Wikipedia - Benutzerbeiträge [de]

Quadratwurzel aus 5

2020-01-02T03:39:05Z

Tali64^2: /* Continued fraction */

{{short description|Positive real number which when multiplied by itself gives 5}}
{| class="infobox bordered" cellpadding=5
| colspan="2" align="center" | {{Irrational numbers}}
|-
|[[Binary numeral system|Binary]]
| {{gaps|10.0011|1100|0110|1110|…}}
|-
| [[Decimal]]
| {{gaps|2.23606|79774|99789|69…}}
|-
| [[Hexadecimal]]
| {{gaps|2.3C6E|F372|FE94|F82C|…}}
|-
| [[Continued fraction]]
| <math>2 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \ddots}}}}</math>
|}

The '''square root of 5''' is the positive [[real number]] that, when multiplied by itself, gives the prime number [[5 (number)|5]]. It is more precisely called the '''principal square root of 5''', to distinguish it from the negative number with the same property. This number appears in the fractional expression for the [[golden ratio]]. It can be denoted in [[nth root|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:

:{{gaps|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. The approximation {{sfrac|161|72}} (≈ 2.23611) for the square root of five can be used. Despite having a [[denominator]] of only 72, it differs from the correct value by less than {{sfrac|1|10,000}} (approx. {{val|4.3e-5}}). As of November 2019, its numerical value in decimal has been computed to at least 2,000,000,000,000 digits.<ref>{{cite web |last1=Yee |first1=Alexander |title=Records Set by y-cruncher |url=http://numberworld.org/y-cruncher/records.html}}</ref>

==Proofs of irrationality==
'''1'''. This irrationality proof for the square root of 5 uses [[Fermat]]'s method of [[infinite descent]]:

:Suppose that {{sqrt|5}} is rational, and express it in lowest possible terms (i.e., as a [[fully reduced fraction]]) as {{math|{{sfrac|''m''|''n''}}}} for natural numbers {{math|''m''}} and {{math|''n''}}. Then {{sqrt|5}} can be expressed in lower terms as {{math|{{sfrac|5''n'' − 2''m''|''m'' − 2''n''}}}}, which is a contradiction.<ref name=Grant>Grant, Mike, and Perella, Malcolm, "Descending to the irrational", ''Mathematical Gazette'' 83, July 1999, pp.263-267.</ref> (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives {{math|5''n''{{sup|2}} {{=}} ''m''{{sup|2}}}} and {{math|{{sfrac|''m''|''n''}} {{=}} {{sqrt|5}}}}, which is true by the premise. The second fractional expression for {{sqrt|5}} is in lower terms since, comparing denominators, {{math|''m'' − 2''n'' < ''n''}} since {{math|''m'' < 3''n''}} since {{math|{{sfrac|''m''|''n''}} < 3}} since {{math|{{sqrt|5}} < 3}}. And both the numerator and the denominator of the second fractional expression are positive since {{math|2 < {{sqrt|5}} < {{sfrac|5|2}}}} and {{math|{{sfrac|''m''|''n''}} {{=}} {{sqrt|5}}}}.)

'''2'''. This irrationality proof is also a proof by contradiction:

:Suppose that {{math|{{sqrt|5}} {{=}} {{sfrac|''a''|''b''}}}} where {{math|{{sfrac|''a''|''b''}}}} is in reduced form.

:Thus {{math|5 {{=}} {{sfrac|''a''{{sup|2}}|''b''{{sup|2}}}}}} and {{math|5''b''{{sup|2}} {{=}} ''a''{{sup|2}}}}. If {{math|''b''}} were even, {{math|''b''{{sup|2}}}}, {{math|''a''{{sup|2}}}}, and {{math|''a''}} would be even making the fraction {{math|{{sfrac|''a''|''b''}}}} ''not'' in reduced form. Thus {{math|''b''}} is odd, and by following a similar process, {{math|''a''}} is odd.

:Now, let {{math|''a'' {{=}} 2''m'' + 1}} and {{math|''b'' {{=}} 2''n'' + 1}} where {{math|''m''}} and {{math|''n''}} are integers.

:Substituting into {{math|5''b''{{sup|2}} {{=}} ''a''{{sup|2}}}} we get:
::<math>5(2n+1)^2=(2m+1)^2</math>
:which simplifies to:
::<math>5\left(4n^2+4n+1\right)=4m^2+4m+1</math>
:making:
::<math>20n^2+20n+5=4m^2+4m+1</math>
:By subtracting 1 from both sides, we get:
::<math>20n^2+20n+4=4m^2+4m</math>
:which reduces to:
::<math>5n^2+5n+1=m^2+m</math>
:In other words:
::<math>5n(n+1)+1=m(m+1)</math>

:The expression {{math|''x''(''x'' + 1)}} is even for any integer {{math|''x''}} (since either {{math|''x''}} or {{math|''x'' + 1}} is even). So this says that {{nowrap|5 × even + 1 {{=}} even}}, or {{nowrap|odd {{=}} even}}. Since there is no integer that is both even and odd, we have reached a contradiction and {{sqrt|5}} is irrational.

==Continued fraction==
It can be expressed as the [[continued fraction]]

: <math> [2; 4, 4, 4, 4, 4,\ldots] = 2 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + \dots}}}}. </math> {{OEIS|id=A040002}}

The convergents and [[continued fraction#Semiconvergents|semiconvergents]] of this continued fraction are as follows (the black terms are the semiconvergents):
:<math>{\color{red}{\frac{2}{1}}}, \frac{7}{3} , {\color{red}{\frac{9}{4}}} , \frac{20}{9} , \frac{29}{13} , {\color{red}{\frac{38}{17}}} , \frac{123}{55} , {\color{red}{\frac{161}{72}}} , \frac{360}{161} , \frac{521}{233} , {\color{red}{\frac{682}{305}}} , \frac{2207}{987} , {\color{red}{\frac{2889}{1292}}}, \dots</math>
[[Convergent (continued fraction)|Convergent]]s of the continued fraction are colored red; their numerators are 2, 9, 38, 161, ... {{OEIS|id=A001077}}, and their denominators are 1, 4, 17, 72, ... {{OEIS|id=A001076}}.

Each of these is the [[Continued fraction#Best rational approximations|best rational approximation]] of {{sqrt|5}}; in other words, it is closer to {{sqrt|5}} than any rational with a smaller denominator.

==Babylonian method==
When {{sqrt|5}} is computed with the [[Methods of computing square roots#Babylonian method|Babylonian method]], starting with {{math|''r''0 {{=}} 2}} and using {{math|''r''''n''+1 {{=}} {{sfrac|1|2}}{{big|{{big|(}}}}''r''''n'' + {{sfrac|5|''r''''n''}}{{big|{{big|)}}}}}}, the {{math|''n''}}th approximant {{math|''r''''n''}} is equal to the {{math|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>

==Nested square expansions==
The following nested square expressions converge to <math> \sqrt{5} </math>:
:<math>
\begin{align}
\sqrt{5} & = 3 - 10 \left( \frac{1}{5}+ \left( \frac{1}{5}+\left( \frac{1}{5}+ \left( \frac{1}{5}+ \cdots \right)^2 \right)^2 \right)^2 \right)^2 \\
& = \frac{9}{4} - 4 \left( \frac{1}{16}- \left( \frac{1}{16}-\left( \frac{1}{16}- \left( \frac{1}{16}- \cdots \right)^2 \right)^2 \right)^2 \right)^2 \\
& = \frac{9}{4} - 5 \left( \frac{1}{20}+ \left( \frac{1}{20}+\left( \frac{1}{20}+ \left( \frac{1}{20}+ \cdots \right)^2 \right)^2 \right)^2 \right)^2
\end{align}
</math>

==Relation to the golden ratio and Fibonacci numbers==
[[File:Golden Rectangle Construction.svg|thumb|The {{sfrac|{{sqrt|5}}|2}} diagonal of a half square forms the basis for the geometrical construction of a [[golden rectangle]].]]
The [[golden ratio]] {{math|φ}} is the [[arithmetic mean]] of [[1 (number)|1]] and {{sqrt|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 {{sqrt|5}}, the golden ratio and the [[golden ratio#Golden ratio conjugate|conjugate of the golden ratio]] ({{math|Φ {{=}} {{sfrac|–1|''φ''}} {{=}} 1 − ''φ''}}) is expressed in the following formulae:

: <math>
\begin{align}
\sqrt{5} & = \varphi - \Phi = 2\varphi - 1 = 1 - 2\Phi \\[5pt]
\varphi & = \frac{1 + \sqrt{5}}{2} \\[5pt]
\Phi & = \frac{1 - \sqrt{5}}{2}.
\end{align}
</math>
(See the section below for their geometrical interpretation as decompositions of a {{sqrt|5}} rectangle.)

{{sqrt|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(n) = \frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}.</math>

The quotient of {{sqrt|5}} and {{math|''φ''}} (or the product of {{sqrt|5}} and {{math|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>
\begin{align}
\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, \ldots] \\[5pt]
\frac{\varphi}{\sqrt{5}} = \frac{1}{\Phi \cdot \sqrt{5}} = \frac{5 + \sqrt{5}}{10} & = 0.72360679774997896964\ldots \\
& = [0; 1, 2, 1, 1, 1, 1, 1, 1, \ldots].
\end{align}
</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 vice versa, respectively:

: <math>
\begin{align}
& {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}}, \ldots \ldots [1; 2, 1, 1, 1, 1, 1, 1, 1, \ldots] \\[8pt]
& {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].
\end{align}
</math>

==Geometry==
[[File:pinwheel 1.svg|250px|thumb|right|[[Pinwheel tiling|Conway triangle]] decomposition into homothetic smaller triangles.]]

[[geometry|Geometrically]], {{sqrt|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 {{sqrt|5}} and {{math|''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 {{math|''φ''}}).

Forming a [[dihedral angle|dihedral]] [[right angle]] with the two equal squares that halve a 1:2 rectangle, it can be seen that {{sqrt|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).{{Citation needed|date=August 2007}}

The number {{sqrt|5}} can be algebraically and geometrically related to [[square root of 2|{{sqrt|2}}]] and [[square root of 3|{{sqrt|3}}]], as it is the length of the [[hypotenuse]] of a right triangle with [[cathetus|catheti]] measuring {{sqrt|2}} and {{sqrt|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 {{nowrap|{{sqrt|2}}:{{sqrt|3}}:{{sqrt|5}}.}} This follows from the geometrical relationships between a cube and the quantities {{sqrt|2}} (edge-to-face-diagonal ratio, or distance between opposite edges), {{sqrt|3}} (edge-to-cube-diagonal ratio) and {{sqrt|5}} (the relationship just mentioned above).

A rectangle with side proportions 1:{{sqrt|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 {{nowrap|1={{sqrt|1}} (= 1), {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}} (= 2), {{sqrt|5}}…}} and successively constructed using the diagonal of the previous root rectangle, starting from a square.<ref>{{Citation | url = https://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&dq=intitle:%22Geometry+of+Design%22+%22root+5%22 | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1-56898-249-6 }}</ref> A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions {{nowrap|{{math|Φ}} × 1}}), or into two golden rectangles of different sizes (of dimensions {{nowrap|{{math|Φ}} × 1}} and {{nowrap|1 × {{math|''φ''}}}}).<ref>{{Citation | title = The Elements of Dynamic Symmetry | author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0-486-21776-0 | url = https://books.google.com/books?id=VYJK2F-dh2oC&pg=PA26&dq=%22root+five+rectangle%22++section+inauthor:hambidge }}</ref> It can also be decomposed as the union of two equal golden rectangles (of dimensions {{nowrap|1 × {{math|φ}}}}) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between {{sqrt|5}}, {{math|''φ''}} and {{math|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 {{sfrac|{{sqrt|5}}|2}} to both sides.

==Trigonometry==
Like {{sqrt|2}} and {{sqrt|3}}, the square root of 5 appears extensively in the formulae for [[exact trigonometric constants]], including in the sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.<ref>[http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Julian D. A. Wiseman, "Sin and cos in surds"]</ref> The simplest of these are
:<math>\begin{align}
\sin\frac{\pi}{10} = \sin 18^\circ &= \tfrac{1}{4}(\sqrt5-1) = \frac{1}{\sqrt5+1}, \\[5pt]
\sin\frac{\pi}{5} = \sin 36^\circ &= \tfrac{1}{4}\sqrt{2(5-\sqrt5)}, \\[5pt]
\sin\frac{3\pi}{10} = \sin 54^\circ &= \tfrac{1}{4}(\sqrt5+1) = \frac{1}{\sqrt5-1}, \\[5pt]
\sin\frac{2\pi}{5} = \sin 72^\circ &= \tfrac{1}{4}\sqrt{2(5+\sqrt5)}\, . \end{align}</math>

As such the computation of its value is important for [[generating trigonometric tables]].{{Citation needed|date=August 2007}} Since {{sqrt|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]].{{Citation needed|date=August 2007}}

==Diophantine approximations==
[[Hurwitz's theorem (number theory)|Hurwitz's theorem]] in [[Diophantine approximations]] states that every [[irrational number]] {{math|''x''}} can be approximated by infinitely many [[rational number]]s {{math|{{sfrac|''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 {{sqrt|5}} is best possible, in the sense that for any larger constant than {{sqrt|5}}, there are some irrational numbers {{math|''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. |mr=0080682 | year=1956}}</ref>

Closely related to this is the theorem<ref name=khinchin/> that of any three consecutive [[convergent (continued fraction)|convergent]]s {{math|{{sfrac|''p''''i''|''q''''i''}}}}, {{math|{{sfrac|''p''''i''+1|''q''''i''+1}}}}, {{math|{{sfrac|''p''''i''+2|''q''''i''+2}}}}, of a number {{math|''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 {{sqrt|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|ℤ[{{sqrt|−5}}]}} contains numbers of the form {{math|''a'' + ''b''{{sqrt|−5}}}}, where {{math|''a''}} and {{math|''b''}} are [[integer]]s and {{math|{{sqrt|−5}}}} is the [[imaginary number]] {{math|''i''{{sqrt|5}}}}. This ring is a frequently cited example of an [[integral domain]] that is not a [[unique factorization domain]].{{Citation needed|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|ℚ[{{sqrt|−5}}]}}, 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^{\frac{2\pi}{5}i} - e^{\frac{4\pi}{5}i} - e^{\frac{6\pi}{5}i} + e^{\frac{8\pi}{5}i}. \, </math>

==Identities of Ramanujan==
The square root of 5 appears in various identities discovered by [[Srinivasa Ramanujan]] involving [[continued fraction]]s.<ref>{{Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction |mr=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^{\frac{2\pi}{5}} = e^{\frac{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 + \sqrt[5]{5^{\frac34}(\varphi - 1)^{\frac52} - 1}} - \varphi \right)e^{\frac{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}}

{{Algebraic numbers}}
{{Irrational number}}

[[Category:Mathematical constants]]
[[Category:Irrational numbers]]
[[Category:Quadratic irrational numbers]]
[[Category:Algebraic numbers]]

Quadratwurzel aus 5

2020-01-02T03:38:28Z

Tali64^2: /* Continued fraction */

{{short description|Positive real number which when multiplied by itself gives 5}}
{| class="infobox bordered" cellpadding=5
| colspan="2" align="center" | {{Irrational numbers}}
|-
|[[Binary numeral system|Binary]]
| {{gaps|10.0011|1100|0110|1110|…}}
|-
| [[Decimal]]
| {{gaps|2.23606|79774|99789|69…}}
|-
| [[Hexadecimal]]
| {{gaps|2.3C6E|F372|FE94|F82C|…}}
|-
| [[Continued fraction]]
| <math>2 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \ddots}}}}</math>
|}

The '''square root of 5''' is the positive [[real number]] that, when multiplied by itself, gives the prime number [[5 (number)|5]]. It is more precisely called the '''principal square root of 5''', to distinguish it from the negative number with the same property. This number appears in the fractional expression for the [[golden ratio]]. It can be denoted in [[nth root|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:

:{{gaps|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. The approximation {{sfrac|161|72}} (≈ 2.23611) for the square root of five can be used. Despite having a [[denominator]] of only 72, it differs from the correct value by less than {{sfrac|1|10,000}} (approx. {{val|4.3e-5}}). As of November 2019, its numerical value in decimal has been computed to at least 2,000,000,000,000 digits.<ref>{{cite web |last1=Yee |first1=Alexander |title=Records Set by y-cruncher |url=http://numberworld.org/y-cruncher/records.html}}</ref>

==Proofs of irrationality==
'''1'''. This irrationality proof for the square root of 5 uses [[Fermat]]'s method of [[infinite descent]]:

:Suppose that {{sqrt|5}} is rational, and express it in lowest possible terms (i.e., as a [[fully reduced fraction]]) as {{math|{{sfrac|''m''|''n''}}}} for natural numbers {{math|''m''}} and {{math|''n''}}. Then {{sqrt|5}} can be expressed in lower terms as {{math|{{sfrac|5''n'' − 2''m''|''m'' − 2''n''}}}}, which is a contradiction.<ref name=Grant>Grant, Mike, and Perella, Malcolm, "Descending to the irrational", ''Mathematical Gazette'' 83, July 1999, pp.263-267.</ref> (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives {{math|5''n''{{sup|2}} {{=}} ''m''{{sup|2}}}} and {{math|{{sfrac|''m''|''n''}} {{=}} {{sqrt|5}}}}, which is true by the premise. The second fractional expression for {{sqrt|5}} is in lower terms since, comparing denominators, {{math|''m'' − 2''n'' < ''n''}} since {{math|''m'' < 3''n''}} since {{math|{{sfrac|''m''|''n''}} < 3}} since {{math|{{sqrt|5}} < 3}}. And both the numerator and the denominator of the second fractional expression are positive since {{math|2 < {{sqrt|5}} < {{sfrac|5|2}}}} and {{math|{{sfrac|''m''|''n''}} {{=}} {{sqrt|5}}}}.)

'''2'''. This irrationality proof is also a proof by contradiction:

:Suppose that {{math|{{sqrt|5}} {{=}} {{sfrac|''a''|''b''}}}} where {{math|{{sfrac|''a''|''b''}}}} is in reduced form.

:Thus {{math|5 {{=}} {{sfrac|''a''{{sup|2}}|''b''{{sup|2}}}}}} and {{math|5''b''{{sup|2}} {{=}} ''a''{{sup|2}}}}. If {{math|''b''}} were even, {{math|''b''{{sup|2}}}}, {{math|''a''{{sup|2}}}}, and {{math|''a''}} would be even making the fraction {{math|{{sfrac|''a''|''b''}}}} ''not'' in reduced form. Thus {{math|''b''}} is odd, and by following a similar process, {{math|''a''}} is odd.

:Now, let {{math|''a'' {{=}} 2''m'' + 1}} and {{math|''b'' {{=}} 2''n'' + 1}} where {{math|''m''}} and {{math|''n''}} are integers.

:Substituting into {{math|5''b''{{sup|2}} {{=}} ''a''{{sup|2}}}} we get:
::<math>5(2n+1)^2=(2m+1)^2</math>
:which simplifies to:
::<math>5\left(4n^2+4n+1\right)=4m^2+4m+1</math>
:making:
::<math>20n^2+20n+5=4m^2+4m+1</math>
:By subtracting 1 from both sides, we get:
::<math>20n^2+20n+4=4m^2+4m</math>
:which reduces to:
::<math>5n^2+5n+1=m^2+m</math>
:In other words:
::<math>5n(n+1)+1=m(m+1)</math>

:The expression {{math|''x''(''x'' + 1)}} is even for any integer {{math|''x''}} (since either {{math|''x''}} or {{math|''x'' + 1}} is even). So this says that {{nowrap|5 × even + 1 {{=}} even}}, or {{nowrap|odd {{=}} even}}. Since there is no integer that is both even and odd, we have reached a contradiction and {{sqrt|5}} is irrational.

==Continued fraction==
It can be expressed as the [[continued fraction]]

: <math> [2; 4, 4, 4, 4, 4,\ldots] = 2 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + \ddots}}}}. </math> {{OEIS|id=A040002}}

The convergents and [[continued fraction#Semiconvergents|semiconvergents]] of this continued fraction are as follows (the black terms are the semiconvergents):
:<math>{\color{red}{\frac{2}{1}}}, \frac{7}{3} , {\color{red}{\frac{9}{4}}} , \frac{20}{9} , \frac{29}{13} , {\color{red}{\frac{38}{17}}} , \frac{123}{55} , {\color{red}{\frac{161}{72}}} , \frac{360}{161} , \frac{521}{233} , {\color{red}{\frac{682}{305}}} , \frac{2207}{987} , {\color{red}{\frac{2889}{1292}}}, \dots</math>
[[Convergent (continued fraction)|Convergent]]s of the continued fraction are colored red; their numerators are 2, 9, 38, 161, ... {{OEIS|id=A001077}}, and their denominators are 1, 4, 17, 72, ... {{OEIS|id=A001076}}.

Each of these is the [[Continued fraction#Best rational approximations|best rational approximation]] of {{sqrt|5}}; in other words, it is closer to {{sqrt|5}} than any rational with a smaller denominator.

==Babylonian method==
When {{sqrt|5}} is computed with the [[Methods of computing square roots#Babylonian method|Babylonian method]], starting with {{math|''r''0 {{=}} 2}} and using {{math|''r''''n''+1 {{=}} {{sfrac|1|2}}{{big|{{big|(}}}}''r''''n'' + {{sfrac|5|''r''''n''}}{{big|{{big|)}}}}}}, the {{math|''n''}}th approximant {{math|''r''''n''}} is equal to the {{math|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>

==Nested square expansions==
The following nested square expressions converge to <math> \sqrt{5} </math>:
:<math>
\begin{align}
\sqrt{5} & = 3 - 10 \left( \frac{1}{5}+ \left( \frac{1}{5}+\left( \frac{1}{5}+ \left( \frac{1}{5}+ \cdots \right)^2 \right)^2 \right)^2 \right)^2 \\
& = \frac{9}{4} - 4 \left( \frac{1}{16}- \left( \frac{1}{16}-\left( \frac{1}{16}- \left( \frac{1}{16}- \cdots \right)^2 \right)^2 \right)^2 \right)^2 \\
& = \frac{9}{4} - 5 \left( \frac{1}{20}+ \left( \frac{1}{20}+\left( \frac{1}{20}+ \left( \frac{1}{20}+ \cdots \right)^2 \right)^2 \right)^2 \right)^2
\end{align}
</math>

==Relation to the golden ratio and Fibonacci numbers==
[[File:Golden Rectangle Construction.svg|thumb|The {{sfrac|{{sqrt|5}}|2}} diagonal of a half square forms the basis for the geometrical construction of a [[golden rectangle]].]]
The [[golden ratio]] {{math|φ}} is the [[arithmetic mean]] of [[1 (number)|1]] and {{sqrt|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 {{sqrt|5}}, the golden ratio and the [[golden ratio#Golden ratio conjugate|conjugate of the golden ratio]] ({{math|Φ {{=}} {{sfrac|–1|''φ''}} {{=}} 1 − ''φ''}}) is expressed in the following formulae:

: <math>
\begin{align}
\sqrt{5} & = \varphi - \Phi = 2\varphi - 1 = 1 - 2\Phi \\[5pt]
\varphi & = \frac{1 + \sqrt{5}}{2} \\[5pt]
\Phi & = \frac{1 - \sqrt{5}}{2}.
\end{align}
</math>
(See the section below for their geometrical interpretation as decompositions of a {{sqrt|5}} rectangle.)

{{sqrt|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(n) = \frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}.</math>

The quotient of {{sqrt|5}} and {{math|''φ''}} (or the product of {{sqrt|5}} and {{math|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>
\begin{align}
\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, \ldots] \\[5pt]
\frac{\varphi}{\sqrt{5}} = \frac{1}{\Phi \cdot \sqrt{5}} = \frac{5 + \sqrt{5}}{10} & = 0.72360679774997896964\ldots \\
& = [0; 1, 2, 1, 1, 1, 1, 1, 1, \ldots].
\end{align}
</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 vice versa, respectively:

: <math>
\begin{align}
& {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}}, \ldots \ldots [1; 2, 1, 1, 1, 1, 1, 1, 1, \ldots] \\[8pt]
& {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].
\end{align}
</math>

==Geometry==
[[File:pinwheel 1.svg|250px|thumb|right|[[Pinwheel tiling|Conway triangle]] decomposition into homothetic smaller triangles.]]

[[geometry|Geometrically]], {{sqrt|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 {{sqrt|5}} and {{math|''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 {{math|''φ''}}).

Forming a [[dihedral angle|dihedral]] [[right angle]] with the two equal squares that halve a 1:2 rectangle, it can be seen that {{sqrt|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).{{Citation needed|date=August 2007}}

The number {{sqrt|5}} can be algebraically and geometrically related to [[square root of 2|{{sqrt|2}}]] and [[square root of 3|{{sqrt|3}}]], as it is the length of the [[hypotenuse]] of a right triangle with [[cathetus|catheti]] measuring {{sqrt|2}} and {{sqrt|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 {{nowrap|{{sqrt|2}}:{{sqrt|3}}:{{sqrt|5}}.}} This follows from the geometrical relationships between a cube and the quantities {{sqrt|2}} (edge-to-face-diagonal ratio, or distance between opposite edges), {{sqrt|3}} (edge-to-cube-diagonal ratio) and {{sqrt|5}} (the relationship just mentioned above).

A rectangle with side proportions 1:{{sqrt|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 {{nowrap|1={{sqrt|1}} (= 1), {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}} (= 2), {{sqrt|5}}…}} and successively constructed using the diagonal of the previous root rectangle, starting from a square.<ref>{{Citation | url = https://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&dq=intitle:%22Geometry+of+Design%22+%22root+5%22 | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1-56898-249-6 }}</ref> A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions {{nowrap|{{math|Φ}} × 1}}), or into two golden rectangles of different sizes (of dimensions {{nowrap|{{math|Φ}} × 1}} and {{nowrap|1 × {{math|''φ''}}}}).<ref>{{Citation | title = The Elements of Dynamic Symmetry | author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0-486-21776-0 | url = https://books.google.com/books?id=VYJK2F-dh2oC&pg=PA26&dq=%22root+five+rectangle%22++section+inauthor:hambidge }}</ref> It can also be decomposed as the union of two equal golden rectangles (of dimensions {{nowrap|1 × {{math|φ}}}}) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between {{sqrt|5}}, {{math|''φ''}} and {{math|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 {{sfrac|{{sqrt|5}}|2}} to both sides.

==Trigonometry==
Like {{sqrt|2}} and {{sqrt|3}}, the square root of 5 appears extensively in the formulae for [[exact trigonometric constants]], including in the sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.<ref>[http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Julian D. A. Wiseman, "Sin and cos in surds"]</ref> The simplest of these are
:<math>\begin{align}
\sin\frac{\pi}{10} = \sin 18^\circ &= \tfrac{1}{4}(\sqrt5-1) = \frac{1}{\sqrt5+1}, \\[5pt]
\sin\frac{\pi}{5} = \sin 36^\circ &= \tfrac{1}{4}\sqrt{2(5-\sqrt5)}, \\[5pt]
\sin\frac{3\pi}{10} = \sin 54^\circ &= \tfrac{1}{4}(\sqrt5+1) = \frac{1}{\sqrt5-1}, \\[5pt]
\sin\frac{2\pi}{5} = \sin 72^\circ &= \tfrac{1}{4}\sqrt{2(5+\sqrt5)}\, . \end{align}</math>

As such the computation of its value is important for [[generating trigonometric tables]].{{Citation needed|date=August 2007}} Since {{sqrt|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]].{{Citation needed|date=August 2007}}

==Diophantine approximations==
[[Hurwitz's theorem (number theory)|Hurwitz's theorem]] in [[Diophantine approximations]] states that every [[irrational number]] {{math|''x''}} can be approximated by infinitely many [[rational number]]s {{math|{{sfrac|''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 {{sqrt|5}} is best possible, in the sense that for any larger constant than {{sqrt|5}}, there are some irrational numbers {{math|''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. |mr=0080682 | year=1956}}</ref>

Closely related to this is the theorem<ref name=khinchin/> that of any three consecutive [[convergent (continued fraction)|convergent]]s {{math|{{sfrac|''p''''i''|''q''''i''}}}}, {{math|{{sfrac|''p''''i''+1|''q''''i''+1}}}}, {{math|{{sfrac|''p''''i''+2|''q''''i''+2}}}}, of a number {{math|''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 {{sqrt|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|ℤ[{{sqrt|−5}}]}} contains numbers of the form {{math|''a'' + ''b''{{sqrt|−5}}}}, where {{math|''a''}} and {{math|''b''}} are [[integer]]s and {{math|{{sqrt|−5}}}} is the [[imaginary number]] {{math|''i''{{sqrt|5}}}}. This ring is a frequently cited example of an [[integral domain]] that is not a [[unique factorization domain]].{{Citation needed|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|ℚ[{{sqrt|−5}}]}}, 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^{\frac{2\pi}{5}i} - e^{\frac{4\pi}{5}i} - e^{\frac{6\pi}{5}i} + e^{\frac{8\pi}{5}i}. \, </math>

==Identities of Ramanujan==
The square root of 5 appears in various identities discovered by [[Srinivasa Ramanujan]] involving [[continued fraction]]s.<ref>{{Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction |mr=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^{\frac{2\pi}{5}} = e^{\frac{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 + \sqrt[5]{5^{\frac34}(\varphi - 1)^{\frac52} - 1}} - \varphi \right)e^{\frac{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}}

{{Algebraic numbers}}
{{Irrational number}}

[[Category:Mathematical constants]]
[[Category:Irrational numbers]]
[[Category:Quadratic irrational numbers]]
[[Category:Algebraic numbers]]