Talk:0.999...

Uncommon misconception?

Is there a common misconception that 0.999...<1? You can comment at Talk:List of common misconceptions#0.999.... Tkuvho (talk) 21:01, 15 January 2011 (UTC)

Yep, and you can see how great those misconceptions are simply by looking at Talk:0.999.../Arguments. It's rife witrh people who believe in that misconception, and others who try to persuade them in the error of their belief. --JB Adder | Talk 04:42, 23 January 2011 (UTC)

One simple error

To make things easier, I would like to point just one simple thing shortly:

The fact that this topic never really points out that .333... isn't really a third. The correct form of describing a third is 1/3. —Preceding unsigned comment added by 83.248.233.85 (talk) 03:07, 13 February 2011 (UTC)

Actually, both are correct ways of describing a third. One is the decimal representation (illustration here), the other is a fractional representation. 28bytes (talk) 03:33, 13 February 2011 (UTC)

I'd bet this misconception comes from fixed precision computing (and the usage of calculators) where 1/3 becomes 0.3333... but with a fixed finite number of digits. 75.18.188.35 (talk) 01:01, 13 March 2011 (UTC)

To be a little more precise, 0.333... is an approximation of a third, but it's "as close as you like" to the actual value of 1/3; kind of like how differentiation takes the smallest interval on a certain scale to measure change. Teamabby (talk) 05:05, 24 March 2011 (UTC)

That's a very intuitive way of approaching both infinite decimals and the derivative. To formalize it as correct mathematics, one needs an infinitesimal-enriched number system such as the hyperreals. Some of the editors in this space are hostile to this approach, however; don't be surprised to get condescending comments sooner or later. Tkuvho (talk) 06:06, 24 March 2011 (UTC)

No, 0.333... is 1/3 exactly. Decimal expansions were invented to represent real numbers, and are unsuited to the hyperreals because you can't represent all hyperreals unambiguously. Your characterization of differentiation is incorrect because there is no "smallest interval" -- even in the hyperreals. You can use hyperreals to define differentiation, however, in a way that turns out to be equivalent to the usual one. Eric119 (talk) 16:40, 24 March 2011 (UTC)

Why can't you represent hyperreals unambiguously? Tkuvho (talk) 17:04, 24 March 2011 (UTC)

Sorry, I meant to say that you can't represent all hyperreals unambiguously,using decimal expansions. For example, which hyperreal does 0.333... represent? Whichever one you pick, there will be many other hyperreals infinitesimally close to it, which have just as much reason to be called 0.333... . Unless there's a natural method of picking a preferred one, which I suppose, there could be.

But, really, the idea of understanding decimal expansions in terms of hyperreals is trying to answer the "wrong" question, because it treats decimal expansions as the primary objects. For that matter, a lot of the discussion of 0.999... seems to be based on treating decimal expansions as primary, and viewing the reals, hyperreals, etc., as potential ways of understanding them. But, both historically and philosophically, the situation is the complete reverse: it is the real numbers that are the main topic of interest, and the purpose of decimals is to help understand the reals. We can also invent the hyperreals to try to understand the notion of infinitesimal, and then we can wonder if we can contrive some kind of decimal expansions to understand the hyperreals. If we can, they probably wouldn't be exactly the same as those for the reals. But using hyperreals to understand decimal expansions (which are fine-tuned for the real numbers) is backwards. Eric119 (talk) 18:05, 25 March 2011 (UTC)

A quick historical correction: it is altogether untrue that historically speaking, the reals came first and "the purpose of the decimals is to help understand the reals". The opposite is true. The decimals were introduced by Simon Stevin a century before Newton and Leibniz (or Isaac Barrow, if you prefer) invented calculus.

As far as the hyperreals are concerned, the point is not so much to use them so as to understand decimal expansions, but rather to find a mathematical expression for persistent student intuitions that "zero, dot, followed by an infinity of 9s" can be a meaningful entity falling short of 1. This is precisely the case in the context of the hyperreals, but can be accomplished in other number systems, as well. Thus, such student intuitions are not "erroneous" but rather "non-standard", as argued in a recent article by Robert Ely in a leading (perhaps the leading) mathematics education journal. Tkuvho (talk) 16:46, 29 March 2011 (UTC)

There is a way to represent hyperreal numbers by a variant of decimal expansions; the article mentions it under 0.999...#Infinitesimals. Of course, the number 0.999... where all digits are nines is still equal to 1. Huon (talk) 19:29, 25 March 2011 (UTC)

Long division and hyperreals

I've reverted Tkuvho's claim that long division yields a hyperreal. I was taught long division in primary school, and I'm pretty sure hyperreals didn't occur back then. Besides, I believe Tkuvho's claim that

a simple division of integers like 1⁄9 produces the sequence (0.1, 0.11, 0.111, ...). What one does with such a sequence has nothing to do with long division. Thus, if one takes its equivalence class [0.1, 0.11, 0.111, ...] in the ultrapower construction, one obtains a hyperreal that falls infinitesimally short of 1⁄9, and adequal to the latter, typically represented by the recurring decimal, 0.111...

is wrong on several levels. First of all, long division indeed produces a recurring decimal, not a sequence. Secondly, the result of a division of integers in the hyperreals should not depend on the choice of an ultrafilter, but the equivalence class of the sequence (0.1, 0.11, 0.111, ...) does. Even more obviously, the result of the division 1⁄9 in the hyperreals is still 1⁄9, not infinitesimally short of 1⁄9. I'd also be interested in what hyperreal, in Lightstone's system of representation, 1⁄9 is supposed to be according to Tkuvho. What's the first digit where 1⁄9 deviates from 0.111... (in the hyperreal sense)? Huon (talk) 10:28, 27 March 2011 (UTC)

It was added to the FAQ too, so I removed it for the same reason, along with a further addition to another question which contradicted the rest of the answer so would only cause confusion. The FAQ is best left to straightforward (as possible) answers to basic questions. Digressions and examinations of the topic in terms of non-standard number systems should be confined to the relevant part of the article.--JohnBlackburne^words_deeds 13:43, 27 March 2011 (UTC)

Again the answer to the question 'Can't "1 - 0.999..." be expressed as "0.000...1"?' is 'No', as it now says with a detailed explanation. Adding a sentence that flatly contradicts that will confuse readers, defeating the point of the FAQ. I'm not even sure what it meant: "infinitesimal enriched" is hardly a mathematical term.--JohnBlackburne^words_deeds 17:44, 27 March 2011 (UTC)

Welcome to a pluralistic society. Sometimes it can be confusing to have more than one point of view represented. However, that's no reason to suppress all points of view other than your own. Rather, we should work toward expressing properly sourced viewpoints in such a way as not to cause confusion. An "infinitesimal-enriched" continuum is a continuum containing infinitesimals. You may not like this, but this page is not your private back yard. Tkuvho (talk) 18:29, 27 March 2011 (UTC)

Suggested addition

I suggest the following material be added to either the main page or the FAQ: "Using long division, a simple division of integers like 1⁄9 produces the sequence (0.1, 0.11, 0.111, ...). What one does with such a sequence afterwards has nothing to do with long division 'per se'. Thus, if one takes the equivalence class [0.1, 0.11, 0.111, ...] of the above sequence in the ultrapower construction, one obtains a hyperreal that falls infinitesimally short of 1⁄9, and is adequal to the latter. The identification of 0.999... with 1 results from a commitment to the real number system, and is not a consequence of long division." Tkuvho (talk) 15:12, 27 March 2011 (UTC)

I disagree. I assume you accept that a simple division of integers like 1⁄8 produces 0.125, not the sequence (0.1, 0.12, 0.125, 0.125, ...). In that case, all such divisions should result in the same type of object, not sometimes in a number and sometimes in a sequence.

Furthermore, you probably agree that a division of integeres like 1⁄9 should result, no matter whether we are in the real numbers or the hyperreals, in a number which, when multiplied by 9, gives 1, and not a little less. Otherwise our division algorithm would produce patently wrong results. I have detailed further objections to that line of reasoning above; I don't think we need two sections for the same proposed addition. Huon (talk) 15:35, 27 March 2011 (UTC)

No, I don't accept your claim about 1/8. In the case of 1/8, the algorithm terminates. In the case of 1/9, the algorithm does not terminate. By no stretch of imagination can one claim that long division already "knows" about the real number system. You can continue long division indefinitely, but what you construct is a sequence of digits. Tkuvho (talk) 16:11, 27 March 2011 (UTC)

Technically, the algorithm does not terminate with 1/8 either, it just gives nothing but zeroes after a certain point. Or you could argue that it terminates in both cases, with the termination event being the second occurrence of the same remainder, that is, once we have ascertained what precisely will be repeating.

Anyway, you mean long division produces an infinite string of digits, ie a repeating decimal? I agree. It happens to be the repeating decimal which represents the correct real number. I still don't see the use of having an algorithm of long division which produces wrong results. Could you please clarify? Huon (talk) 17:34, 27 March 2011 (UTC)

The algorithm of long division does not produce wrong results. It produces a string of digits. At the next stage, this string of digits is interpreted in a certain way. We are in agreement that one useful way of interpreting the outcome is in such a way that when you multiply back by 9, you get the same fraction you started with. But relying on this as evidence for the fact that long division must give such an outcome, amounts to circular reasoning. There is an infinitesimal ambiguity in interpreting the result of the long division. Afficionadoes of the real number system are free to suppress all infinitesimal differences and proclaim there is no life after real numbers. However, there are other points of view, which should not be suppressed in a pluralistic society. Tkuvho (talk) 18:24, 27 March 2011 (UTC)

We agree that long division gives a string of digits in which every digit has a finite distance from the first. This string of digits obviously is the decimal representation of a real number, and a meaningful real number at that. Of course you can interpret it otherwise, but I would like to see a reliable source for someone actually doing that. I'm pretty sure that any school textbook explaining long division will not mention hyperreals, and it will present the result as one number, not an equivalence class of infinitesimally close numbers. Anything else is a fringe position. Huon (talk) 18:53, 27 March 2011 (UTC)

You are absolutely right that we could define the result of long division to be a hyperreal. The fact of the matter is, though, that we don't. We define it to be a real number. Real numbers are a very useful way of modelling all sorts of things in our lives. Hyperreals aren't (they do have their uses, but not anywhere near as many as reals do). --Tango (talk) 19:16, 27 March 2011 (UTC)

Tango, this is exactly the problem I was pointing out. We define the result of long division to be a real number. Therefore purported "proofs" of .9=1 based on long division are merely brow-beating the students and their useful intuitions. I was certainly not proposing that we should redefine long division. Huon's "fringe" claims are WP:OR and should not be allowed to dominate this page. Tkuvho (talk) 13:45, 28 March 2011 (UTC)

Long division, as it is universally taught and understood, is about numbers. You divide one number with another number and get a third number which can be a terminating decimal or a recurring one. If can be extended to other things, polynomials for example, or other bases, but the common sense is the one about decimal numbers. Your proposed digressions into number systems that are little known even among mathematicians are fringe, and the consensus is clearly against their suggested inclusion.--JohnBlackburne^words_deeds 14:05, 28 March 2011 (UTC)

For example, our article on long division cites this paper, which in turn on page 4 cites a report published in the Notices of the AMS: "To understand that rational numbers correspond to repeating decimals essentially means understanding the structure of division of decimals as embodied in the division algorithm." So it's the official position of the AMS that long division of integers yields repeating decimals. The paper itself discusses this in greater detail, including a variant of this very proof. Huon (talk) 14:10, 28 March 2011 (UTC)

The paper you cited is a fine paper, and I think everybody here is in agreement that repeating decimals correspond to rational numbers. If you read the article carefully you will also notice that it deals with pre-calculus highschool education. At the pre-calculus level, there is no need for either infinitesimals or a nonzero entity "1-.999...". At the calculus level, there is a rich education literature about the advantages of the infinitesimal approach that you cannot write off with the stroke of your "fringe" pen. The point is not whether infinitesimals are useful or not, but rather if there is a legitimate literature to that effect. Supression of such literature is what is causing student frustrations which you seem interested in perpetuating. Tkuvho (talk) 16:52, 28 March 2011 (UTC)

suggested addition to FAQ

I suggest we add the following material to the FAQ: "The string '0.000...1' can be assigned definite meaning in the context of an infinitesimal-enriched number system, see the discussion of A. H. Lightstone's extended decimal notation in the main article." This is properly sourced in R. Ely's article referenced on the main page. Tkuvho (talk) 18:21, 27 March 2011 (UTC)

See my comments two sections above.--JohnBlackburne^words_deeds 18:27, 27 March 2011 (UTC)

To repeat, this page is not your private back yard where you can impose your opinion on everybody. Tkuvho (talk) 18:30, 27 March 2011 (UTC)

I agree with JohnBlackburne. There is no need to cover hyperreals in that question. Besides, we already discuss h

yperreals in another question, where we argue (rightly so, imo) that the hyperreal number with the best claim to being called 0.999... is still 1, which would imply that while a hyperreal number 0.000...;...0001 exists, it's not 1-0.999... Huon (talk) 18:57, 27 March 2011 (UTC)

I have to agree with JohnBlackburne and Huon that this proposed addition is outside the scope of the article. 28bytes (talk) 19:04, 27 March 2011 (UTC)

Apart from the issue of the relation of 0.000...01 to "1-.999...", there is a separate issue that the current version of the FAQ makes in incorrect claim to the effect that "0.000...01" cannot be meaningfully interpreted. This is inaccurate, as documented in the literature. The suppression of this point of view is not helpful to the students. Tkuvho (talk) 16:55, 28 March 2011 (UTC)

The FAQ says "[t]he string "0.000...1" is not a meaningful real decimal", which is correct. That it may be interpreted a hyperreal with no connection to 0.999... does not help the reader. Huon (talk) 17:03, 28 March 2011 (UTC)

Technically speaking the claim is correct but most readers have interpreted it and will interpret it as saying "[t]he string "0.000...1" is not a meaningful decimal". By the way, the way the question is formulated does not imply a commitment to the real number system on the part of the questioner. Tkuvho (talk) 17:08, 28 March 2011 (UTC)

It isn't a meaningful decimal – which digit is supposed to be 1? There is no "next" digit just after the standard naturals in any nonstandard model of PA (since 0 is the only non-successor natural number), so it isn't clear from the notation 0.000...1 which power of 10 is supposed to be multiplied by that 1. — Carl (CBM · talk) 20:33, 31 March 2011 (UTC)

There are two separate items here: (a) a pre-mathematical student's intuition of a tiny number with an infinity of zeros before 1; (b) a hyperreal infinitesimal 10^{-H} for an infinite H. What you seem to be pointing out is that there is no relation between (a) and (b), and that the student intuition is merely erroneous. This is precisely the claim contested by Robert Ely's paper. Ely argues that the student intuition is not erroneous, but rather nonstandard. What he means by that is that the intuition is robust, withstands challenges from the traditional interpretations, and provides a fruitful way of learning calculus. I happen to agree with Ely, but this is irrelevant. What is relevant is that this viewpoint is properly sourced. Tkuvho (talk) 20:47, 31 March 2011 (UTC)

All I am saying is that 0.0001 has a clear meaning of $10^{-4}$ but 0.000...1 has no meaning, even in the hyperreals, because there is no canonical candidate for which nonstandard power of 10 it is supposed to represent. — Carl (CBM · talk) 20:54, 31 March 2011 (UTC)

Again, 0.000...1 is not a precise mathematical notion, but a pre-mathematical intuition. Compare it to naive concepts students have of the continuity of functions. A teacher's role is not to discourage their naive concepts, but help them connect those concepts with precise mathematical structures. Note that there is no canonical candidate for 10^{-H} in the hyperreals, either. But it has an infinite number of zeros nontheless! The student in Ely's study naturally developed an arithmetic of infinite numbers to accomodate her infinitesimal intuitions. This does not mean she developed nonstandard analysis, or that she cannot benefit from her insights until she learns the details of the ultrapower construction. Tkuvho (talk) 21:06, 31 March 2011 (UTC)

There is a canonicial 10^{-H} for every nonstandard integer H. Peano arithmetic proves that for 10^{H} exists and is unique, by induction, and the nonstandard integers model PA because the standard ones do, via transfer of all the first-order axioms of PA. Then 10^{-H} is the reciprocal of 10^{H}. Alternatively, you can just adjoin the exponential function 10^x to the structure before you take the ultrapower, and then you can use the nonstandard extension of this function to give a unique value 10^x for every nonstandard x. — Carl (CBM · talk) 21:15, 31 March 2011 (UTC)

You pointed out earlier that there is no canonical choice for "0.000...1". I was merely pointing out that there is no canonical choice of an infinite hypernatural, either, in the sense that it depends on the choice of the ultrafilter (one can choose the nearly canonical sequence (1, 2, 3, ...), but the properties of the hypernatural it generates will depend on the ultrafilter). Of course transfer implies that exponentiation is well-defined. We are talking about the transition from student intuition to formal mathematics, not about the technicalities of the ultrapower construction. The transition from "0.000...1" to a formal infinitesimal is a very interesting one, and one that is documented in the recent education literature, in a leading math education journal. I fail to see why such a hullabaloo is being made about including a short sentence about this at the FAQ. Are we maintaining an ideological purity here? Tkuvho (talk) 04:37, 1 April 2011 (UTC)

(od) There is simply no need to mention hyperreals in this context at all. You may call that "ideological purity", but I'd call it "staying on topic". Besides, I don't think a field study involving a single student can be generalized to suggest changing mathematics education for all students, no matter where it has been published. Huon (talk) 11:59, 1 April 2011 (UTC)

(←) I was really responding to Huon's comment of 17:03, 28 March 2011 which suggested that the notation "0.000...1" did have a well defined value in each model of the hyperreals. I wanted to point out that that notation doesn't have a well-defined value in any model of the hyperreals.

As for adding the sentence to the FAQ, my personal opinion is that it's out of place. But even if it was added, we can't claim that the notation "0.000...1" has a value in the hyperreals. The strings "0.1", "0.01", "0.001" can be assigned definite meaning, but this doesn't extend to strings like "0.000...1" and "0.000...01". — Carl (CBM · talk) 11:51, 1 April 2011 (UTC)

Nonzero or non-zero

I would be writing the term as "nonzero." Any opinions, or arguments against that? Twipley (talk) 01:14, 30 March 2011 (UTC)

My dictionaries show "nonzero" as the standard spelling, so go for it. 28bytes (talk) 03:29, 30 March 2011 (UTC)

New thread

1/9 does not equal .111.... It is an approximation. —Preceding unsigned comment added by 165.155.196.69 (talk) 15:26, 14 April 2011 (UTC)

More precisely, it is an adequality. Tkuvho (talk) 15:32, 14 April 2011 (UTC)

The article could devote more attention to the student intuition that "0.999..." is adequal, rather than equal, to 1. Before the number system is specified (e.g. as being the real numbers), such intuitions are not erroneous but rather nonstandard, see R. Ely's article. Tkuvho (talk) 18:34, 16 April 2011 (UTC)

That's precisely what the article already says. Giving more prominence to the Ely article would be undue weight. Regarding the re-addition of 165.155.196.69's comment, please have a look at WP:TALK: "Article talk pages should not be used by editors as platforms for their personal views on a subject", and "[i]rrelevant discussions are subject to removal." This clearly includes your n-th attempt to proclaim that students somehow intuitively think of hyperreals instead of reals. I'd say the arguments page is a nice case study to the contrary: Hardly anybody who disagrees with the equality instead thinks of the hyperreals - the hyperreals just violate their intuition in other places than the reals. Huon (talk) 18:44, 16 April 2011 (UTC)

OK, I agree with what you wrote. At any rate, it was a legitimate issue to be raised, and I see no reason to delete it summarily as some kind of an expletive. Tkuvho (talk) 19:48, 16 April 2011 (UTC)

The mention of adequality was just deleted from the lede, which is a pity. The comment deleted nicely summarized the infinitesimal section. I have mentioned numerous times that it is not hyperreals that the students intuit (that would be remarkable indeed!), but rather a number system containing infinitesimals, as envisioned by Fermat and Leibniz. The latter certainly were not thinking in terms of hyperreals. Tkuvho (talk) 12:42, 17 April 2011 (UTC)

Well, in the decades in which I taught mathematics, I never once came across a student who appeared to "intuit" either hypereals or a number system containing infinitesimals. What I saw year after year was students who perceived decimals as strings of figures, rather than as an abstract concept which has strings of numbers as a convenient concrete representation. If your concept of what a decimal number actually is is a string of figures, then clearly 0.99999... is not the same as 1. And if you think of the order relation on decimal numbers being defined in terms of that string in some such tems as "Compare the numbers digit by digit from the left until you find the first difference: then the number which has the larger digit in that place is the larger number" then clearly 0.99999... < 1. Since these students have learnt the properties of decimals by rote from a very early age, they have internalised some process along those lines. From the point of view of a person with a high level of mathematical understanding it is clear that logically the difference in that case would be infinitesimal, but in my experience that is not how it is perceived by the vast majority of people who cannot accept that 0.999... = 1. They simply are not thinking in such terms. Yes, you can push them in the diection of thinkinbg in such terms by, for example, asking them what 1 - 0.999... is, but I have never seen any evidence that they think in such terms spontaneously. However, in my opinion this is not the main point. The main point is that the article essentially is about what 0.999... represents in the real number system, about the popular miconception that it does not represent the number 1 in that system, and about what reasons there are for accepting that in fact does. To insert stuff about hyperreals or infinitesimals into such an article confuses and muddies the issue for the average reader, and detracts from the clarity of the essential point that the article seeks to convey. Mathematicians are not the principal readership. JamesBWatson (talk) 18:11, 17 April 2011 (UTC)

Speaking directly about the article, I'm not sure that adequality is worth mentioning in the lede. We do link the term lower in the article, in the section on infinitesimals. Because infinitesimals are already not directly the subject of this article, I personally prefer to keep the lede sentences about them very tight. — Carl (CBM · talk) 18:12, 17 April 2011 (UTC)

Yes, that is very much in keeping with my own thoughts. It is mentioned, but it is not germane to the central point of the article, and should be resricted to the place where it is most relevant to the context. Certainly not in the lead. JamesBWatson (talk) 18:20, 17 April 2011 (UTC)

OK, whatever consensus emerges here is fine. Tkuvho (talk) 07:32, 18 April 2011 (UTC)

To respond briefly to JamesBWatson's detailed remarks above: Huon pointed out correctly that R. Ely's paper should not be given undue weight. On the other hand, it cannot be ignored altogether, either. JamesBWatson argues that student misconceptions about .999... result from their thinking of a number as being a string of digits. This is one possible interpretation. However, his claim that there is no evidence for any other interpretation is not correct, as Ely documents in his field study. The claim that some students think of 1-0.999... as an infinitesimal is documented in the education literature and is no longer in the realm of pure speculation. Tkuvho (talk) 10:17, 20 April 2011 (UTC)

I did not say that "there is no evidence for any other interpretation". I said that I had not seen such evidence. JamesBWatson (talk) 22:42, 20 April 2011 (UTC)

fuzzy rendering... (not the same old question)

In the section on non-standard systems, the N renders on my browser as fuzzy. not clear or sharp. Anybody know why this might be? Cliff (talk) 16:55, 22 April 2011 (UTC)

Shows up as a blackboard bold N on my browser. 28bytes (talk) 17:03, 22 April 2011 (UTC)

mine too, but it's not clear. It's fuzzy. Cliff (talk) 17:06, 22 April 2011 (UTC)

same person, different browser. It renders fuzzy in chrome and in firefox. 134.29.231.11 (talk) 17:11, 22 April 2011 (UTC)