5}{7}, \frac{8}{11}, \frac{13}{18}, \frac{21}{29}, \frac{34}{47}, \frac{55}{76}, \frac{89}{123}}, \dots \dots [0; 1, 2, 1, 1, 1, 1, 1, 1,\dots].</math>

== Geometry ==

[[geometry|Geometrically]], the square root of 5 corresponds to the [[diagonal]] of a [[rectangle]] whose sides are of length [[1 (number)|1]] and [[2 (number)|2]], as is evident from the [[Pythagorean theorem]]. Such a rectangle can be obtained by halving a [[Square (geometry)|square]], or by placing two equal squares side by side. Together with the algebraic relationship between √5 and φ, this forms the basis for the geometrical construction of a [[golden rectangle]] from a square, and for the construction of a regular [[pentagon]] given its side (since the side-to-diagonal ratio in a regular pentagon is φ).

Forming a [[dihedral angle|dihedral]] [[right angle]] with the two equal squares that halve a 1:2 rectangle, it can be seen that √5 corresponds also to the ratio between the length of a [[cube]] [[edge (geometry)|edge]] and the shortest distance from one of its [[vertex (geometry)|vertices]] to the opposite one, when traversing the cube ''surface'' (the shortest distance when traversing through the ''inside'' of the cube corresponds to the length of the cube diagonal, which is the [[square root of three]] times the edge).{{Fact|date=August 2007}}

The number √5 can be algebraically and geometrically related to the [[square root of 2]] and the [[square root of 3]], as it is the length of the [[hypotenuse]] of a right triangle with [[cathetus|catheti]] measuring √2 and √3 (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the [[centre (geometry)|centre]] point of a cube, one of its vertices, and the middle point of a side located on one the faces containing that vertex and opposite to it, are in the ratio √2:√3:√5. This follows from the geometrical relationships between a cube and the quantities √2 (edge-to-face-diagonal ratio, or distance between opposite edges), √3 (edge-to-cube-diagonal ratio) and √5 (the relationship just mentioned above).

A rectangle with side proportions 1:√5 is called a ''root-five rectangle'' and is part of the series of root rectangles, a subset of [[dynamic rectangle]]s, which are based on √1 (= 1), √2, √3, √4 (= 2), √5... and successively constructed using the diagonal of the previous root rectangle, starting from a square.<ref>{{cite book | url = http://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&ots=8ZNc5ZKfTG&dq=intitle:%22Geometry+of+Design%22+%22root+5%22&sig=YitS7tv3b4_r87coR4s7EcjL4kk | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1568982496 }}</ref> A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ).<ref>{{cite book | title = The Elements of Dynamic Symmetry
| author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0486217760 | url = http://books.google.com/books?id=VYJK2F-dh2oC&pg=PA26&ots=MqxrsVLmIH&dq=%22root+five+rectangle%22++section+inauthor:hambidge&sig=meu0juFja5gpsjHKk_gG1stMbYo#PPA27,M1 }}</ref> It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between √5, φ and Φ mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length {{fraction|√5|2}} to both sides.

== Trigonometry ==

Like √2 and √3, the square root of five appears extensively in the formulae for [[exact trigonometric constants]], and as such the computation of its value is important for [[generating trigonometric tables]].{{Fact|date=August 2007}} Since √5 is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a [[dodecahedron]].{{Fact|date=August 2007}}

== Diophantine approximations ==
[[Hurwitz's theorem]] in [[Diophantine approximations]] states that every [[irrational number]] ''x'' can be approximated by infinitely many [[rational number]]s ''m''/''n'' in [[lowest terms]] in such a way that

:<math> \left|x - \frac{m}{n}\right| < \frac{1}{\sqrt{5}\,n^2} </math>

and that √5 is best possible, in the sense that for any larger constant than √5, there are some irrational numbers ''x'' for which only finitely many such approximations exist.<ref>{{Citation | last1=LeVeque | first1=William Judson | title=Topics in number theory | publisher=Addison-Wesley Publishing Co., Inc., Reading, Mass. | id={{MathSciNet | id = 0080682}} | year=1956}}</ref>

Closely related to this is the theorem<ref name=khinchin/> that of any three consecutive [[convergent (continued fraction)|convergent]]s
''p''''i''/''q''''i'',
''p''''i''+1/''q''''i''+1,
''p''''i''+2/''q''''i''+2,
of a number α, at least one of the three inequalities holds:

:<math>\left|\alpha - {p_i\over q_i}\right| < {1\over \sqrt5 q_i^2}, \qquad
\left|\alpha - {p_{i+1}\over q_{i+1}}\right| < {1\over \sqrt5 q_{i+1}^2}, \qquad
\left|\alpha - {p_{i+2}\over q_{i+2}}\right| < {1\over \sqrt5 q_{i+2}^2}.
</math>

And the √5 in the denominator is the best bound possible since the convergents of the [[golden ratio]] make the difference on the left-hand side arbitrarily close to the value on the right-hand side. In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.<ref name=khinchin>{{Citation | last1=[[A. Ya. Khinchin|Khinchin]] | first1=Aleksandr Yakovlevich | title=Continued Fractions | publisher = University of Chicago Press, Chicago and London | year = 1964}}</ref>

== Algebra ==

The [[ring (mathematics)|ring]] <math>\scriptstyle\mathbb{Z}\left[\,\sqrt{-5}\,\right]</math> contains numbers of the form <math>\scriptstyle a\, +\, b\sqrt{-5}</math>, where ''a'' and ''b'' are [[integer]]s and <math>\scriptstyle \sqrt{-5}</math> is the [[imaginary number]] <math>\scriptstyle i\sqrt{5}</math>. This ring is a frequently cited example of an [[integral domain]] that is not a [[unique factorization domain]].{{Fact|date=August 2007}} The number 6 has two inequivalent factorizations within this ring:

: <math>6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}).</math>

The [[field (mathematics)|field]] <math>\scriptstyle\mathbb{Q}\left[\,\sqrt{5}\,\right]</math>, like any other [[quadratic field]], is an [[abelian extension]] of the rational numbers. The [[Kronecker–Weber theorem]] therefore guarantees that the square root of five can be written as a rational linear combination of [[roots of unity]]:

:<math>\sqrt5 = e^{2\pi i/5} - e^{4\pi i/5} - e^{6\pi i/5} + e^{8\pi i/5}.</math>

== Identities of Ramanujan ==

The square root of 5 appears in various identities of [[Srinivasa Ramanujan|Ramanujan]] involving [[continued fraction]]s.<ref>{{Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction | id={{MathSciNet | id = 813071}} | year=1984 | journal=Indian Academy of Sciences. Proceedings. Mathematical Sciences | issn=0253-4142 | volume=93 | issue=2 | pages=67--77 | doi=10.1007/BF02840651}}</ref><ref>{{Citation | url=http://mathworld.wolfram.com/RamanujanContinuedFractions.html | author=Eric W. Weisstein | title=Ramanujan Continued Fractions}} at [[MathWorld]]</ref>

For example, this case of the [[Rogers–Ramanujan continued fraction]]:

:<math>
\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1 + \ddots}}}}
= \left( \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2} \right)e^{2\pi/5} = e^{2\pi/5}\left( \sqrt{\varphi\sqrt{5}} - \varphi \right).
</math>



:<math>
\cfrac{1}{1 + \cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1 + \cfrac{e^{-6\pi\sqrt{5}}}{1 + \ddots}}}}
= \left( {\sqrt{5} \over 1 + \left[5^{3/4}(\varphi - 1)^{5/2} - 1\right]^{1/5}} - \varphi \right)e^{2\pi/\sqrt{5}}.
</math>



:<math>
4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx
= \cfrac{1}{1 + \cfrac{1^2}{1 + \cfrac{1^2}{1 + \cfrac{2^2}{1 + \cfrac{2^2}{1 + \cfrac{3^2}{1 + \cfrac{3^2}{1 + \ddots}}}}}}}.
</math>

==See also==


*[[Golden ratio]]
*[[Square root]]
*[[Square root of 2]]
*[[Square root of 3]]

==References==
{{reflist}}

[[Category:Mathematical constants]]
[[Category:Irrational numbers]]
[[Category:Algebraic numbers]]

[[es:Raíz cuadrada de 5]]
[[fr:Racine carrée de cinq]]
[[zh:5的平方根]]

Quadratwurzel aus 5

2009-10-18T13:45:42Z

Paul August: undo my last edit

{| border="1" style="float: right; border-collapse: collapse; margin-left: 1em"
| colspan="2" align="center" | [[List of numbers]] [[Euler-Mascheroni constant|γ]] - [[Apéry's constant|ζ(3)]] - [[Square root of 2|√2]] - [[Square root of 3|√3]] - <math>\sqrt{5}</math> - [[Golden ratio|φ]] - [[Feigenbaum constants|α]] - [[E (mathematical constant)|e]] - [[Pi|π]] - [[Feigenbaum constants|δ]]
|-
|[[Binary numeral system|Binary]]
| 10.0011110001101111...
|-
| [[Decimal]]
| 2.23606797749978969...
|-
| [[Hexadecimal]]
| 2.3C6EF372FE94F82C...
|-
| [[Continued fraction]]
| <math>2 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \ddots\qquad{}}}}}</math>
|}

The '''square root of 5''' is the positive [[real number]] that, when multiplied by itself, gives the prime number [[5 (number)|5]]. This number appears in the formula for the [[golden ratio]]. It can be denoted in [[nth root#Working with surds|surd]] form as:

:<math>\sqrt{5}. \, </math>

It is an [[irrational number|irrational]] [[algebraic number]].<ref>Dauben, Joseph W. (June 1983) [[Scientific American]] ''Georg Cantor and the origins of transfinite set theory.'' Volume 248; Page 122.</ref> The first sixty significant digits of its [[decimal expansion]] are:

:2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089... {{OEIS|id=A002163}}

which can be rounded down to 2.236 to within 99.99% accuracy. As of April 1994, its numerical value in decimal had been computed to at least one million digits.<ref>R. Nemiroff and J. Bonnell: ''[http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt5.1mil The first 1 million digits of the square root of 5]''</ref>

== Continued fraction ==

It can be expressed as the [[continued fraction]] [2; 4, 4, 4, 4, 4...] {{OEIS|id=A040002}}. The sequence of [[continued fraction#Best rational approximations|best rational approximations]] is:

:<math>{\color{OliveGreen}\frac{2}{1}}, \frac{7}{3} , {\color{OliveGreen}\frac{9}{4}} , \frac{20}{9} , \frac{29}{13} , {\color{OliveGreen}\frac{38}{17}} , \frac{123}{55} , {\color{OliveGreen}\frac{161}{72}} , \frac{360}{161} , \frac{521}{233} , {\color{OliveGreen}\frac{682}{305}} , \frac{2207}{987} , {\color{OliveGreen}\frac{2889}{1292}}, \dots</math>

[[Convergent (continued fraction)|Convergent]]s of the continued fraction are colored; their numerators are sequence {{OEIS url|id=A001077}}, and their denominators are sequence {{OEIS url|id=A001076}}. The other (non-colored) terms are [[continued fraction#Semiconvergents|semiconvergents]].

==Babylonian method==

When <math>\sqrt{5}</math> is computed with the [[Methods of computing square roots#Babylonian method|Babylonian method]], starting with ''r''0 = 2 and using ''r''''n''+1 = (''r''''n'' + 5/''r''''n'') / 2, the ''n''th approximant ''r''''n'' is equal to the 2''n''-th convergent of the convergent sequence:

:<math>\frac{2}{1} = 2.0,\quad \frac{9}{4} = 2.25,\quad \frac{161}{72} = 2.23611\dots,\quad \frac{51841}{23184} = 2.2360679779 \ldots</math>

== Relation to the golden ratio and Fibonacci numbers==
[[Image:Golden Rectangle Construction.svg|thumb|The {{fraction|√5|2}} diagonal of a half square forms the basis for the geometrical construction of a [[golden rectangle]].]]
This [[golden ratio]] φ is the [[arithmetic mean]] of [[1 (number)|1]] and the square root of 5.<ref>Browne, Malcolm W. (July 30, 1985) [[New York Times]] ''Puzzling Crystals Plunge Scientists into Uncertainty.'' Section: C; Page 1. (Note - this is a widely cited article).</ref> The [[algebra]]ic relationship between the square root of 5, the golden ratio and the [[golden ratio#Conjugate golden ratio|conjugate golden ratio]] (Φ = {{fraction|1|φ}} = φ − 1) are expressed in the following formulae:{{Fact|date=August 2007}}

:<math>\sqrt{5} = \varphi + \Phi = 2\varphi - 1 = 2\Phi + 1</math>

:<math>\varphi = \frac{1 + \sqrt{5}}{2}</math>

:<math>\Phi = \frac{\sqrt{5} - 1}{2}.</math>

(See section below for their geometrical interpretation as decompositions of a root-5 rectangle.)

The square root of 5 then naturally figures in the closed form expression for the [[Fibonacci number]]s, a formula which is usually written in terms of the golden ratio:
:<math>F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}.</math>

The quotient of √5 and φ (or the product of √5 and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the [[Lucas number]]s:<ref>[[Richard K. Guy]]: "The Strong Law of Small Numbers". ''[[American Mathematical Monthly]]'', vol. 95, 1988, pp. 675–712</ref>

:<math>\frac{\sqrt{5}}{\varphi} = \Phi \cdot \sqrt{5} = \frac{5 - \sqrt{5}}{2} = 1.3819660112501051518\dots = [1; 2, 1, 1, 1, 1, 1, 1, 1, \dots]</math>

:<math>\frac{\varphi}{\sqrt{5}} = \frac{1}{\Phi \cdot \sqrt{5}} = \frac{2}{5 - \sqrt{5}} = 0.72360679774997896964\dots = [0; 1, 2, 1, 1, 1, 1, 1, 1, \dots].</math>

The series of convergents to these values feature the series of Fibonacci numbers and the series of [[Lucas number]]s as numerators and denominators, and viceversa, respectively:

:<math>{1, \frac{3}{2}, \frac{4}{3}, \frac{7}{5}, \frac{11}{8}, \frac{18}{13}, \frac{29}{21}, \frac{47}{34}, \frac{76}{55}, \frac{123}{89}}, \dots \dots [1; 2, 1, 1, 1, 1, 1, 1, 1, \dots]</math>

:<math>{1, \frac{2}{3}, \frac{3}{4}, \frac{5}{7}, \frac{8}{11}, \frac{13}{18}, \frac{21}{29}, \frac{34}{47}, \frac{55}{76}, \frac{89}{123}}, \dots \dots [0; 1, 2, 1, 1, 1, 1, 1, 1,\dots].</math>

== Geometry ==

[[geometry|Geometrically]], the square root of 5 corresponds to the [[diagonal]] of a [[rectangle]] whose sides are of length [[1 (number)|1]] and [[2 (number)|2]], as is evident from the [[Pythagorean theorem]]. Such a rectangle can be obtained by halving a [[Square (geometry)|square]], or by placing two equal squares side by side. Together with the algebraic relationship between √5 and φ, this forms the basis for the geometrical construction of a [[golden rectangle]] from a square, and for the construction of a regular [[pentagon]] given its side (since the side-to-diagonal ratio in a regular pentagon is φ).

Forming a [[dihedral angle|dihedral]] [[right angle]] with the two equal squares that halve a 1:2 rectangle, it can be seen that √5 corresponds also to the ratio between the length of a [[cube]] [[edge (geometry)|edge]] and the shortest distance from one of its [[vertex (geometry)|vertices]] to the opposite one, when traversing the cube ''surface'' (the shortest distance when traversing through the ''inside'' of the cube corresponds to the length of the cube diagonal, which is the [[square root of three]] times the edge).{{Fact|date=August 2007}}

The number √5 can be algebraically and geometrically related to the [[square root of 2]] and the [[square root of 3]], as it is the length of the [[hypotenuse]] of a right triangle with [[cathetus|catheti]] measuring √2 and √3 (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the [[centre (geometry)|centre]] point of a cube, one of its vertices, and the middle point of a side located on one the faces containing that vertex and opposite to it, are in the ratio √2:√3:√5. This follows from the geometrical relationships between a cube and the quantities √2 (edge-to-face-diagonal ratio, or distance between opposite edges), √3 (edge-to-cube-diagonal ratio) and √5 (the relationship just mentioned above).

A rectangle with side proportions 1:√5 is called a ''root-five rectangle'' and is part of the series of root rectangles, a subset of [[dynamic rectangle]]s, which are based on √1 (= 1), √2, √3, √4 (= 2), √5... and successively constructed using the diagonal of the previous root rectangle, starting from a square.<ref>{{cite book | url = http://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&ots=8ZNc5ZKfTG&dq=intitle:%22Geometry+of+Design%22+%22root+5%22&sig=YitS7tv3b4_r87coR4s7EcjL4kk | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1568982496 }}</ref> A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ).<ref>{{cite book | title = The Elements of Dynamic Symmetry
| author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0486217760 | url = http://books.google.com/books?id=VYJK2F-dh2oC&pg=PA26&ots=MqxrsVLmIH&dq=%22root+five+rectangle%22++section+inauthor:hambidge&sig=meu0juFja5gpsjHKk_gG1stMbYo#PPA27,M1 }}</ref> It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between √5, φ and Φ mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length {{fraction|√5|2}} to both sides.

== Trigonometry ==

Like √2 and √3, the square root of five appears extensively in the formulae for [[exact trigonometric constants]], and as such the computation of its value is important for [[generating trigonometric tables]].{{Fact|date=August 2007}} Since √5 is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a [[dodecahedron]].{{Fact|date=August 2007}}

== Diophantine approximations ==
[[Hurwitz's theorem]] in [[Diophantine approximations]] states that every [[irrational number]] ''x'' can be approximated by infinitely many [[rational number]]s ''m''/''n'' in [[lowest terms]] in such a way that

:<math> \left|x - \frac{m}{n}\right| < \frac{1}{\sqrt{5}\,n^2} </math>

and that √5 is best possible, in the sense that for any larger constant than √5, there are some irrational numbers ''x'' for which only finitely many such approximations exist.<ref>{{Citation | last1=LeVeque | first1=William Judson | title=Topics in number theory | publisher=Addison-Wesley Publishing Co., Inc., Reading, Mass. | id={{MathSciNet | id = 0080682}} | year=1956}}</ref>

Closely related to this is the theorem<ref name=khinchin/> that of any three consecutive [[convergent (continued fraction)|convergent]]s
''p''''i''/''q''''i'',
''p''''i''+1/''q''''i''+1,
''p''''i''+2/''q''''i''+2,
of a number α, at least one of the three inequalities holds:

:<math>\left|\alpha - {p_i\over q_i}\right| < {1\over \sqrt5 q_i^2}, \qquad
\left|\alpha - {p_{i+1}\over q_{i+1}}\right| < {1\over \sqrt5 q_{i+1}^2}, \qquad
\left|\alpha - {p_{i+2}\over q_{i+2}}\right| < {1\over \sqrt5 q_{i+2}^2}.
</math>

And the √5 in the denominator is the best bound possible since the convergents of the [[golden ratio]] make the difference on the left-hand side arbitrarily close to the value on the right-hand side. In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.<ref name=khinchin>{{Citation | last1=[[A. Ya. Khinchin|Khinchin]] | first1=Aleksandr Yakovlevich | title=Continued Fractions | publisher = University of Chicago Press, Chicago and London | year = 1964}}</ref>

== Algebra ==

The [[ring (mathematics)|ring]] <math>\scriptstyle\mathbb{Z}\left[\,\sqrt{-5}\,\right]</math> contains numbers of the form <math>\scriptstyle a\, +\, b\sqrt{-5}</math>, where ''a'' and ''b'' are [[integer]]s and <math>\scriptstyle \sqrt{-5}</math> is the [[imaginary number]] <math>\scriptstyle i\sqrt{5}</math>. This ring is a frequently cited example of an [[integral domain]] that is not a [[unique factorization domain]].{{Fact|date=August 2007}} The number 6 has two inequivalent factorizations within this ring:

: <math>6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}).</math>

The [[field (mathematics)|field]] <math>\scriptstyle\mathbb{Q}\left[\,\sqrt{5}\,\right]</math>, like any other [[quadratic field]], is an [[abelian extension]] of the rational numbers. The [[Kronecker–Weber theorem]] therefore guarantees that the square root of five can be written as a rational linear combination of [[roots of unity]]:

:<math>\sqrt5 = e^{2\pi i/5} - e^{4\pi i/5} - e^{6\pi i/5} + e^{8\pi i/5}.</math>

== Identities of Ramanujan ==

The square root of 5 appears in various identities of [[Srinivasa Ramanujan|Ramanujan]] involving [[continued fraction]]s.<ref>{{Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction | id={{MathSciNet | id = 813071}} | year=1984 | journal=Indian Academy of Sciences. Proceedings. Mathematical Sciences | issn=0253-4142 | volume=93 | issue=2 | pages=67--77 | doi=10.1007/BF02840651}}</ref><ref>{{Citation | url=http://mathworld.wolfram.com/RamanujanContinuedFractions.html | author=Eric W. Weisstein | title=Ramanujan Continued Fractions}} at [[MathWorld]]</ref>

For example, this case of the [[Rogers–Ramanujan continued fraction]]:

:<math>
\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1 + \ddots}}}}
= \left( \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2} \right)e^{2\pi/5} = e^{2\pi/5}\left( \sqrt{\varphi\sqrt{5}} - \varphi \right).
</math>



:<math>
\cfrac{1}{1 + \cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1 + \cfrac{e^{-6\pi\sqrt{5}}}{1 + \ddots}}}}
= \left( {\sqrt{5} \over 1 + \left[5^{3/4}(\varphi - 1)^{5/2} - 1\right]^{1/5}} - \varphi \right)e^{2\pi/\sqrt{5}}.
</math>



:<math>
4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx
= \cfrac{1}{1 + \cfrac{1^2}{1 + \cfrac{1^2}{1 + \cfrac{2^2}{1 + \cfrac{2^2}{1 + \cfrac{3^2}{1 + \cfrac{3^2}{1 + \ddots}}}}}}}.
</math>

==See also==


*[[Golden ratio]]
*[[Square root]]
*[[Square root of 2]]
*[[Square root of 3]]

==References==
{{reflist}}

[[Category:Mathematical constants]]
[[Category:Irrational numbers]]
[[Category:Algebraic numbers]]

[[es:Raíz cuadrada de 5]]
[[fr:Racine carrée de cinq]]
[[zh:5的平方根]]

Quadratwurzel aus 5

2009-10-18T13:43:12Z

Paul August: /* Identities of Ramanujan */ combine refs

{| border="1" style="float: right; border-collapse: collapse; margin-left: 1em"
| colspan="2" align="center" | [[List of numbers]] [[Euler-Mascheroni constant|γ]] - [[Apéry's constant|ζ(3)]] - [[Square root of 2|√2]] - [[Square root of 3|√3]] - <math>\sqrt{5}</math> - [[Golden ratio|φ]] - [[Feigenbaum constants|α]] - [[E (mathematical constant)|e]] - [[Pi|π]] - [[Feigenbaum constants|δ]]
|-
|[[Binary numeral system|Binary]]
| 10.0011110001101111...
|-
| [[Decimal]]
| 2.23606797749978969...
|-
| [[Hexadecimal]]
| 2.3C6EF372FE94F82C...
|-
| [[Continued fraction]]
| <math>2 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \ddots\qquad{}}}}}</math>
|}

The '''square root of 5''' is the positive [[real number]] that, when multiplied by itself, gives the prime number [[5 (number)|5]]. This number appears in the formula for the [[golden ratio]]. It can be denoted in [[nth root#Working with surds|surd]] form as:

:<math>\sqrt{5}. \, </math>

It is an [[irrational number|irrational]] [[algebraic number]].<ref>Dauben, Joseph W. (June 1983) [[Scientific American]] ''Georg Cantor and the origins of transfinite set theory.'' Volume 248; Page 122.</ref> The first sixty significant digits of its [[decimal expansion]] are:

:2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089... {{OEIS|id=A002163}}

which can be rounded down to 2.236 to within 99.99% accuracy. As of April 1994, its numerical value in decimal had been computed to at least one million digits.<ref>R. Nemiroff and J. Bonnell: ''[http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt5.1mil The first 1 million digits of the square root of 5]''</ref>

== Continued fraction ==

It can be expressed as the [[continued fraction]] [2; 4, 4, 4, 4, 4...] {{OEIS|id=A040002}}. The sequence of [[continued fraction#Best rational approximations|best rational approximations]] is:

:<math>{\color{OliveGreen}\frac{2}{1}}, \frac{7}{3} , {\color{OliveGreen}\frac{9}{4}} , \frac{20}{9} , \frac{29}{13} , {\color{OliveGreen}\frac{38}{17}} , \frac{123}{55} , {\color{OliveGreen}\frac{161}{72}} , \frac{360}{161} , \frac{521}{233} , {\color{OliveGreen}\frac{682}{305}} , \frac{2207}{987} , {\color{OliveGreen}\frac{2889}{1292}}, \dots</math>

[[Convergent (continued fraction)|Convergent]]s of the continued fraction are colored; their numerators are sequence {{OEIS url|id=A001077}}, and their denominators are sequence {{OEIS url|id=A001076}}. The other (non-colored) terms are [[continued fraction#Semiconvergents|semiconvergents]].

==Babylonian method==

When <math>\sqrt{5}</math> is computed with the [[Methods of computing square roots#Babylonian method|Babylonian method]], starting with ''r''0 = 2 and using ''r''''n''+1 = (''r''''n'' + 5/''r''''n'') / 2, the ''n''th approximant ''r''''n'' is equal to the 2''n''-th convergent of the convergent sequence:

:<math>\frac{2}{1} = 2.0,\quad \frac{9}{4} = 2.25,\quad \frac{161}{72} = 2.23611\dots,\quad \frac{51841}{23184} = 2.2360679779 \ldots</math>

== Relation to the golden ratio and Fibonacci numbers==
[[Image:Golden Rectangle Construction.svg|thumb|The {{fraction|√5|2}} diagonal of a half square forms the basis for the geometrical construction of a [[golden rectangle]].]]
This [[golden ratio]] φ is the [[arithmetic mean]] of [[1 (number)|1]] and the square root of 5.<ref>Browne, Malcolm W. (July 30, 1985) [[New York Times]] ''Puzzling Crystals Plunge Scientists into Uncertainty.'' Section: C; Page 1. (Note - this is a widely cited article).</ref> The [[algebra]]ic relationship between the square root of 5, the golden ratio and the [[golden ratio#Conjugate golden ratio|conjugate golden ratio]] (Φ = {{fraction|1|φ}} = φ − 1) are expressed in the following formulae:{{Fact|date=August 2007}}

:<math>\sqrt{5} = \varphi + \Phi = 2\varphi - 1 = 2\Phi + 1</math>

:<math>\varphi = \frac{1 + \sqrt{5}}{2}</math>

:<math>\Phi = \frac{\sqrt{5} - 1}{2}.</math>

(See section below for their geometrical interpretation as decompositions of a root-5 rectangle.)

The square root of 5 then naturally figures in the closed form expression for the [[Fibonacci number]]s, a formula which is usually written in terms of the golden ratio:
:<math>F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}.</math>

The quotient of √5 and φ (or the product of √5 and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the [[Lucas number]]s:<ref>[[Richard K. Guy]]: "The Strong Law of Small Numbers". ''[[American Mathematical Monthly]]'', vol. 95, 1988, pp. 675–712</ref>

:<math>\frac{\sqrt{5}}{\varphi} = \Phi \cdot \sqrt{5} = \frac{5 - \sqrt{5}}{2} = 1.3819660112501051518\dots = [1; 2, 1, 1, 1, 1, 1, 1, 1, \dots]</math>

:<math>\frac{\varphi}{\sqrt{5}} = \frac{1}{\Phi \cdot \sqrt{5}} = \frac{2}{5 - \sqrt{5}} = 0.72360679774997896964\dots = [0; 1, 2, 1, 1, 1, 1, 1, 1, \dots].</math>

The series of convergents to these values feature the series of Fibonacci numbers and the series of [[Lucas number]]s as numerators and denominators, and viceversa, respectively:

:<math>{1, \frac{3}{2}, \frac{4}{3}, \frac{7}{5}, \frac{11}{8}, \frac{18}{13}, \frac{29}{21}, \frac{47}{34}, \frac{76}{55}, \frac{123}{89}}, \dots \dots [1; 2, 1, 1, 1, 1, 1, 1, 1, \dots]</math>

:<math>{1, \frac{2}{3}, \frac{3}{4}, \frac{5}{7}, \frac{8}{11}, \frac{13}{18}, \frac{21}{29}, \frac{34}{47}, \frac{55}{76}, \frac{89}{123}}, \dots \dots [0; 1, 2, 1, 1, 1, 1, 1, 1,\dots].</math>

== Geometry ==

[[geometry|Geometrically]], the square root of 5 corresponds to the [[diagonal]] of a [[rectangle]] whose sides are of length [[1 (number)|1]] and [[2 (number)|2]], as is evident from the [[Pythagorean theorem]]. Such a rectangle can be obtained by halving a [[Square (geometry)|square]], or by placing two equal squares side by side. Together with the algebraic relationship between √5 and φ, this forms the basis for the geometrical construction of a [[golden rectangle]] from a square, and for the construction of a regular [[pentagon]] given its side (since the side-to-diagonal ratio in a regular pentagon is φ).

Forming a [[dihedral angle|dihedral]] [[right angle]] with the two equal squares that halve a 1:2 rectangle, it can be seen that √5 corresponds also to the ratio between the length of a [[cube]] [[edge (geometry)|edge]] and the shortest distance from one of its [[vertex (geometry)|vertices]] to the opposite one, when traversing the cube ''surface'' (the shortest distance when traversing through the ''inside'' of the cube corresponds to the length of the cube diagonal, which is the [[square root of three]] times the edge).{{Fact|date=August 2007}}

The number √5 can be algebraically and geometrically related to the [[square root of 2]] and the [[square root of 3]], as it is the length of the [[hypotenuse]] of a right triangle with [[cathetus|catheti]] measuring √2 and √3 (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the [[centre (geometry)|centre]] point of a cube, one of its vertices, and the middle point of a side located on one the faces containing that vertex and opposite to it, are in the ratio √2:√3:√5. This follows from the geometrical relationships between a cube and the quantities √2 (edge-to-face-diagonal ratio, or distance between opposite edges), √3 (edge-to-cube-diagonal ratio) and √5 (the relationship just mentioned above).

A rectangle with side proportions 1:√5 is called a ''root-five rectangle'' and is part of the series of root rectangles, a subset of [[dynamic rectangle]]s, which are based on √1 (= 1), √2, √3, √4 (= 2), √5... and successively constructed using the diagonal of the previous root rectangle, starting from a square.<ref>{{cite book | url = http://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&ots=8ZNc5ZKfTG&dq=intitle:%22Geometry+of+Design%22+%22root+5%22&sig=YitS7tv3b4_r87coR4s7EcjL4kk | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1568982496 }}</ref> A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ).<ref>{{cite book | title = The Elements of Dynamic Symmetry
| author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0486217760 | url = http://books.google.com/books?id=VYJK2F-dh2oC&pg=PA26&ots=MqxrsVLmIH&dq=%22root+five+rectangle%22++section+inauthor:hambidge&sig=meu0juFja5gpsjHKk_gG1stMbYo#PPA27,M1 }}</ref> It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between √5, φ and Φ mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length {{fraction|√5|2}} to both sides.

== Trigonometry ==

Like √2 and √3, the square root of five appears extensively in the formulae for [[exact trigonometric constants]], and as such the computation of its value is important for [[generating trigonometric tables]].{{Fact|date=August 2007}} Since √5 is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a [[dodecahedron]].{{Fact|date=August 2007}}

== Diophantine approximations ==
[[Hurwitz's theorem]] in [[Diophantine approximations]] states that every [[irrational number]] ''x'' can be approximated by infinitely many [[rational number]]s ''m''/''n'' in [[lowest terms]] in such a way that

:<math> \left|x - \frac{m}{n}\right| < \frac{1}{\sqrt{5}\,n^2} </math>

and that √5 is best possible, in the sense that for any larger constant than √5, there are some irrational numbers ''x'' for which only finitely many such approximations exist.<ref>{{Citation | last1=LeVeque | first1=William Judson | title=Topics in number theory | publisher=Addison-Wesley Publishing Co., Inc., Reading, Mass. | id={{MathSciNet | id = 0080682}} | year=1956}}</ref>

Closely related to this is the theorem<ref name=khinchin/> that of any three consecutive [[convergent (continued fraction)|convergent]]s
''p''''i''/''q''''i'',
''p''''i''+1/''q''''i''+1,
''p''''i''+2/''q''''i''+2,
of a number α, at least one of the three inequalities holds:

:<math>\left|\alpha - {p_i\over q_i}\right| < {1\over \sqrt5 q_i^2}, \qquad
\left|\alpha - {p_{i+1}\over q_{i+1}}\right| < {1\over \sqrt5 q_{i+1}^2}, \qquad
\left|\alpha - {p_{i+2}\over q_{i+2}}\right| < {1\over \sqrt5 q_{i+2}^2}.
</math>

And the √5 in the denominator is the best bound possible since the convergents of the [[golden ratio]] make the difference on the left-hand side arbitrarily close to the value on the right-hand side. In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.<ref name=khinchin>{{Citation | last1=[[A. Ya. Khinchin|Khinchin]] | first1=Aleksandr Yakovlevich | title=Continued Fractions | publisher = University of Chicago Press, Chicago and London | year = 1964}}</ref>

== Algebra ==

The [[ring (mathematics)|ring]] <math>\scriptstyle\mathbb{Z}\left[\,\sqrt{-5}\,\right]</math> contains numbers of the form <math>\scriptstyle a\, +\, b\sqrt{-5}</math>, where ''a'' and ''b'' are [[integer]]s and <math>\scriptstyle \sqrt{-5}</math> is the [[imaginary number]] <math>\scriptstyle i\sqrt{5}</math>. This ring is a frequently cited example of an [[integral domain]] that is not a [[unique factorization domain]].{{Fact|date=August 2007}} The number 6 has two inequivalent factorizations within this ring:

: <math>6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}).</math>

The [[field (mathematics)|field]] <math>\scriptstyle\mathbb{Q}\left[\,\sqrt{5}\,\right]</math>, like any other [[quadratic field]], is an [[abelian extension]] of the rational numbers. The [[Kronecker–Weber theorem]] therefore guarantees that the square root of five can be written as a rational linear combination of [[roots of unity]]:

:<math>\sqrt5 = e^{2\pi i/5} - e^{4\pi i/5} - e^{6\pi i/5} + e^{8\pi i/5}.</math>

== Identities of Ramanujan ==

The square root of 5 appears in various identities of [[Srinivasa Ramanujan|Ramanujan]] involving [[continued fraction]]s.<ref>{{Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction | id={{MathSciNet | id = 813071}} | year=1984 | journal=Indian Academy of Sciences. Proceedings. Mathematical Sciences | issn=0253-4142 | volume=93 | issue=2 | pages=67--77 | doi=10.1007/BF02840651}}; {{Citation | url=http://mathworld.wolfram.com/RamanujanContinuedFractions.html | author=Eric W. Weisstein | title=Ramanujan Continued Fractions}} at [[MathWorld]]</ref>

For example, this case of the [[Rogers–Ramanujan continued fraction]]:

:<math>
\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1 + \ddots}}}}
= \left( \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2} \right)e^{2\pi/5} = e^{2\pi/5}\left( \sqrt{\varphi\sqrt{5}} - \varphi \right).
</math>



:<math>
\cfrac{1}{1 + \cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1 + \cfrac{e^{-6\pi\sqrt{5}}}{1 + \ddots}}}}
= \left( {\sqrt{5} \over 1 + \left[5^{3/4}(\varphi - 1)^{5/2} - 1\right]^{1/5}} - \varphi \right)e^{2\pi/\sqrt{5}}.
</math>



:<math>
4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx
= \cfrac{1}{1 + \cfrac{1^2}{1 + \cfrac{1^2}{1 + \cfrac{2^2}{1 + \cfrac{2^2}{1 + \cfrac{3^2}{1 + \cfrac{3^2}{1 + \ddots}}}}}}}.
</math>

==See also==


*[[Golden ratio]]
*[[Square root]]
*[[Square root of 2]]
*[[Square root of 3]]

==References==
{{reflist}}

[[Category:Mathematical constants]]
[[Category:Irrational numbers]]
[[Category:Algebraic numbers]]

[[es:Raíz cuadrada de 5]]
[[fr:Racine carrée de cinq]]
[[zh:5的平方根]]

Quadratwurzel aus 5

2009-10-18T13:41:35Z

Paul August: /* Continued fraction */create new section, as the information in that section has nothing to do with continued fractions

0,999…

2009-05-27T00:23:44Z

Paul August: Reverted edits by 24.20.72.92 (talk) to last version by AdjustShift

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math>0.\bar{9} , 0.\dot{9}</math> or <math> 0.(9)\,\!</math> denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|one]]. In other words: the notations ''0.999…'' and ''1 '' actually represent the same real number. This [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. [[mathematical proof|Proofs]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

The fact that certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies 1/9 = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==Alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
{{Col-2-of-3}}
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2009-05-13T04:18:32Z

Paul August: Reverted edits by 71.86.91.129 (talk) to last version by CardinalDan

[[Image:999 Perspective.svg|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math>0.\bar{9} , 0.\dot{9}</math> or <math> 0.(9)\,\!</math> denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|one]]. In other words: the notations ''0.999…'' and ''1 '' actually represent the same real number. This [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. [[mathematical proof|Proofs]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

The fact that certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies 1/9 = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

====Dedekind cuts====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==Alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
{{Col-2-of-3}}
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|2}}

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
</div>

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
{{featured article}}



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[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
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George Stibitz

2008-12-17T15:49:12Z

Paul August: /* Computer */ italics

[[Image:Stibitz3 lg.jpg|right|200px|thumb|George Stibitz]]
'''George Robert Stibitz''' ([[April 20]], [[1904]] – [[January 31]], [[1995]]) is internationally recognized as a father of the modern digital computer. He was a [[Bell Labs]] researcher known for his 1930s and 1940s work on the realization of [[Boolean logic]] digital circuits using electromechanical [[relay]]s as the switching element.

Born in [[York, Pennsylvania]], he received his [[bachelor's degree]] from [[Denison University]] in [[Granville, Ohio]], his [[master's degree]] from [[Union College]] in 1927, and his [[Ph.D.]] in mathematical physics in 1930 from [[Cornell University]].

==Computer==
In November of 1937, George Stibitz, then working at [[Bell Labs]], completed a relay-based computer he dubbed the "Model K" (for "'''k'''itchen table", on which he had assembled it), which calculated using binary addition. Bell Labs subsequently authorized a full research program in late 1938 with Stibitz at the helm. Their '''''Complex Number Calculator''''', completed [[January 8]], [[1940]], was able to calculate [[complex numbers]]. In a demonstration to the [[American Mathematical Society]] conference at [[Dartmouth College]] on [[September 11]], [[1940]], Stibitz used a [[teletype]] to send commands to the Complex Number Calculator in New York over telephone lines. It was the first computing machine ever used remotely over a phone line. (See the commemorative plaque and the hall where this event took place in the photos below.)

Stibitz held 38 patents, in addition to those he earned at Bell Labs. He became a member of the faculty at [[Dartmouth College]] in 1964 to build bridges between the fields of computing and medicine, and retired from research in 1983. Replicas of the "Model K" reside in both the [[Smithsonian Institution]] and the William Howard Doane Library at Denison University.

==Awards==
* [[Harry H. Goode Memorial Award]] in 1965 (together with [[Konrad Zuse]])

[[Image:Stibitz plaque 0708 edited-1.jpg|middle|400px|thumb|This bronze plaque is located in the entryway of McNutt Hall at Dartmouth College]]

[[Image:Stibitz McNutt Hall Dartmouth 0709 edited-1.jpg|right|200px|thumb|The site of the first long-distance communication of man and computer: McNutt Hall at Dartmouth College, September 9, 1940. The bronze commemorative plaque is mounted on the left wall in the entryway of the hall.]]

*Commemorative Bronze plaque located in McNutt Hall, Dartmouth College, Hanover NH, USA:
:"IN THIS BUILDING ON SEPTEMBER 9, 1940, GEORGE ROBERT STIBITZ, THEN A MATHEMATICIAN WITH BELL TELEPHONE LABORATORIES, FIRST DEMONSTRATED THE REMOTE OPERATION OF AN ELECTRICAL DIGITAL COMPUTER. STIBITZ, WHO CONCEIVED THE ELECTRICAL DIGITIAL COMPUTER IN 1937 AT BELL LABS, DESCRIBED HIS INVENTION OF THE "COMPLEX NUMBER CALCULATOR" AT A MEETING OF THE MATHEMATICAL ASSOCIATION OF AMERICA HELD HERE. MEMBERS OF THE AUDIENCE TRANSMITTED PROBLEMS TO THE COMPUTER AT BELL LABS IN NEW YORK CITY, AND IN SECONDS RECEIVED SOLUTIONS TRANSMITTED FROM THE COMPUTER TO A TELETYPEWRITER IN THIS HALL."

==Computer art==
In his later years, George "turned to non-verbal uses of the computer". Specifically, he used an [[Amiga]] to create computer art.

In a 1990 letter written to the department chair of the Mathematics and Computer Science department of Denison University he said:

<blockquote>I have turned to non-verbal uses of the computer, and have made a display of computer "art". The quotes are obligatory, for the result of my efforts is not to create important art but to show that this activity is fun, much as the creation of computers was fifty years ago.</blockquote>

The Mathematics and Computer Science department at Denison University has enlarged and displayed some of his artwork.

==See also==
* [[John Vincent Atanasoff]]
* [[Charles Babbage Institute]]

==External links==
===Patents===
* {{US patent|2668661}} "Complex Computer" filed April 1941, issued February 1954 (102 pages)

===Other===
*[http://ei.cs.vt.edu/~history/Stibitz.html Obituary by Kip Crosby of the Computing History Association of California]
*[http://www.denison.edu/mathsci/stibitz/index.html George R. Stibitz website at Denison University]
*[http://www.kerryr.net/pioneers/stibitz.htm Biography of Stibitz on the ''Pioneers'' website] – By Kerry Redshaw, Brisbane, Australia

== References ==
* Melina Hill, Valley News Correspondent, ''A Tinkerer Gets a Place in History'', Valley News West Lebanon NH, Thursday March 31, 1983, page 13.

* Andrew Hodges (1983), ''[[Alan Turing]]: The Enigma'', Simon and Schuster, New York, ISBN 0-671-49207-1. Stibitz is mentioned briefly on pages 299 and 326. Hodges refers to Stibitz's machine as one of two "big relay calculators" (Aiken's being the other one, p.326).
::"The second American project [Aiken's being the first] was underway at Bell Laboratories. Here the engineer G. Stibiz had first only thought of designing relay machines to perform decimal arithmetic with complex numbers, but after the outbreak of war had incorporated the facility to carry out a fixed sequence of aithmetical operations. His 'Model III' [sic] was under way in the New York building at the time of [[Alan Turing]]'s stay there, but it had not drawn his attention." (p. 299)
: Stibitz's work with binary addition has a peculiar (i.e. apparently simultaneous) overlap with some experimenting Alan Turing did in 1937 while a PhD student at Princeton. The following is according to a Dr. Malcolm McPhail "who became involved in a sideline that Alan took up" (p. 137)); Turing built his own relays and "actually designed an electric multiplier and built the first three or four stages to see if it could be made to work" (p. 138). It is unknown whether Stibitz and/or McPhail had any influence on this work of Turing's; McPhail's implication is that Turing's "[alarm]about a possible war with Germany" (p. 138) caused him to become interested in crytanalysis, and this interest led to discussions with McPhail, and these discussions led to the relay-multiplier experiments (the pertinent part of McPhail's letter to Hodges is quoted in Hodges p. 138).

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[[Category:Computer pioneers]]
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[[Category:Union College, New York alumni]]
[[Category:Scientists at Bell Labs]]
[[Category:People from York, Pennsylvania]]

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George Stibitz

2008-12-17T15:45:01Z

Paul August: /* Computer */ "Complex Number Calculator" redirects to this page

[[Image:Stibitz3 lg.jpg|right|200px|thumb|George Stibitz]]
'''George Robert Stibitz''' ([[April 20]], [[1904]] – [[January 31]], [[1995]]) is internationally recognized as a father of the modern digital computer. He was a [[Bell Labs]] researcher known for his 1930s and 1940s work on the realization of [[Boolean logic]] digital circuits using electromechanical [[relay]]s as the switching element.

Born in [[York, Pennsylvania]], he received his [[bachelor's degree]] from [[Denison University]] in [[Granville, Ohio]], his [[master's degree]] from [[Union College]] in 1927, and his [[Ph.D.]] in mathematical physics in 1930 from [[Cornell University]].

==Computer==
In November of 1937, George Stibitz, then working at [[Bell Labs]], completed a relay-based computer he dubbed the "Model K" (for "'''k'''itchen table", on which he had assembled it), which calculated using binary addition. Bell Labs subsequently authorized a full research program in late 1938 with Stibitz at the helm. Their '''Complex Number Calculator''', completed [[January 8]], [[1940]], was able to calculate [[complex numbers]]. In a demonstration to the [[American Mathematical Society]] conference at [[Dartmouth College]] on [[September 11]], [[1940]], Stibitz used a [[teletype]] to send commands to the Complex Number Calculator in New York over telephone lines. It was the first computing machine ever used remotely over a phone line. (See the commemorative plaque and the hall where this event took place in the photos below.)

Stibitz held 38 patents, in addition to those he earned at Bell Labs. He became a member of the faculty at [[Dartmouth College]] in 1964 to build bridges between the fields of computing and medicine, and retired from research in 1983. Replicas of the "Model K" reside in both the [[Smithsonian Institution]] and the William Howard Doane Library at Denison University.

==Awards==
* [[Harry H. Goode Memorial Award]] in 1965 (together with [[Konrad Zuse]])

[[Image:Stibitz plaque 0708 edited-1.jpg|middle|400px|thumb|This bronze plaque is located in the entryway of McNutt Hall at Dartmouth College]]

[[Image:Stibitz McNutt Hall Dartmouth 0709 edited-1.jpg|right|200px|thumb|The site of the first long-distance communication of man and computer: McNutt Hall at Dartmouth College, September 9, 1940. The bronze commemorative plaque is mounted on the left wall in the entryway of the hall.]]

*Commemorative Bronze plaque located in McNutt Hall, Dartmouth College, Hanover NH, USA:
:"IN THIS BUILDING ON SEPTEMBER 9, 1940, GEORGE ROBERT STIBITZ, THEN A MATHEMATICIAN WITH BELL TELEPHONE LABORATORIES, FIRST DEMONSTRATED THE REMOTE OPERATION OF AN ELECTRICAL DIGITAL COMPUTER. STIBITZ, WHO CONCEIVED THE ELECTRICAL DIGITIAL COMPUTER IN 1937 AT BELL LABS, DESCRIBED HIS INVENTION OF THE "COMPLEX NUMBER CALCULATOR" AT A MEETING OF THE MATHEMATICAL ASSOCIATION OF AMERICA HELD HERE. MEMBERS OF THE AUDIENCE TRANSMITTED PROBLEMS TO THE COMPUTER AT BELL LABS IN NEW YORK CITY, AND IN SECONDS RECEIVED SOLUTIONS TRANSMITTED FROM THE COMPUTER TO A TELETYPEWRITER IN THIS HALL."

==Computer art==
In his later years, George "turned to non-verbal uses of the computer". Specifically, he used an [[Amiga]] to create computer art.

In a 1990 letter written to the department chair of the Mathematics and Computer Science department of Denison University he said:

<blockquote>I have turned to non-verbal uses of the computer, and have made a display of computer "art". The quotes are obligatory, for the result of my efforts is not to create important art but to show that this activity is fun, much as the creation of computers was fifty years ago.</blockquote>

The Mathematics and Computer Science department at Denison University has enlarged and displayed some of his artwork.

==See also==
* [[John Vincent Atanasoff]]
* [[Charles Babbage Institute]]

==External links==
===Patents===
* {{US patent|2668661}} "Complex Computer" filed April 1941, issued February 1954 (102 pages)

===Other===
*[http://ei.cs.vt.edu/~history/Stibitz.html Obituary by Kip Crosby of the Computing History Association of California]
*[http://www.denison.edu/mathsci/stibitz/index.html George R. Stibitz website at Denison University]
*[http://www.kerryr.net/pioneers/stibitz.htm Biography of Stibitz on the ''Pioneers'' website] – By Kerry Redshaw, Brisbane, Australia

== References ==
* Melina Hill, Valley News Correspondent, ''A Tinkerer Gets a Place in History'', Valley News West Lebanon NH, Thursday March 31, 1983, page 13.

* Andrew Hodges (1983), ''[[Alan Turing]]: The Enigma'', Simon and Schuster, New York, ISBN 0-671-49207-1. Stibitz is mentioned briefly on pages 299 and 326. Hodges refers to Stibitz's machine as one of two "big relay calculators" (Aiken's being the other one, p.326).
::"The second American project [Aiken's being the first] was underway at Bell Laboratories. Here the engineer G. Stibiz had first only thought of designing relay machines to perform decimal arithmetic with complex numbers, but after the outbreak of war had incorporated the facility to carry out a fixed sequence of aithmetical operations. His 'Model III' [sic] was under way in the New York building at the time of [[Alan Turing]]'s stay there, but it had not drawn his attention." (p. 299)
: Stibitz's work with binary addition has a peculiar (i.e. apparently simultaneous) overlap with some experimenting Alan Turing did in 1937 while a PhD student at Princeton. The following is according to a Dr. Malcolm McPhail "who became involved in a sideline that Alan took up" (p. 137)); Turing built his own relays and "actually designed an electric multiplier and built the first three or four stages to see if it could be made to work" (p. 138). It is unknown whether Stibitz and/or McPhail had any influence on this work of Turing's; McPhail's implication is that Turing's "[alarm]about a possible war with Germany" (p. 138) caused him to become interested in crytanalysis, and this interest led to discussions with McPhail, and these discussions led to the relay-multiplier experiments (the pertinent part of McPhail's letter to Hodges is quoted in Hodges p. 138).

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[[Category:1904 births]]
[[Category:1995 deaths]]
[[Category:Computer pioneers]]
[[Category:Cornell University alumni]]
[[Category:Denison University alumni]]
[[Category:Union College, New York alumni]]
[[Category:Scientists at Bell Labs]]
[[Category:People from York, Pennsylvania]]

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0,999…

2008-12-05T03:32:46Z

Paul August: Reverted edits by 99.233.224.98 (talk) to last version by 132.70.50.117

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[recurring decimal]] '''0.999…''', which is also written as <math>0.\bar{9} , 0.\dot{9}</math> or <math> 0.(9)\,\!</math>, denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|1]]. In other words, the notations "0.999…" and "1" represent the same real number. The [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. Various [[mathematical proof|proofs]] of this identity have been formulated with various degrees of [[mathematical rigour]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The non-uniqueness of real expansions such as 0.999… is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is the [[real number]] which is the limit of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

==Proofs==
===Algebraic===
==== Fractions ====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies 1/9 = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

An even easier version of the same proof is based on the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, by the [[transitive property]], 0.999… must equal 1. Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

==== Digit manipulation ====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

=== Analytic ===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

=== Based on the construction of the real numbers ===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

==== Dedekind cuts ====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

==== Cauchy sequences ====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

== Alternative number systems ==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:{{Failed verification|date=December 2008}}
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245–247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since 0.999… = 1 (in the reals) and …999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
{{Col-2-of-3}}
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|2}}

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format= |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format= |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format= |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format= |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format= |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format= |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format= |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format= |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format= |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format= |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2008-12-04T17:39:03Z

Paul August: revert as apparently unsourced and as possible original research, per Huon and Silly Rabbit and discussion on talk page. Please do not re add without a consensus on the talk page.

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[recurring decimal]] '''0.999…''', which is also written as <math>0.\bar{9} , 0.\dot{9}</math> or <math> 0.(9)\,\!</math>, denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|1]]. In other words, the notations "0.999…" and "1" represent the same real number. The [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. Various [[mathematical proof|proofs]] of this identity have been formulated with various degrees of [[mathematical rigour]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The non-uniqueness of real expansions such as 0.999… is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is the [[real number]] which is the limit of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

==Proofs==
===Algebraic===
==== Fractions ====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

Another form of this proof multiplies 1/9 = 0.111… by 9.

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

An even easier version of the same proof is based on the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, by the [[transitive property]], 0.999… must equal 1. Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

==== Digit manipulation ====

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''x''. Then 10''x'' − ''x'' = 9. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name="CME"/> Written as a sequence of equations,

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

=== Analytic ===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

=== Based on the construction of the real numbers ===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

==== Dedekind cuts ====
{{further|[[Dedekind cut]]}}

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

==== Cauchy sequences ====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999… = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

== Alternative number systems ==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245–247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation "0.999… + ''x'' = 1"
has no solution.<ref>Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>

===''p''-adic numbers===
{{main|p-adic number}}

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since 0.999… = 1 (in the reals) and …999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
{{Col-2-of-3}}
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|2}}

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9–11)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format= |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format= |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format= |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format= |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format= |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format= |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format= |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*:Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format= |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*:This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format= |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format= |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999… = 1 ?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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