Talk:0.999.../Archive 14

Removal of dubious content
I just removed the following addition by User:132.70.50.117:


 * If $$\infty$$ denotes an unbounded hyperinteger, then the transfer principle implies the identity $$\sum_{i=0}^{\infty}\frac{9}{10^{i+1}}=.\underset{\infty}{\underbrace{999\ldots}}\; = 1\;-^{*}\;\frac{1}{10^{\infty}}\;$$, yielding a strict inequality $$\quad .\underset{\infty}{\underbrace{999\ldots}} < 1.$$

To the best of my knowledge, the transfer principle doesn't apply because $$\sum_{i=0}^{\infty}$$ is not a function on the real numbers; the sum to an infinite hyperreal number is undefined. And definitely $$.\underset{\infty}{\underbrace{999\ldots}}$$ is undefined because the number of nines must be a cardinal number, not an infinite hyperreal. Even if the statement were true, it would need a much more detailed explanation of what number system the objects are supposed to live in and how they're defined. Huon (talk) 20:38, 1 December 2008 (UTC)

The 'transfer principle'? - This has never been proven. It was conjectured and is being used by Wikipedia sysops/admins as accepted fact. It is based on the ill-defined concept of infinitesimal. How can you use such a phrase at all when discussing what you like to refer to as real numbers. Real numbers have never been well-defined. A cauchy sequence does not define a real number. It is time for you to start thinking on your own. Anyone who read wikipedia must take everything read with a pinch of salt and a bit of laughter. 98.201.123.22 (talk) 23:13, 2 April 2009 (UTC)


 * Transfer principle states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.
 * For every finite positive integer n, the sentence
 * $$\sum_{i=1}^n\frac{9}{10^i}= 1\;-\;\frac{1}{10^n}\;$$
 * is certainly true of real numbers, so it must be true of hyperreal numbers. Sentence
 * $$\forall n \in \mathbb{N} \quad \sum_{i=1}^n\frac{9}{10^i}= 1\;-\;\frac{1}{10^n}\;$$
 * is not expressible in a 'certain formal language'. It is usually proven with axiom of induction (see Peano axioms), which is a second order statement. When transfer principle is applied to a second order statement, subsets are replaced with internal subsets. If S is the set of all hyperintegers such that above equality is true (assuming the sentence makes any sense for infinite integers), why is S an internal subset? If it is, then by induction it must contain all positive hyperintegers. Tlepp (talk) 00:40, 2 December 2008 (UTC)
 * We are talking first order here:
 * $$\forall n \in \mathbb{N^*} \quad \sum_{i=1}^n\frac{9}{10^i}= 1\;-\;\frac{1}{10^n}\;$$
 * 132.70.50.117 (talk) 05:45, 2 December 2008 (UTC)
 * Sigma $$\Sigma$$ is not a symbol in first order logic. If n is a constant finite integer it's a shorthand. Here n is a variable, so the sentence is not first order. Tlepp (talk) 10:40, 2 December 2008 (UTC)

I might be persuaded to believe that for an infinite hyperinteger x, we have $$\sum_{i=1}^x\frac{9}{10^i}=1-\frac{1}{10^x}$$, though that would still require some work (because unless I'm mistaken, that's a sum over an uncountable index set, which is certain to make quite a lot of trouble). But I definitely don't see the connection to 0.999... Of all the possible uncountably many hyperintegers, which one is supposed to be the one you call $$\infty$$? Or are there supposed to be uncountably many different versions of 0.999...? That would make the notation useless, if it no longer denotes a single number. In any case, I don't think that those lines add significant content or clarity to the section on infinitesimals. If someone actually uses 0.999... for a hyperreal number different from 1, a source might be useful. Huon (talk) 12:56, 2 December 2008 (UTC)
 * Lightstone's hyperreal decimal formulas are prominently displayed just above, would you argue for their deletion, as well? 132.70.50.117 (talk) 13:16, 3 December 2008 (UTC)
 * I find your comments extremely unhelpful. How about discussing why your additions are useful instead of whether something else is? Lightstone is sourced (and looks relevant for discussing 0.333...;...333...), and while I don't think he actually discusses hyperreals,
 * He certainly does!
 * I know where to read it up. Can you give a source for your additions? Huon (talk) 13:26, 3 December 2008 (UTC)
 * My source is Lightstone. I think you misunderstood him.  The point of the second equality is that even for hyperreal decimals, decimal representation is not unique.  Namely, the "full" one (with 9's everywhere) equals 1 on the nose.  On the other hand, all the other ones (the ones that "exist", as he puts it) are strictly smaller, though of course by an infinitesimal amount. 132.70.50.117 (talk) 14:04, 3 December 2008 (UTC)
 * The function $$f(n)=\sum_{i=1}^n\frac{9}{10^i}$$ satisfies the identity above, therefore by the transfer principle its natural extension f* does, as well. Please refrain from reverting a legitimate edit for the fourth time. 132.70.50.117 (talk) 13:01, 2 December 2008 (UTC)
 * P.S. I copied over your formula but now I notice there is an error in it: the summation should start at i=0, while the exponent of 10 should be i+1 instead of i (as in the main article). 132.70.50.117 (talk) 13:13, 2 December 2008 (UTC)

I won't engage in a revert war unless there's some broader consensus, but: Yours, Huon (talk) 14:24, 2 December 2008 (UTC)
 * 1) The function f is not a function on the real numbers. I doubt that statements on integers are covered by the transfer principle. At least our article doesn't say so. Can you provide a source?
 * 2) I still don't see the connection to 0.999..., and you didn't explain it.
 * 3) My formula is correct, yours is off by one: $$\sum_{i=0}^n\frac{9}{10^{i+1}}=1-\frac{1}{10^{n+1}}$$, so if the transfer principle works the way you claim, then $$\sum_{i=0}^\infty\frac{9}{10^{i+1}}=1-\frac{1}{10^{\infty+1}}$$, which is well-defined and different from $$1-\frac{1}{10^\infty}$$ if $$\infty$$ is some given hyperreal.

I am going to remove the disputed content. I have no opinion whether it is true or not, but no reference has been provided, and this is a featured article which means that it must strictly adhere to content policies. In this case, it appears that the information is not verifiable. If you wish to restore the content, then you must give a reference that specifically discusses this example. siℓℓy rabbit (  talk  ) 13:44, 3 December 2008 (UTC)

Anon, you claim that your source is Lightstone. Please provide a page number for the formula you keep adding. A direct quote implicating the transfer principle also seems to be necessary. siℓℓy rabbit (  talk  ) 14:16, 3 December 2008 (UTC)
 * The formulas are on page 247 as already noted. The changes in question are in a section on infinitesimals.  Please leave it to editors knowledgeable about them to resolve this.  132.70.50.117 (talk) 15:39, 3 December 2008 (UTC)


 * I've read up on Lightstone's paper. In his notation, 0.999...;...999... equals 1. Lightstone is not explicit in mentioning that, but he does say (as we already quote) that 0.333...;...333... equals 1/3. (Lightstone also says that every real number has a unique decimal expansion, which is simply wrong, but that's beside the point.) So if I interpret 132.70.50.117's addition in view of Lightstone, he states that 0.999...;...999000... does not equal 1. That's not really surprising, and I don't think it merits explicit mention in the article.

I don't see why it should merit explicit mention any less than the formulas from Lightstone reproduced immediately above, or the more contrived constructions of the strict inequliaty reproduced immediately below, including subtraction-breaking. R* is after all an honest to Robinson field 132.70.50.117 (talk) 04:03, 4 December 2008 (UTC)


 * If we mention it, we should use Lightstone's notation and not invent some notation of our own (and to me, anon's notation obscures what's actually happening by making it look as if some sort of 0.999... does not equal 1, while the opposite is true).
 * Lightstone's extended real numbers are indeed hyperreals, though Lightstone doesn't call them that. Lightstone also doesn't use the term "transfer principle", though that's the main object of study in his paper. I believe he simply predates modern terminology. Huon (talk) 16:05, 3 December 2008 (UTC)


 * The formula in question is not on page 247 of Lightstone's paper. Here is what he says at the top of page 247 (to which I assume you are referring):
 * "...conjecture is false, consider the multiplicative inverse of 3. Since 1/3=.333... in R, it follows that 1/3=.333...;...3... in R*.  So 1/3=a+e where a=1/3 and e=0.  Returning to our conjecture, if .000...;...3... &isin; R* so does 3&times;.000...;...3... i.e., .000...;...9... is a number.  But we have already pointed out .000...;...9...&notin;R*.  Moreover, since the difference of two numbers is also a number, we conclude that neither .000...;...3..., nor .333...;...0..., is a number.  Therefore we cannot break up the decimal expansion of 1/3 in the simple manner suggested by intuition."
 * As far as I can tell, this is the only place on the indicated page where Lightstone mentions anything vaguely relevant to the formula in question. However, since it is obvious from the above that Lightstone does not write the formula Anon claims he does, I am going to flag the content as verification failed.   siℓℓy rabbit  (  talk  ) 17:30, 3 December 2008 (UTC)


 * (reply to Huon) I haven't re-read Lightstone recently but I believe he avoids repeating 9s, which is why he can talk about unique expansions, and why it's not straightforward to say that he essentially deals with 0.999...;...999.... Now, although I am not an expert in hyperreals, from what I've learned I suspect that he could have allowed repeating nines without too much difficulty, in which case you would be right that 0.999...;...999... = 1 and 0.999...;...999000... < 1. However, he doesn't, and we don't know of an author who does make such an analysis. So as much as I'd like to include the material in the article, I think it's best left out. Melchoir (talk) 23:30, 3 December 2008 (UTC)


 * If I understand it correctly, Lightstone says that every internal function from $${^*\mathbb{N}}$$ to the set of digits {0,1,2,3,4,5,6,7,8,9} corresponds to a hyperreal number in the interval [0,1]. The extended decimal notation with one semicolon is the simplest case. If n is an infinite hyperinteger, so are $$2n, n^2, \left \lfloor n^{\pi} \right \rfloor, n^n$$. The extended notation for
 * $$10^{-n} + 2*10^{-2*n} + 3*10^{-n^2} + 4*10^{-\left \lfloor n^{\pi} \right \rfloor} + 5*10^{-n^n}$$
 * is
 * 0.000...;...010...;...020...;...030...;...040...;...050...
 * For every part after semicolon it is necessary to specify a hyperinteger index of the digits. (1 in ...010... correspond to 'n'th digit, and so on...)
 * 0.333...;...000... is not a number, since there is no location where 3's end and 0's begin. (It's an external function)
 * 0.333...;...333...;...333...;...333... is equal to 1/3, but everything after first semicolon is redundant, 0.333... is enough. I find
 * $$.\underset{\infty}{\underbrace{999\ldots}}$$
 * a bad notation, out of the all possible symbols, why use infinity symbol for a hyperinteger. There is a last 9, so 0.999...;...999 or 0.999...;...999000... is better. Tlepp (talk) 13:19, 4 December 2008 (UTC)

Why is such notation unreasonable? Let's take an exaggerated example to make a point: if it is reasonable to denote
 * $$\sum_{i=0}^{{\rm google}-1}\frac{9}{10^{i+1}}$$

by
 * $$ 0.\underset{\underbrace{999\ldots}}$$

couldn't one argue plausibly in favor of denoting
 * $$\sum_{i=0}^{{g\infty gle}-1}\frac{9}{10^{i+1}}$$

by
 * $$0.\underset{\underbrace{999\ldots}}$$

as well? The transfer principle then yields an identity
 * $$0.\underset{\underbrace{999\ldots}}\; = 1\;-^{*}\;\frac{1}{10^{g\infty gle}}$$

and a strict inequality
 * $$0.\underset{\underbrace{999\ldots}}\; < 1$$

with an initial string of an unbounded number of 9's. 132.70.50.117 (talk) 13:50, 4 December 2008 (UTC)
 * I meant googol, not google, sorry. 132.70.50.117 (talk) 13:51, 4 December 2008 (UTC)

There seems to be consensus that this formula does not appear in the cited source, and that it should be removed pending a proper reference which actually discusses the case in question. Have I inaccurately gauged the above discussion? There seems to be only one editor keen on pushing this into the article. Perhaps a wider consensus can be obtained by seeking input from the WikiProject Mathematics? siℓℓy rabbit (  talk  ) 14:08, 4 December 2008 (UTC)
 * I agree with Silly rabbit; since a broad consensus considers these remarks inappropriate for the article, I will remove them. To answer anon's latest question: The notation is unreasonable because firstly the ellipsis suggests that there's no end to the nines. In the finite case, $$\sum_{i=0}^{{\rm googol}-1}\frac{9}{10^{i+1}}$$ could be written as $$ 0.\underset{\underbrace{999\ldots 9}}$$, but not denoting that there's a last 9 is misleading. Secondly, the number under the brace gives the cardinality of the nines: "There are googol nines." But for the sum over all non-negative hyperintegers less than or equal to some given hyperinteger (say &omega;) there are not actually &omega; nines because &omega; is not a cardinal. Thirdly, there is no reason to denote any specific hyperinteger as ∞ because there are lots of infinite hyperintegers, and there's nothing special about any given one. Finally, none of these notations concerning hyperreals (including the "sum to infinity") are used by Lightstone, so they're original research, too. Huon (talk) 15:26, 4 December 2008 (UTC)
 * I do not see a broad consensus at this talk page. The thrust of the mathematical comments has been that Lightstone's notation is preferable; this can certainly be discussed, but please gentlemen do control your revert instinct and concentrate on the mathematical superego. 132.70.50.117 (talk) 15:51, 4 December 2008 (UTC)
 * P.S. Concerning the issue of the last digit "9", I have no objection to including it if you think the notation is misleading without it. Is that a reason to delete an entire section under discussion?  Why do you feel this particular strict inequality fare worse than the broken addition one? 132.70.50.117 (talk) 16:01, 4 December 2008 (UTC)
 * Concerning consensus: Melchoir has agreed that your formula is unsuitable, and Silly rabbit and myself have reverted you. I don't see anyone but you arguing for inclusion.
 * Concerning the reasons for removal: Firstly, it's original research. Your inequality does not appear in Lightstone's paper, and no other reference has been suggested. Secondly, if we use Lightstone's notation, it amounts to: "For a specific infinite natural number ∞, the number $$0.999...;...99\hat{9}000...$$ does not equal 1, where ^ indicates the ∞-th place." While true, I still don't see the connection to 0.999... - a number that's not 0.999... nor its generalization 0.999...;...999... is also not 1. So what? Should we also note that 0.9999 does not equal 1? Why should we include that inequality? I don't think you ever explained.
 * Finally, I begin to get amused by your reversals followed by "Hey, please don't revert me!" Huon (talk) 17:10, 4 December 2008 (UTC)

I've removed the content. Please do not re add the content without a consensus here. Paul August &#9742; 17:40, 4 December 2008 (UTC)
 * Dear Huon, I have argued above that the number thus constructed should be included, because its expansion starts with an unbounded number of 9's. You feel that you "still don't see the connection to 0.999..."  but it seems to me that other people may see such a connection.  The OR charge is unfounded as the editors who have bothered to comment on the actual mathematics, have only expressed reservations about the notation, not the mathematical facts.  You have yourself named one of your edits "Lightstone is properly sourced", which apparently should have settled the OR issue.  How this particular construction of the strict inequality is worse than "addition breaking" has not been explained.  How is it helpful to resolve a mathematical issue by fiat of administrative intervention? 132.70.50.117 (talk) 19:00, 4 December 2008 (UTC)
 * The "Breaking subtraction" section is sourced to Richman, who discusses exactly what we say he does. Especially, he constructs a number 0.999... and discusses its properties.
 * But Lightstone does not actually discuss either 0.999...;...999... or 0.999...;...999000... So anything we say about these numbers is original research. That doesn't mean it's wrong, but that it's not published anywhere. What I meant when I said that "Lightstone is properly sourced" was that the paragraph currently remaining - the one discussing Lightstone's work which you tagged as "not in citation given" - was properly sourced. I was able to find and read Lightstone's article. But I don't see what in that paragraph is supposed not to be in the citation given: The excerpt of Lightstone given by Silly rabbit above explicitly states that 1/3=0.333...;...333... and that 0.333...;...000... is not a hyperreal. How are we misrepresenting Lightstone? Is your issue with the claimed uniqueness of the expanded decimal extensions? I won't fight for that, though I'd have to look up Lightstone once more to see what exactly he says on that issue. Huon (talk) 20:11, 4 December 2008 (UTC)
 * (I've removed the fv tag; it's not appropriate. Even if it were, surely the archives of this talk page teach us not to attempt argument-by-template?) Melchoir (talk) 09:52, 5 December 2008 (UTC)
 * Looking back at Lightstone, he's awfully vague on uniqueness. There's a similar approach in Infinitesimal Calculus By James M. Henle that's explicit about avoiding repeating 9s, but it makes fewer relevant conclusions, and I'm not sure if it's actually more rigorous or not. Melchoir (talk) 10:46, 5 December 2008 (UTC)
 * For the record, Lightstone explicitly claims that decimal representations are unique. Page 245, line 5 from bottom: "Each real number has a unique decimal expansion, so each member of R* has a unique decimal expansion." I believe this is simply wrong because for Lightstone, a decimal expansion of a number x between 0 and 1 is "a mapping d of N into the digits such that x=∑N dn10n, where dn = d(n)" (also p. 245, a little farther down). He doesn't make any provisions to avoid repeating nines (or repeating zeroes); instead, he even writes that "∑N dn10n$$\in$$ R whenever d is a mapping of N into the digits" (p. 246, l. 14), which means that both repeating nines and repeating zeroes must be allowed. Thus, despite his explicit claims his decimal expansions are not unique. Thus, we should possibly remove mention of uniqueness of Lightstone's decimals. Huon (talk) 16:35, 5 December 2008 (UTC)
 * Right, do you think this is sufficient? Melchoir (talk) 21:22, 5 December 2008 (UTC)
 * Looks fine to me. Huon (talk) 22:49, 5 December 2008 (UTC)

A couple of comments to 132.70.50.117: You wrote "do control your revert instinct and concentrate on the mathematical superego"! Now, this is an encyclopedia, not a research forum. There is a clear consensus (excluding one anon user - yourself) that this material is not properly sourced, and especially in a profiled and "controversial" article like this one, that is a problem. (It's not really controversial, but some people find it to be.) Your mathematical ego may be convinced it's right and relevant, and you may even be right about that (I don't know) - but that doesn't matter. Without a source, it's OR. I've occasionally added OR to wikipedia myself when I thought something was missing and I couldn't be bothered to locate a good source. Either it's been sourced by someone else, or it's been left alone because noone found it objectional (or noone bothered to read it...), or it's been removed - and then I haven't fought for it's inclusion, unless I could provide a source. (Well, in one case where I committed OR, I created a neologism, Non-standard positional numeral system, and I got away with that - it seems to have been picked up by mathematical literature, so NOW it could be sourced... I'm not sure whether to be proud or ashamed of that...). If it is really important for you, maybe you should try to look for a brief statement that is unobjectionable, if that is possible (I suppose it isn't in this case).

Another comment: When you post on a talk page, don't do so in the middle of someone else's post - that's considered a bit rude, and may make the thread difficult to read. Generally, under each heading, posts should be in cronological order. Occasionally, one may deviate from this, and e.g. try to make the deviation clear by using indentation. If you need to comment on a specific sentence in someone else's post, you could quote that sentence in your post.--Noe (talk) 10:24, 5 December 2008 (UTC)

A strict non-standard inequality .999... < 1
see here Count Iblis (talk) 17:23, 7 December 2008 (UTC)
 * What about it? It's basically arguing that ".999..." is ambiguous, and when approached in a hyperreal context, could be interpreted to refer to a number other than 1. If you're suggesting that should be added to the article page, I really don't think arXiv posts satisfy WP:N, especially considering how many inaccurate and incorrect things get posted there. Furthermore, I think this is what is being argued in . --Zarel (talk) 06:00, 9 December 2008 (UTC)


 * Mikhail Katz doesn't look like one of those quacks who would post nonsense on the arXiv. Count Iblis (talk) 14:25, 9 December 2008 (UTC)


 * I think the anon in the section above is one of the authors of this paper :) Count Iblis (talk) 14:32, 9 December 2008 (UTC)


 * I dunno. Why don't you ask User:Katzmik?   siℓℓy rabbit  (  talk  ) 15:06, 9 December 2008 (UTC)
 * The text on the arxiv is an expository introduction to Lightstone's ideas (incidentally he has a joint book with Robinson that came out later, but I have not had a chance to see it yet). Note that broken subtraction is thoroughly documented by Richman.  Without doing violence to arithmetic operations, Lightstone similarly documents the following facts: (1) there are infinitely many hyperreals smaller than 1/3 but infinitely close to it; (2) each of them has a decimal expansion starting with an infinite string of 3's, in other words we get
 * $$0.333\ldots <^{*} 1/3.$$

On the other hand, multiplying this by 3 so as to obtain
 * $$0.999\ldots <^{*} 1$$

seems to run into OR problems according to a majority of editors on this talkpage, to whose opinion I defer. Noe raised the issue of mathematical egos. So as to spare his, I have not yet taken this to WPM to get some input from editors actually knowledgeable about NSA. Katzmik (talk) 15:43, 9 December 2008 (UTC)
 * P.S. The anonymous edits were mine; now that I have made the comment above, I certainly don't want to give the impression that more than one editor is arguing in favor of including the hypperreal derivation of .99<1 Katzmik (talk) 15:46, 9 December 2008 (UTC)
 * I think the key point here is whether or not it is meaningful to associate decimal expansions to hyperreals, if it is then certainly you can get the relevant inequality. I haven't read Lightstone's paper, so I won't pass judgement on whether the method given there is a good one or not. Even if it is, though, the fact remains that when people talk about 0.999... they are not talking about hyperreals, they are talking about numbers that are useful in describing the real world, ie. real numbers. --Tango (talk) 17:25, 9 December 2008 (UTC)
 * To be clear, Lighstone does not document the existence of any hyperreals smaller than 1/3 but infinitely close to it, nor does he deal with any decimal expansions starting with an infinite string of 3's except the two that are already mentioned in the present article.
 * Obviously nobody here is denying the existence of these objects -- indeed their existence is painfully obvious -- but let's not attribute them to earlier publications unless they're actually mentioned in those publications. This saves us from having to invent new notation, such as "0.333... <* 1/3". I for one think this is bad notation, but since it isn't attributable, that's a moot argument. Melchoir (talk) 18:45, 9 December 2008 (UTC)
 * There is, of course, the problem already discussed that "0.333...", insofar as it refers to a bona fide hyperreal, is exactly equal to 1/3. But as you say, the point is moot.  siℓℓy rabbit  (  talk  ) 18:48, 9 December 2008 (UTC)
 * The ambiguity seems to be a little artificial. That's as if I were denoting 0.999...9 with, say, more than googol nines (but finitely many) as "0.999..." - look, I've created multiple real numbers denoted as 0.999..., all but one of which are actually less than 1. That's an abuse of notation, and similarly it's an abuse of notation to denote a hyperreal of the type 0.999...;...999000... as "0.999...". Huon (talk) 19:20, 9 December 2008 (UTC)
 * Would the claim of the existence of such hyperreals amount to OR in your opinion? Katzmik (talk) 17:08, 10 December 2008 (UTC)
 * Not in itself, no. Melchoir (talk) 22:14, 10 December 2008 (UTC)
 * Why are we going into such details of Lightsone's ideas here? As we stated above he does not directly apply them to 0.999….  In so far as they do apply they would show that 0.999…;…999…=1. (which is the most reasonable in my opinion candidate for an interpretation of 0.999… in Lightstone's expansion.  Though as Katzmik's article points out there are other ways to interpret 0.999… in the non standard setting.)  I think we this may be a  bit too much of a tangent from the subject of the article. Thenub314 (talk) 11:40, 12 December 2008 (UTC)

error in featured article
As I mentioned in an earlier edit, the current version of the article contains an absurd misrepresentation of Lightstone's paper. I placed a flag to note this, which was removed by one of the editors on the grounds of "avoiding a flag war" or something of that sort. Perhaps we can agree to restore the flag, with an eye eventually to correcting the error.

To explain what the error is, note that one need not even know anything about NSA to catch it. The article (1) states that Lighthouse gives two examples, (2) proceeds to present the examples explicitly, and then (3) it turns out that the first example does not exist! It would appear then that Lightstone only gives a unique example, rather than two?

Moreover, the current version of the page seems to suggest that these are (this is?) the only hyperreal(s) close to 1/3, whereas we have already agreed above that it is not OR to assert that there are infinitely many (this fact is of course documented on numerous NSA pages).

Before we agree on an actual edit to correct this error in what is supposed to be FA, can we agree to place a flag signaling a problem with the text? Katzmik (talk) 08:28, 11 December 2008 (UTC)


 * I don't see this as much of a problem. Lighstone explicitly discusses 0.333...;...000... and shows that it's not an element of R*, compare the excerpt given by Silly Rabbit above. We might make it clearer and only report that Lightstone says that 0.333...;...333... equals 1/3 while 0.333...;...000... does not exist. Lightstone is not interested in discussing representations of numbers "close to" 1/3, but only in the representation of 1/3 precisely. Might I suggest the following formulation:
 * In his formalism, the real number 1/3 is represented by 0.333…;…333…, while 0.333…;…000… does not represent any hyperreal number. (Footnote about Lightstone not discussing repeating nines)
 * That should be a faithful summary carrying Lightstone's meaning without any implications he doesn't make. Huon (talk) 09:32, 11 December 2008 (UTC)
 * Such an edit is certainly preferable, as it removes an apparent logical contradiction. Furthermore, it does not really matter whether or not Lightstone is interested in this particular article in discussing other hyperdecimals starting with an infinite string of 3's.  The main point of his article is that hyperdecimal expansions EXIST.  In particular, the ones for hyperreals infinitely close to 1/3 will necessarily start with an infinite string of 3's.  We seem to have agreed above that the existence of such hyperreals is not OR.  It seems odd not to include a remark concerning them in an article that discusses numerous other constructions of the strict inequality, particularly since Lightstone, through the fundamental fact of the existence of hyperdecimals, achieves such a construction without doing violence to arithmetic operations. Katzmik (talk) 16:02, 11 December 2008 (UTC)
 * I'm fine with Huon's text. Katzmik, I don't know what you mean by "constructions of the strict inequality". Melchoir (talk) 06:18, 12 December 2008 (UTC)
 * The real question is how would "0.999..." be interpreted as a hyperdecimal? I can only see two plausible interpretations, 0.999...;000... or 0.999...;999..., the former doesn't exist and the latter is 1 (assuming I'm understanding these correctly). There is no interpretation that gives a number strictly less than 1. Sure, there are hyperdecimals that start with an infinite sequence of nines and are infinitely close to 1 by strictly less than it, but you would never write any of them as "0.999..." since the notation would be horribly ambiguous, or otherwise completely arbitrary. --Tango (talk) 12:46, 12 December 2008 (UTC)
 * Melchoir above wrote: "I don't know what you mean by "constructions of the strict inequality" ". I would like to clarify that every hyperdecimal expansion other than the one mentioned in the article will satisfy the strict inequality .333... < 1/3 (and therefore, multiplying by 3, we obtain a strict inequality .999... < 1).  Here, of course, ".333..." is shorthand for a more complex hyperdecimal containing semicolons ";" as in Lightstone's notation.  An arxiv article obviously cannot be used as a reference (not until it is published anyway), but if you are interested you can read a more detailed explanation at the arXiv link above.  In particular, given a hyperinteger H, we get an example that I presented a week ago using the underbrace notation.  The latter does not seem to have found favor with editors here, so it need not be included.  At any rate,  Lightstone's presentation of the existence of hyperdecimal expansions does imply immediately that one can construct an expansion starting with an infinite string of 9's, which produces a hyperreal strictly smaller than 1.  This is done, of course, in the context of a totally ordered field *R, with arithmetic operations intact.  Katzmik (talk) 20:40, 13 December 2008 (UTC)
 * As per Tango's comment: I think an 11th grade student can legitimately ask: What does Teacher mean to happen exactly after nine, nine, nine when he writes dot, dot, dot? As professional mathematicians we have gotten used to the idea that .999... is a real number, but after all this is only a convention.  Strictly speaking, we are talking about "the limit of sum from 1 to n as n tends to infinity of the string .99...9 of length n".  If we allow a wider interpretation of the ellipsis (dot, dot, dot), a lot of student frustration would disappear.  Katzmik (talk) 20:47, 13 December 2008 (UTC)
 * It is not accepted practice to write ".333..." for any and every decimal that starts with 3s but eventually contains other digits. If we allowed this, we would be able to interpret all three statements .333... < 1/3, .333... = 1/3, and .333... > 1/3 as true -- not only in the hyperreals but in the reals as well. This would render ellipses useless for doing and communicating mathematics. Melchoir (talk) 22:02, 13 December 2008 (UTC)
 * And yet 3.14159... contains all sorts of "other digits", almost all of which are unknown. What exactly is the reason that I can't use "3.14159..." to describe a number that is 10-99 less than π? (Other then the obvious that people would assume I was talking about π.) Algr (talk)
 * Why do people ask questions like that? "Why can't I use '3.14...' to describe a number other than pi, other than that people would assume I'm talking about pi?" "What is the answer, other than the answer?" It would presumably be the null set.
 * In any case, to be more specific: it's because ".333..." is recognizably a pattern that continues with 3's and is extremely doubtful to refer to any other pattern, while "3.14..." is not recognizable as any pattern other than the digits of pi. --Zarel (talk) 12:38, 14 December 2008 (UTC)
 * Again, the point is that the precise meaning of the expression ".999...9, n times" is that the digit 9 occurs precisely n times. Meanwhile, the expression "infinitely many 9's" is only a figure of speach, as "infinity" is not a number in standard analysis.  Whenever a precise meaning is attributed to the phrase "infinitely many 9's", it is almost invariably in terms of limits.  In the hyperreal line, there is a notion of an infinite (sometimes called unbounded) hyperinteger.  Denoting such an entity H, we can consider a hyperreal repeated decimal where the digit 9 occurs precisely H times.  Such a number can be denoted suggestively by .999... with an underbrace indicating that 9 occurs H times, resulting in a strict inequality .999... < 1 (with the underbrace indicating that we are not talking about the standard real).  In Lightstone's notation, this hyperreal would be expressed by the hyperdecimal .999...;...9, the last digit 9 occurring in the H-th position.  As far as limits are concerned, from the hyperreal viewpoint we have
 * $$\lim_{n\to \infty} u_n = {\rm st}(u_H),$$
 * where "st" is the standard part function which "strips off" the infinitesimal part. What is bothering the students is the unacknowledged application of the standard part function, resulting in a loss of an infinitesimal.  If you see Schmidt's calculations on the "Arguments" page, you will see a perfect illustration of this.  Katzmik (talk) 13:05, 14 December 2008 (UTC)
 * What do you mean infinitely many is a figure of speech? There are perfectly well defined notions of finite and infinite.  Momentarily viewing a decimal expansion simply as a list of digits it is clear what infinitely many 9's means.  Describing why this decimal expansion represents the number 1 does involve a limit, but so does any non-terminating decimal.  If you want to get very technical you also use a limit in interpreting 1.000… (where I of course mean all digits not listed are 0.). Thenub314 (talk) 16:31, 14 December 2008 (UTC)
 * I'm unsure what you mean by "What is bothering the students." If you were referring to the students with disagreements to the idea that 0.999...=1, I don't think many of them would be bothered by the standard part function, partially because few of them have heard of hyperreal numbers, and of those, fewer still would think of 0.999... as an unbounded hyperintegral number of 9's. I mean, nothing about a set of ellipses really shouts "hyperreals number", you know. :) --Zarel (talk) 14:14, 14 December 2008 (UTC)
 * Zarel: You are absolutely 100% wrong here.  Your assumption illustrates why there is no other subject in math that has such widespread resistance as .999...=1.  ANYONE who rejects .999...=1 is thinking about infinitesimals, even if they have never heard of the words "infinitesimal" or "hyperreal".   When an academic responds to 0.999...<1 by trying to lock the subject into the Real set, what they have done is told the questioner that no one has ever though of infinitesimals before.  If an academic demonstrates that he doesn't even understand the question, why should the student then accept the answer?  Academics only have their own arrogance to blame for the current situation with .999...  Algr (talk) 19:36, 14 December 2008 (UTC)
 * Try not to say "absolutely 100% wrong"; that kind of belligerent language is uncalled for - I don't mind, but it's not a good habit to get into. Regardless, I have made no assumption - in fact, I state explicitly that "I'm unsure" and "I... think".
 * In any case, I admit that it is likely that someone who rejects 0.999...=1 is thinking about infinitesimals, but I meant that I doubt they are thinking of hyperreals, specifically. After all, hyperreals are not the only kind of infinitesimals.
 * Further, someone who locks 0.999... to the reals isn't necessarily ignoring the existence of infinitesimals - he/she could have a perfectly valid reason for disregarding them (such as by convention). Although it's true that educators could be better at identifying that a student is thinking of infinitesimals and explaining why the notation of "0.999..." does not refer to any infinitesimals, it's a bit harsh to to blame their arrogance. Mathematicians are not psychologists - they might not realize that students don't understand this.
 * And I don't understand how criticism of mathematical educators (nor the usage of the term "academic" as some sort of insult) belongs on the discussion of a Wikipedia article. --Zarel (talk) 07:26, 16 December 2008 (UTC)
 * I'm sorry about coming across that way Zarel. This discussion can be very fatiguing, and concepts of reasonable discussion tend not to apply to people who are "wrong" such as myself.  For example, notice below that Huon had no problem with you speculating about what students who reject .999...=1 think, but even though I am IN that group he demands that I need "Some sort of backup." for my explanation.  As for "criticism of mathematical educators", this article is doing precisely what makes smart people reject it's answer.  It is at best ill advised, and at worst, intentionally dishonest.  An article about .999... serves no purpose for those who already understand Reals and Infinitesimals, so any article to be useful must understand what the reader really wants and needs to know. Algr (talk) 08:37, 10 February 2009 (UTC)
 * Algr, time and again you've been asked what hyperreal number 0.999... should be if it's not 1. I still can't find an answer. By Katzmik's suggestion 0.999... is not a number at all, but a class of closely related numbers - a class that does contain 1, by the way. Furthermore, the claim that "anyone thinks of infinitesimals" would need some sort of backup - and even that wouldn't mean people think of hyperreals. Maybe they prefer surreals? On an unrelated note, I strongly object to Katzmik's terminology when he says that "the digit 9 occurs precisely H times". Given numbers xH and xH+1 with nines in all digits up to the one corresponding to 10-H and 10-(H+1), respectively, for an unbounded hyperinteger H, there's a bijection between the sets of "9" digits. Thus, xH and xH+1 contain the same amount of nines. Huon (talk) 21:43, 14 December 2008 (UTC)
 * Here are two answers for you, Huon: .999... = WR(W) = 0_R1_-1 ≠ 1.  Happy now? Algr (talk)
 * Unfortunately not, because I've never seen such notation before. Could you point me to an explanation of what 0_R1_-1 is defined to be? I suppose WR(W) is a recurring Hackenstring? Then you have either given up on hyperreals, or you'd have to provide a map from Hackenstrings into the hyperreals. Huon (talk) 17:35, 10 February 2009 (UTC)
 * At the end of transfer principle article, it says: If n is an infinite integer, then the complement of the image of any internal one-to-one function ƒ from the infinite set {1, ..., n} into {1, ..., n, n + 1, n + 2, n + 3} has exactly three members. So in this context it is acceptable to say 'precisely H times'. On the other hand there's an external bijection between the sets of "9" digits. Thus the cardinality of nines is same. Imho the internal/external thing seems to suggest that the marriage between non-standard analysis and ZFC set theory is forced and unnatural. When you ask what hyperreal number 0.999... should be if it's not 1, I feel it's like asking what is the cardinality of the real numbers. The question is more important than the answer. Tlepp (talk) 11:40, 15 December 2008 (UTC)

(rewrap) Katzmik, you wrote, "As far as limits are concerned, from the hyperreal viewpoint we have
 * $$\lim_{n\to \infty} u_n = {\rm st}(u_H),$$

where "st" is the standard part function which "strips off" the infinitesimal part." This is an important equation, but it holds only if we already know that u converges in the first place -- equivalently, that the right-hand side is a constant function for all infinite hypernatural numbers H. I assume you already knew this, but I want to emphasize for the folks at home that if Alice hands Bob a sequence s and asks for a number that describes its limiting behavior, it's not as if Bob can just hand back the information-rich hyperreal number (*s)(H), and then if Alice narrowmindedly demands a real number, tell her to take the standard part. In other words, we do not have here a hyperreal-valued procedure "truelim" such that lim = st(truelim), which is the implication some people might take from the equation. Melchoir (talk) 00:22, 15 December 2008 (UTC)

Question.
We currently have in the section on alternative number systems the statement: "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 reciprocals."

Which is a fine thing to say about alternative number systems. But how does it relate to 0.999… ? Is there something interesting we can add about this number in this setting? If not, should we have this comment in the article? Thenub314 (talk) 11:15, 12 December 2008 (UTC)


 * I agree, such a comment belongs in infinitesimal, not here, unless those number systems have a meaningful (and commonly used) interpretation of the notation "0.999...". --Tango (talk) 12:41, 12 December 2008 (UTC)

In the mean time I will take it out until it can be made to fit in better with the subject at hand. Thenub314 (talk) 12:46, 12 December 2008 (UTC)


 * I could ask that question of the entire "alternative number systems" section, since none of it actually refers to a value written "0.999...", but rather to tenuously-related numbers. --Zarel (talk) 12:41, 14 December 2008 (UTC)

if Alice narrowmindedly demands a real number
Hi, There are several interesting comments above that I would like to respond to, but I am a little busy now and only have time for one reply, therefore I hasten to correct the impression that Melchoir's comment may have given that the two definitions of limit somehow agree only if the limit already exists. NSA does provide an alternative definition of limit in the full sense thereof. Thus, the limit of f(x) as x tends to zero, equals zero, if and only if the following condition is satisfied: Katzmik (talk) 11:28, 15 December 2008 (UTC)
 * if x≠0 is infinitesimal, then f(x) is infinitesimal, as well.


 * My point, translated to this case, is that the condition demands all x. If you just check f(x) for a single nonzero infinitesimal x, you can't conclude that f converges. Melchoir (talk) 21:35, 15 December 2008 (UTC)

precisely H times
Huon wrote:
 * I strongly object to Katzmik's terminology when he says that "the digit 9 occurs precisely H times".

Perhaps the language I used was misleading. I merely meant to say that the "last" digit 9 occurs at position labeled by hyperinteger H. The value of the hyperdecimal depends on the choice of the hyperinteger H.  One certainly cannot assign a unique hyperreal value to .999..., as Huon correctly points out. Still, the values one can assign do turn out to be strictly smaller than 1. Katzmik (talk) 16:17, 15 December 2008 (UTC)


 * For the ones that are less than one I would object to calling them 0.999… on the grounds that you haven't specified all of the digits. For example to specify the hyper-real 0.999…;…998999… you must tell me which position the 8 occurs in.  Is there a reference we have for referring to 0.999… in the non-standard setting as a number less than 1? Thenub314 (talk) 16:33, 15 December 2008 (UTC)


 * Nobody with any sense will claim .999... is a number less than one or else be called a quack. Yet, the accepted notation .abcde... for a decimal expansion as a shorthand for the limit has had the adverse side effect of 11 graders being unable to follow the discussion very well, as they certainly have not been told about limits, nor of the fact that .abcde... is merely a notational convention.  Incidentally, the article on the arxiv provides a hyperreal explanation, along similar lines, for why a calculator will sometimes return .999999 when you expect it to return 1.  It does not seem an entirely unreasonable model.  Katzmik (talk) 17:04, 15 December 2008 (UTC)


 * So you're arguing that the notation of .abcde... should not refer to the limit. Do you think the article should be changed to reflect that? If so, that's WP:OR unless you have a citation arguing this. If not, I'm not sure why this belongs in the discussion section of a Wikipedia article.
 * In addition, if you're referring to the arXiv article linked earlier in this discussion page, that article doesn't explain why a calculator will sometimes return .999999 when 1 is expected (presumably, it should be obvious to readers that it's because of rounding errors), but instead just mentions it as an introduction to a discussion of theoretical hypercalculators. --Zarel (talk) 07:08, 16 December 2008 (UTC)

The current situation
Algr wrote above:
 * If an academic demonstrates that he doesn't even understand the question, why should the student then accept the answer? Academics only have their own arrogance to blame for the current situation with .999...

I would say that this is taking the argument too far. It certainly cannot be argued that the hyperreal model should replace the reals altogether, no more than it can be argued that the field of the reals should replace the field of the rationals altogether. On the other hand, just as it is convenient to extend the field from Q to R so as to be able to find roots of polynomials, or to describe the length of the circle, it is similarly convenient to extend R to R* with a view to applications in calculus (as well as sorting out the thorny .999... issue, as I have tried to suggest). I am still hoping to get Schmidt's reaction to this discussion. Katzmik (talk) 17:34, 15 December 2008 (UTC)


 * I never argued that the hyperreal model should replace the reals altogether. I am saying that the Reals are an inadequate number space to describe .999...   This is because the Reals are defined as not including infinitesimals, and infinitesimals are central to any objection to .999... = 1.  It is the same as insisting that a discussion of "one divided by two" take place in [natural numbers].  Algr (talk) 08:53, 10 February 2009 (UTC)


 * Infinitesimals aren't needed precisely because we are talking about real numbers. It is clear that you are bringing up a point that absolutely does not apply. Your logic is flawed to a degree that I have to assume that you have no mathematical background. Tparameter (talk) 01:26, 25 March 2009 (UTC)

Plausible interpretations
Tango wrote above:
 * The real question is how would "0.999..." be interpreted as a hyperdecimal? I can only see two plausible interpretations, 0.999...;000... or 0.999...;999...

Frankly I don't see why, if those two are both plausible interpretations, the hyperdecimal 0.999...;...9000... would not be a plausible interpretation, as well. It is true that Lightstone does not mention it in his article, but then again he would rather have his article published than be called a quack who thinks .999... is less than 1 :) Katzmik (talk) 17:45, 15 December 2008 (UTC)
 * How about this application of the transfer principle: For every integer n, the n-th digit of 0.999... is a nine. So for every hyperinteger H, the H-th digit of (0.999...)* is a nine. That makes (0.999...)*=0.999...;...999...=1. And (0.999...)* is the obvious candidate to be named 0.999..., if any hyperreal number should be thus denoted. It has the added advantage of actually being a fixed number, while 0.999...;...9000... is any of an infinite set of different numbers depending on the position of that last nine. Another angle would be this: In the reals, we have $$0.999...=\sum_{n\in\mathbb{N}}\frac{9}{10^n}$$ (if 0 is not a natural number for simplicity's sake). The two obvious hyperreal candidates are $$\sum_{n\in\mathbb{N}}\frac{9}{10^n}=0.999...;...000...$$ and $$\sum_{n\in\mathbb{N}^*}\frac{9}{10^n}=0.999...;...999...$$ . Huon (talk) 18:21, 15 December 2008 (UTC)

Out of curiosity, just so everyone knows what's going on here (myself included), which hyperreal is the one referred to by the notation 0.999...? Thanks in advance, siℓℓy rabbit  (  talk  ) 21:53, 15 December 2008 (UTC)
 * There isn't a set convention, that's the point. --Tango (talk) 22:16, 15 December 2008 (UTC)
 * In that case, isn't there already a convention that 0.999... = 1 in the reals, and since the hyperreals are a superset of the reals, shouldn't any operation of real numbers remain unchanged, and thus 0.999... is still 1? --Zarel (talk) 06:59, 16 December 2008 (UTC)
 * How many nines do you mean after the semi colon? Any number you pick is completely arbitrary. There are logical reasons for choosing ;000... or ;999..., there is no logical reason for choosing anything else that I can see, it's just "pick a number any number". --Tango (talk) 22:16, 15 December 2008 (UTC)
 * True, and there are logical reasons for not choosing ;0000.... because it is not a number in Lightstone's setup. So really I feel the most reasonable interpretation for 0.999... is the number all of whose digits are 9, all of the regardless if they occur at a finite integer or unbounded hyper integer. Thenub314 (talk) 22:33, 15 December 2008 (UTC)

Application of the transfer principle
Huon wrote above:
 * "How about this application of the transfer principle: For every integer n, the n-th digit of 0.999... is a nine. So for every hyperinteger H, the H-th digit of (0.999...)* is a nine. That makes (0.999...)*=0.999...;...999...=1."

That's in fact a very clear explanation of why all the extended digits of the standard real .999... are indeed 9, as Lightstone notes. I have been meaning to point this out, thanks for your help. Using the modern terminology, the explanation is clearer than what appears in Lightstone. Incidentally, I have been meaning to point out that, while it is true that H and H+1 will have the same cardinality, actually finding a bijection is impossible internally. This is in fact the crux of the matter that makes NSA work. Thus, H can be treated as an honest "integer" and everything you can do with an integer, you can do with H. For instance, you can partition a compact interval into H equal subintervals (of course, each will have infinitesimal length). In non-standard calculus, this type of argument replaces tedious epsilon, delta estimates. For instance, one can give a short elegant proof of the extremum value theorem this way. I would like to add that the other claim contained in Lightstone can also be explained more clearly using modern terminology. If .999...;000... existed, we could consider its set of nonzero digits. This set is precisely N sitting inside N*. But N is not an internal subset of N*. Hence such a hyperreal does not exist. Katzmik (talk) 15:36, 16 December 2008 (UTC)
 * Concerning cardinality, I'm obviously too deeply rooted in the "standard world" and not accustomed enough to the non-standard intricacies. Thanks (also to Tlepp) for pointing out how the hyperreals allow to "count" infinities. Concerning 0.999...;...000..., I believe Lightstone actually shows that N is not an internal subset of N*, which he seems to regard as counterintuitive. But this entire discussion, while both fascinating and educational, becomes more and more off-topic. Are there proposed changes to the article? I would object to adding a sentence on calling more-or-less arbitrary hyperreals with infinitely many nines "0.999..." without a reference. If there are no proposed changes, we should probably move the discussion to the arguments page. Huon (talk) 15:50, 16 December 2008 (UTC)
 * I would suggest three edits: (a) explanation in terms of the transfer principle why the hyperreal .999...;...9... is in fact 1, as above; (b) explanation in terms of internal sets why ;000... does not exist; (c) a remark to the effect that if .999... is interpreted as the sum of H-infinitely many summands (for any specific hyperinteger H), then one obtains a hyperreal infinitely close to, but strictly smaller than, 1. Incidentally, I have been meaning to point out that the reason hyperreals are preferable to surreals, etc. is because the former do have a notion of extended decimal expansion.  The latter is essentially a special case of the transfer principle, which is not available for the surreals, making the latter merely an intellectual curiosity with limited relevance to calculus. Katzmik (talk) 16:06, 16 December 2008 (UTC)
 * I think an extensive explanation about properties of hyperreals is out of place in this article, which is about the real number 0.999... I think that the current one-paragraph coverage of nonstandard analysis is the correct amount of coverage. Extending what is already an "extra topic" into a long explanation would give more weight to that area than is due. On the other hand, if there are references available for the three facts just mentioned, I would not oppose adding a single sentence for each of them, with a reference to back it up. It's the idea of long paragraphs of explanation that I don't find ideal. &mdash; Carl (CBM · talk) 16:10, 16 December 2008 (UTC)
 * Fine; according to Huon, the "internal" explanation of ;...0... is already in Lightstone, so that should do. I will have to re-read it again to see how close he comes to giving the "transfer principle" explanation of ;9... Katzmik (talk) 16:20, 16 December 2008 (UTC)
 * Wait, I am not an expert, but I two problems with the current changes. One, I read Lightstone's article and don't recall him talking about internal sets.  Secondly, Lightsone's article predates the article defining internal sets by 5 years.  Also in his setup, he explicitly says that *R has unique decimal expansions, so if we want to talk about the non-uniqueness arising from 0.999...;...999... and 1.000;...000... then we need another source.  Thenub314 (talk) 21:46, 16 December 2008 (UTC)
 * Lightstone simply adopts the standard convention that every terminating decimal is replaced by a tail 999... Furthermore, an administrator has approved of these additions to the article, so I think your revert is conterproductive. Katzmik (talk) 10:11, 17 December 2008 (UTC)
 * I am a bit unsure which administrator you mean, but I see glancing above that Carl is an administrator. Is this the admin your referring to? In all good conscience I cannot leave an explanation that uses internal sets and references Lightstone's article.  I realize that it is a weak point, but Wikipedia is not a storing house for all facts that are true, but only verifiable facts.  I understand the convention that Lightstone is adopting, and I understand from a mathematical point of view that it is completely trivial to connect his article to a discussion of 0.999.... But if we are going to add some explanation about something it should be that the will not find the discussion we are describing in Lightstone  or any other reference.  Carl's comment includes the phrase "if there are references available,"  and Lightstone's article is not the reference for the three facts you give above. Thenub314 (talk) 10:58, 17 December 2008 (UTC)
 * We should keep in mind that admins don't wield special rights in content disputes...
 * I've just reviewed Lightstone (yet again):
 * He doesn't address 0.999...;...999... or any other decimal beginning 0.999....
 * He doesn't use the phrase "transfer principle". He does observe, "...each statement that is true for R, is true for R* when interpreted in the language of R*." on p.244. By p.247 he says "Since 1/3 = .333... in R, it follows that 1/3 = .333... ; ...3... in R*. I think this is safely recognizable as an application of the transfer principle.
 * He doesn't use the phrase "internal set". He does observe that "(P(R))* != P(R*)" in connection with two paradoxes on pp.247-8 and discusses what this means in the next pages. In this sense you could argue that Lightstone would recognize what we call internal sets. However, he never makes the connection back to the decimal paradoxes on pp.246-7. Making the connection for him would be one too many stretches.
 * So, I wouldn't be opposed to adding "(due to the transfer principle)" at the appropriate place. It's five words, which is cheap. Melchoir (talk) 11:31, 17 December 2008 (UTC)

(rewrap) I am also do not have an issue with using the phrase "(due to the transfer principal)". The question I have, just to make sure we are on the same page, do we apply his ideas to 0.999...? (I suppose I feel that is a bit too much of a stretch, but I could be persuaded otherwise.) Thenub314 (talk) 11:50, 17 December 2008 (UTC)
 * If you're asking me, I'd rather not. Melchoir (talk) 12:08, 17 December 2008 (UTC)
 * Sorry, yes I was asking you directly, I should have made that clear. I agree, I also would rather not. Thenub314 (talk) 12:35, 17 December 2008 (UTC)
 * We are not bound to Lightstone when it comes to NSA. If you would like a reference for applying the transfer principle to a real statement such as d_n=3 (or 9), you can find it on pages 28-29 in Keisler's book.  Katzmik (talk) 12:56, 17 December 2008 (UTC)
 * Ok, to keep things civil and calm I wait a bit before taking out comments about internal sets. You asked I speak to and administrator before making changes, will do.  I have asked Carl, since he was following this conversation up to some point.  Thenub314 (talk) 14:38, 17 December 2008 (UTC)

Point of clarification: Administrators have no special authority concerning editorial decisions. I believe Carl was speaking above as a fellow (well-respected) editor not as an administrator. Paul August &#9742; 15:23, 17 December 2008 (UTC)

Thanks Your of course correct. It was clearly stated above, and I was aware of this before hand. Of course, Carl said basically the same thing to me. I just wanted to keep tempers down, so I played along. Thenub314 (talk) 15:42, 17 December 2008 (UTC)
 * I hadn't noticed that part of Melchior's comment above, sorry to be beating a dead horse ;-) Paul August &#9742; 18:00, 17 December 2008 (UTC)
 * I see roughly speaking a consensus about the new edit among Huon, silly rabbit, carl, and myself. Furthermore, I would like to point out that thenub does not seem to realize that there is a difference between internal sets (an integral part of Robinson's theory) and internal set theory (a later innovation by Nelson).  I am happy indeed to see that you are interested in learning more about NSA, but please make edits out of somewhat more complete knowledge. Katzmik (talk) 16:08, 17 December 2008 (UTC)
 * Your correct that I was confused about the origin of the term internal set. (Not particularly its meaning, as I looked that up on wikipedia! :) ).  Do you mean reading Lightstone's article is not enough background to edit the section about Lightstone's ideas?  I don't buy it.  He doesn't use this language and we are pointing the reader to him as a reference. Thenub314 (talk) 18:04, 17 December 2008 (UTC)
 * I propose to add Keisler as a reference, it is much more detailed and clear. Katzmik (talk) 18:22, 17 December 2008 (UTC)
 * If Keisler discusses decimal expansions, I think it would be an excellent reference. Thenub314 (talk) 19:48, 17 December 2008 (UTC)
 * I'm almost ready to agree at this point, so I looked through Keisler. It's available here for reference. Decimal expansions aren't discussed at length, and they aren't in the index. Example 2 on pages 493 and 495 contains a straightforward proof that the (ordinary) sequence of finite decimal expansions of the real number π converges to π, using hyperreals. Sequences of finite decimals reappear in Ex.8, p.500; Table 9.2.1, p.502; and Ex.1, p.503. Infinite decimals appear in Problems 9-13 on p.510; there they are defined as infinite series. As far as I can tell, Keisler doesn't treat the full decimal expansion of a possibly-nonstandard hyperreal; for him decimal sequences are extended to infinite hypernatural numbers only as a calculational tool to find real limits. Compared to Lightstone this is disappointing.
 * There is one alternative that I found, Infinitesimal Calculus by James Henle, that all but spells out the use of the transfer principle in finding the decimal expansion of the hyperreal 1/3, in Exercise 1, p.39 and its Hint. Henle also talks about the decimal expansion of a nonstandard number at the bottom of p.38 and specifically in Exercise 3, p.39. However, the abuse of decimals in Theorem 4.5, pp.39-40 is so utterly bogus that I don't really think of the book as a reliable source on this topic. Henle isn't used in any mathematics course syllabus on the web, which generally isn't a good sign.
 * I really wish I could report more positive findings...! Melchoir (talk) 06:31, 18 December 2008 (UTC)


 * My general opinion is still as expressed above - if there are suitable references, I think a few one-sentence statements would increase the completeness of the article without being too tangential. However I have not consulted any references and have no opinion about the material that was added and then removed. My edit to the article was only to restore the space between two paragraphs. I will be very busy over the next week or two, and so I won't be able to follow this discussion very closely. &mdash; Carl (CBM · talk) 20:57, 17 December 2008 (UTC)


 * My edit was mostly for style, and should not be interpreted as a conclusive endorsement of the present version of the article. However, based on my own understanding of Lighthouse's paper, the present edit is not an unreasonable rendering of the relevant details.   siℓℓy rabbit  (  talk  ) 22:14, 17 December 2008 (UTC)

I say let's work on it here, until we have something we agree on.

Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses). A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)&lowast;. .  Lightstone shows how to associate to each extended real number a sequence of digits 0.d1d2d3…;…d&infin;&minus;1d&infin;d&infin;+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.

Here is one attempt I have made to improve it. More could be done. Let me explain where I think it could be improved. First we should provided a reference for the fact that N is not an internal set. This should be utterly trivial. Secondly, it is not clear that this clause explains something to someone who is not familiar with non-standard analysis. A reader may ask why does the index set need to be an internal set? (And they won't find the answer in Lightstone, so if we could provide a reference that would answer this question I would be much, much happier.)  There could be some confusion what I mean by "indexed by". A reader might wonder why multiplying by 3 results in 0.999…;…999… (that is why in this new system does the arithmetic for decimals work the way we expect.)  But this is rather implicitly assumed in Lightstone. One idea is that we may not need to mention counter examples at all, does it really add to the discussion of 0.999…? (And if we do shouldn't we use 0.000…;…999… since he does specifically talk about this, and it is more to the point?). I hope everyone interested will feel free to edit the above paragraph Thenub314 (talk)

The more I think about it, the more it seems mentioning 0.0000…;…333… 0.333…;…000… is just not part of the subject at hand. So I will try taking it out.Thenub314 (talk) 14:03, 19 December 2008 (UTC)
 * I am not sure what you are referring to. The article does not contain any such example.  Katzmik (talk) 17:56, 20 December 2008 (UTC)

Sorry, simple typo. I corrected it above. Thenub314 (talk) 18:15, 20 December 2008 (UTC)


 * There are several typos in the suggested text given above. There is at least "Lightstone developed of decimal expansion" and "Lightstone shows how to associated to each".  This text has been copied to the article (twice now) and so needs to be fixed.  I would fix it directly in the article but I suspect that this passage, currently in dispute, will be copied back to the article at some point in the future.  So: could Thunub314 (or anyone who claims "ownership" of the passage) please fix the typos both above and in the main page?  (I do not have the context or mathematical knowledge to comment either way on the appropriateness of the passage itself.  My understanding of non-standard analysis is either zero or infinitesimal.) Molinari (talk) 22:37, 27 December 2008 (UTC)


 * Thanks for pointing these out. I know I don't own anything on Wikipedia and I am sorry if I have been a bit rash.  Thenub314 (talk) 23:23, 27 December 2008 (UTC)


 * I apologize. My note was harsher than it should have been.  My point was that there were typos (which were apparently going unnoticed in the various edits), not that you had done anything particularly wrong. Molinari (talk) 21:12, 29 December 2008 (UTC)

Multiplication by 3
In response to thenub's comment above, I would like to mention that multiplication by 3 acts by multiplication by 3 on all finite digits d_n. Therefore by the transfer principle, it also works by multiplication by 3 on all digits in the extended expansion. I freely admit that if you put your mind to it you could find WP:OR fault with this, but I think it may be difficult to document multiplication by 3 in secondary sources. Incidentally, is the multiplication table properly sourced at this encyclopedia ;? Katzmik (talk) 17:11, 21 December 2008 (UTC)
 * As I said above these types of manipulations are implicit in Lightstone's articles, and that is why I left it in the version I put on the page. I was simply playing devils advocate and criticizing my modification.  Wikipedia has an exception for routine calculations, and this is almost a routine calculation. Provided your willing to accept that arithmetic works the way you expect it should, and I just wanted to acknowledge this fact.  I admit it would be hard to find any references that discuss how to this arithmetic works (I turn up no references on google scholar and 1 on mathscinet.)  I have put back the version I had previously put on the page. Let's discuss its merits and faults and fix it. Thenub314 (talk) 23:58, 26 December 2008 (UTC)

edits by Thenub
A few months ago, I proposed some material an a non-standard inequality .9<1 where the repeated 9 occurs H times, H being an infinite hyperinteger. At the time, some editors have objected on the grounds that the material seems to violate WP:OR. I have thought about the matter and have been able to express the argument in a fashion that is irreproachable from this viewpoint. However, when I attempted to add it to the article at the appropriate place, Thenub reacted by repeatedly deleting the material with the explanation that it is "too long" and constitutes "undue weight". There must be an appropriate WP:WEIGHT item that he is referring to, but his approach to working on this article seems less than fully constructive. I would like to obtain input from fellow editors. Katzmik (talk) 15:56, 14 January 2009 (UTC) P.S. I should add that the measly sentence he left in is completely incomprehensible to anyone who has not followed the detailed discussion in this space. For instance, H is not defined at all. The genius kid who can figure out that it stands for an infinite hyperinteger will benefit from this section of the article, but as for the rest of the public, it will be non-plussed. Katzmik (talk) 16:00, 14 January 2009 (UTC)
 * I see that the appropriate definition has been added, and I appreciate the revert of the delete. The succinct explanation of why exactly the strict inequality holds really does not seem to be superfluous here, as the reader is not necessarily used to working with the hyperreals any more than with broken subtraction.  Is two lines really too long?? Katzmik (talk) 16:57, 14 January 2009 (UTC)

Proposed notation change?
I propose that we change


 * $$0.999\ldots := \lim_{n\to \infty} u_n$$

to


 * $$0.999\ldots \equiv \lim_{n\to \infty} u_n$$

which is prettier and more symmetrical. If the intention of the former notation was to represent the "one-wayness" of the definition that establishes the equivalence, we already have the words "is defined by" in the sentence above the equation, and once the definition has been made, the equivalence is two-way, as in the latter notation. -- The Anome (talk) 09:10, 2 February 2009 (UTC)


 * I do not like the $$\equiv$$ notation, if we change it, I suggest simply an equals sign. Thenub314 (talk) 12:16, 2 February 2009 (UTC)


 * An equals sign would be the more traditional notation in mathematics anyway. &mdash; Carl (CBM · talk) 14:41, 2 February 2009 (UTC)


 * I read $$\equiv$$ as "equal, by definition" in this case. -- The Anome (talk) 20:52, 2 February 2009 (UTC)


 * So do I, and that would make sense here - except the article is more accesible using a plain equals sign and letting the text make it clear it's a definition.--Noe (talk) 14:20, 3 February 2009 (UTC)

I've now removed the whole thing, and it reads much better now. We were already saying the same thing in words in the sentence immediately above, and saying exactly the same thing directly afterwards, but this time pulling some notation out of the air in order to do so, without even defining the terms $$u_n$$ precisely, aided neither understanding not rigor. -- The Anome (talk) 12:08, 4 February 2009 (UTC)

Another long division
The result can also be obtained by using long division, albeit with a twist. Divide 1 into 1, but underestimate the first digit of the quotient as 0 instead of 1, and then continue dividing past the decimal point. Visually (I don't know how to show this in TEX): 0.999... -- 1 ) 1.000...    0.    ---    1.0     .9      10       9      ---       10       ... Other examples:     0.999...        0.999...          0.999...   --      --       -- 9 ) 9.000...    7 ) 7.000...    53 ) 53.000...     0.              0. 0.    ---             ---                   9.0             7.0              53.0     8.1             6.3              47.7                              -       90              70              5.30       81              63              4.77       ---             ---             -        90              70               530        ...             ...              ... — Loadmaster (talk) 16:07, 13 February 2009 (UTC)


 * It's an interesting approach, but I'm not sure why someone who fails to accept the 1/3 proof should be willing to accept that this effective abuse of long division should produce a correct result. --72.177.97.222 (talk) 22:30, 13 February 2009 (UTC)


 * I didn't present it so much to convince someone who doesn't believe it, but rather to demonstrate yet another calculation that is consistent with other forms of calculating the same result.
 * The "abuse" you mention is simply an underestimation of the initial digit of the quotient. This also works for other (non-repeating) long division examples, where the second digit of the quotient carries over into, and thereby corrects, the first digit. — Loadmaster (talk) 02:24, 14 February 2009 (UTC)

0.   → first digit, underestimated 1.2  → second digit, correct the first digit with by carrying the 1 5 → 1.25  final quotient --- 4 ) 5.000    0.     ---     5.0   → 12     4.8       20  → 5       20       ---         0 → 0
 * The only difference with a quotient of 0.999... is that the digits following the first never carry over into the digits to the left. Yes, it's an unorthodox bending of the rules, but it's mathematically sound. — Loadmaster (talk) 02:24, 14 February 2009 (UTC)

Books/Reading List
The extensive, commented reading list ist imho totally inappropriate for an encyclopedic article. The inline references plus 2-3 books and the external links are good enough. The additional commented book list looks more like reading list for math or science students and doesn't belong in the article.--Kmhkmh (talk) 10:33, 15 February 2009 (UTC)


 * I don't agree. I think the reading list is quite appropriate to the topic in this case, and is a really good feature of the article, and one of the things you'll see on Wikipedia and not elsewhere. I would have been glad if something like that reading list had been available when I was a high school student. -- The Anome (talk) 00:19, 15 February 2009 (UTC)
 * There is nothing wrong with a good reading, however it has nothing with this article/lemma. And it is debatable whether an encyclopedia should feature general reading lists at all, not everything that is good automatically belongs into wikipedia. A more appropriate place for a reading list would be wikiversity for instance. Even if you feel that wikipedia should have math reading list then the appropriate way to do that, is to create a separate lemma like Calculus Reading List or something along that line. But what we shouldn't do is, adding general reading lists as references to arbitrary math articles. The purpose of the references is to provide the sources the author has explictly used for writing up that article plus a few good sources containing related material or more detailed information about the article's sbject.--Kmhkmh (talk) 02:39, 15 February 2009 (UTC)
 * This article's References section is already restricted to cited sources. There's also such a thing as Further reading, but you'd have to look elsewhere for an example. Melchoir (talk) 05:07, 15 February 2009 (UTC)
 * True, I didn't notice at first glance, that they are actually just the sources from notes section and hence not a reading least but rather commented sources. So nevermind my earlier comment. It still looks a bit excessive though with notes/reference section being almost as long as the article.--Kmhkmh (talk) 10:32, 15 February 2009 (UTC)

How Is This a Featured Article?
Is it just me, or does anyone feel that the fact that not a single statement in the first two paragraphs is backed up by a reliable citation? Asperger, he&#39;ll know. (talk) 21:02, 18 March 2009 (UTC)


 * Those two paragraphs are summary of content further down, where it's extensively sourced. I don't think there's a need to add the same citations to the summary. Huon (talk) 21:06, 18 March 2009 (UTC)


 * Doesn't it seem a little odd that the citations are further down than the original statements? If the first paragraph is indeed a summary of the writing below, why does it exist? It must say the exact same thing in similar words to not require citations. I personally don't understand the need to read the same thing twice, just like you don't understand the need to source twice. Asperger, he&#39;ll know. (talk) 21:31, 18 March 2009 (UTC)


 * The intro exists to make the article easier to read; and no, a summary need not say the exact same thing in similar words to not require a citation, if the relevant material is clearly present and cited later in the same document. At some point, common sense has to take over. -- The Anome (talk) 21:41, 18 March 2009 (UTC)


 * I don't comprehend... How does increasing the length of an article make it easier to read? I think you mean easier to understand, yes? And your statement that common sense has to take over is not included in any of Wikipedia's popular rules. Also, if the summary does not have the same information, it does indeed require different citations. If we follow this and produce a summary that is so close to what is written later on that it can be said to be nearly identical, why have we written it? A summary is supposed to be composed of generalizations on the subject... And introduction, not an investigation. Let us observe Wikipedia article on alligators.


 * "An Alligator is a crocodilian of the genus Alligator of the family Alligatoridae. The name alligator is an anglicized form of the Spanish el lagarto (the lizard), the name by which early Spanish explorers and settlers in Florida called the alligator. There are two living alligator species: the American alligator (Alligator mississippiensis) and the Chinese alligator (Alligator sinensis)."


 * This is the introduction/summary. While the article itself is not a specimen, it must be mentionedf that nowhere in the summary is a particularly detailed transcript of any aspect of an everyday al's life. Nowhere will you find mentions of alligators in Floridan swimming pools, them having rough green/black skin that sparkles in the moonlight, or their contributions to a specific children's television franchise. Juxtapose this with the Wikipedia article on 0.999...


 * " In mathematics, the repeating decimal 0.999… which may also be written as or  denotes a real number equal to one. In other words: the notations 0.999… and 1 actually represent the same real number. This equality has long been accepted by professional mathematicians and taught in textbooks. 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 bases, and mathematicians have also quantified the ways of writing 1 in 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 fractions 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."


 * Should all this really be in the summary? Could we not suffice to say that 0.999... has been the subject of much debate and is SO equal to 1? This summary goes very far in trying to convince people that 1 is 0.999... (Even though 0.999... is a mathematical concept in itself. What is 0. infite 0's followed by one? You don't exactly find 0.999... lying on any street corners), but it does not provide any real insight into the subject, only saying that people who think it isn't one is wrong. Asperger, he&#39;ll know. (talk) 22:39, 18 March 2009 (UTC)


 * Please read Lead section. The lead section is an inverted pyramid overview of the rest of the article, which can be read as a standalone short article if so desired, albeit necessarily short on detail and citations, and long on bold assertion. It's lengthy because the article it has to summarize is extremely lengthy and detailed, on a topic which many people find hard to understand. The rest of the article then covers the same topics in far more detail, including full citations and links to specialist articles that cover individual subtopics in yet more detail.


 * If you want to start discussing such frequently-raised topics as the idea of "infinite 0's followed by one", you might first want to read the FAQ, and then move to the Talk:0.999.../Arguments page if you can't accept the statements there (which are backed up by copious citations to the literature) and want to discuss this further. -- The Anome (talk) 00:11, 19 March 2009 (UTC)


 * ...Whoever gave me a revert warning, not funny. Anyway, I just wanted, in shorter, to explain that I am not, in fact, Satan's illegitimate child, and realise that 0.999... are one and the same, not by any logical argument, but just because they are. I'm just complaining about the structure, and how indeed it does NOT follow Lead section. I feel this way because it does not actually provide an introduction on the subject, but more of a thorough lambasting. It is too detailed, in my opinion. Introduction, not beating. This, for instance, seems highly unecessary. "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". Just say that some people are unconvinced, not that the target audience is a bunch of stupid middleschoolers who are just too stupid to understand. And for the record, I thought I made up the 0.000...1 thing. Didn't know about the controversy page. Asperger, he&#39;ll know. (talk) 01:03, 19 March 2009 (UTC)


 * Regarding, "... I thought I made up the 0.000...1 thing..." Don't worry about it. Every retarded (by definition, not by insult) non-mathematician and their two-pantied granny that stumbles upon this article thinks they've cleverly come up with an original proof against it. Sadly, the article could very easily and irrefutably clarify the truth on the matter, but it's become a behemoth, and those on the correct side of the "debate" usually don't understand the issue really. Tparameter (talk) 06:47, 19 March 2009 (UTC)

lead sentence is awkward
I made this change already, but another editor objected, so I'll propose it here first.

The lead sentence currently says (in part):


 * In mathematics, the repeating decimal 0.999… denotes a real number equal to one.

This is, of course, a surprising fact, which it's safe to assume the first-time reader doesn't know. So to have the surprising fact stated so baldly in the very first sentence is bizarre, and invites disbelief. (It's like saying "black is white. So there.")  I think it would be better to lead the reader into the article ever-so-slightly more gradually, by adding a few words:


 * In mathematics, the repeating decimal 0.999… denotes a real number which can be shown to be exactly equal to one.

Also, I think it's worth an explicit reminder of what the ... means here, by adding the word "infinitely". In combination with the second sentence, I think this reads very well:


 * In mathematics, the infinitely repeating decimal 0.999… denotes a real number which can be shown to be exactly equal to one. In other words: the notations 0.999… and 1  actually represent the same real number.

Any objections? (As I said, one editor objected already, although I confess I don't really understand the objection.) —Steve Summit (talk) 13:47, 23 March 2009 (UTC)


 * I have to say that the statement that it "denotes a real number equal to one" is a bit odd to my ears. I find it more natural to say that it "denotes the real number one".  I'm not sure whether I like your suggestion, since claiming that it "can be shown to be" seems a bit roundabout and makes the claim seem somehow weaker than it ought to be stated. Phiwum (talk) 14:09, 23 March 2009 (UTC)

If you look over the 13 chapters of archives you will see that it has been hashed over repeatedly (which is not to say that you shouldn't bring it up again, just that it has been discussed and the current opening reflects a certain delicate balance). Many people are not approaching the topic for the first time. Those who understand it would (almost all) agree [probably more than 99.99% but not 99.999...% of those people] that 0.999... and 1.000... are notations for the same real number but would be less unanimous on how it can and/or should be shown. I'd like to make the second sentence the first (as you can maybe tell) but I can live with it the way it is.

As far as bald statements, I think exactly equal is more SO THERE!. Sentiment is against the exactly in that you can't be equal without being exactly equal. I am going off topic here a bit, but are 1 and 1.0 and 1/1 exactly equal or is it matter of notation and not precision? --Gentlemath (talk) 18:09, 23 March 2009 (UTC)

Proposed new opening.
0.999… (which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$) is a repeating decimal. The "..." after the nines indicate an infinite number of nines after the decimal point. In mathematics, ) 0.999... is best known for debates between students and teachers about the figure is to be interpreted. At first glance 0.999… appears to suggest a value that is infinitely close to 1, yet somehow falls short of it, in other words, 1 minus an infinitesimal.  But this is not usually the case.

Expressing a number in decimal notation (as opposed to fractions) generally indicates that the set of Real Numbers is in use. In this number space 0.999… is exactly equal to 1. While this result may seem counterintuitive, this equality is based on the nature of real numbers and can be arrived at by a variety of means. (see proofs below) But the fundamental reason is that real numbers adhere to the Archimedean property, which forbids non-zero infinitesimals.

While other number system include infinitesimals, the absence of the Archimedean property can make decimal notation problematic, so 0.999… may have ambiguous meanings, or non at all, in these systems. Hackenstrings includes an analog of .999... which is less then 1. This has proven useful in game theory, and can be used as an introduction to Surreal Numbers.

So far?Algr (talk) 07:17, 27 March 2009 (UTC)


 * I like it, though it might need some tweaking. For example, I'd remove the (as opposed to fractions) - the other probable number sets we have to consider aren't the rationals anyway. I'd also rewrite the first few sentences, for example:
 * 0.999…, which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9),$$ is a repeating decimal. The "..." after the nines indicate an infinite number of nines after the decimal point. 0.999… is best known for debates about how the figure is to be interpreted. At first glance...
 * Especially in the context of the internet, the discussions aren't merely between students and teachers. Maybe we should substitute something else for "debates", since it's not the mathematicians who debate it - would "confusion" be too strong? Huon (talk) 10:50, 27 March 2009 (UTC)

That's not bad at all. It certainly "teaches the controversy" right at the start, which I believe is appropriate in this case: 0.999... is generally only discussed in the context of this particular issue, and the controversy is what makes it notable. How about this:


 * In the real number system 0.999… is exactly equal to 1. 0.999… (which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$) is a repeating decimal.  The "..." after the nines indicate an infinite number of nines after the decimal point.  0.999... is best known for debates between students and teachers about the figure is to be interpreted.  At first glance 0.999… appears to suggest a value that is infinitely close to 1, yet somehow falls short of it, in other words, 1 minus an infinitesimal. However, this is not the case.


 * While this result may seem counterintuitive, this equality is based on the nature of real numbers and can be arrived at by a variety of means. (see proofs below) But the fundamental reason is that real numbers adhere to the Archimedean property, which forbids non-zero infinitesimals.

I don't think we should mention non-standard systems right at the top of the article, but they should definitely be mentioned in the next step down the inverted pyramid.

-- The Anome (talk) 10:55, 27 March 2009 (UTC)


 * I would point out that the fundamental reason for the equality is that the limit of the infinite sum represented by the number 0.999... is exactly 1 (within the reals). There is no need to mention to the Archimedean property in the intro, because the decimal representation 0.999... really has nothing to do with that property. Furthermore, all mention of non-standard number systems belong farther down in the body of the article. — Loadmaster (talk) 14:48, 27 March 2009 (UTC)


 * Ick, no. That just confuses the reader. I like the current opening far more. In addition, yours is a bit NPOV. It sounds like whoever wrote it is trying to suggest that infinitesimals are more important than they are. That infinitesimals do not exist in the real line is no more fundamental than any other reason that 0.999... = 1. I mean, it's math; it all boils down to tautology. In addition, the "(see proofs below)" is awkward and should be rewritten.
 * The idea that there are number systems in which 0.999... does not equal 1 is not important enough to have two entire paragraphs in the lead. I mean, in those number systems, the concept of the limit of an infinite sequence of numbers that tend infinitesimally close to 1 but whose limit can be proven not to be equal 1 generally isn't referred to as "0.999..." At most, I would call it "a concept similar to 0.999... in Hackenstrings" - something that doesn't deserve to be mentioned in the lead section at all.
 * "Best known for debates between students and teachers" is ridiculously silly (I especially dislike that "best" - it sounds a bit NPOV). It should be more like "Well known for being counterintuitive." Kind of like Banach-Tarski.
 * Minor nitpicks: "Real Numbers" and "Surreal Numbers" should not be capitalized. --Zarel (talk) 18:16, 27 March 2009 (UTC)


 * Zarel, are you aware that you just deleted three people's posts? Algr (talk)
 * Whoops. I wrote the comment back when it was the only one, but forgot to save the page. When I came back the next day, I refreshed the page, and saw no new comments. Apparently Firefox doesn't refresh the contents of text boxes. :/ --Zarel (talk) 00:42, 28 March 2009 (UTC)


 * I've restored the deleted comments above. Try this:


 * In the real number system 0.999… is exactly equal to 1. 0.999… (which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$) is a repeating decimal.  The "..." after the nines indicate an infinite number of nines after the decimal point.  0.999... is best known for debates between students and teachers about the figure is to be interpreted.  At first glance 0.999… appears to suggest a value that is infinitely close to 1, yet somehow falls short of it, in other words, 1 minus an infinitesimal. However, this is not the case.


 * While this result may seem counterintuitive, this equality is based on the idea of limit which is central to the development of the real number system, and can be arrived at by a variety of means. (see proofs below)


 * -- The Anome (talk) 21:22, 27 March 2009 (UTC)


 * Better. Let me try to further revise it.
 * 0.999… (which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$) is a repeating decimal equal to 1.
 * Hmm, this is sounding a lot like the current lead section. Let me go check it.
 * In mathematics, the repeating decimal 0.999… which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$ denotes a real number equal to one. In other words: the notations 0.999… and 1  actually represent the same real number.
 * Though I agree that the current lead section might benefit from rewriting, I actually like these first two sentences. I don't think these should be changed significantly, except maybe adding parentheses. Let me try my hand and writing a lead.
 * In mathematics, the repeating decimal 0.999… (which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$) denotes a real number equal to one. In other words: the notations 0.999… and 1 represent the same real number. The fact that 0.999… equals 1 is well known for being counterintuitive, and students who are taught this equation often disbelieve it. This is because at first glance, 0.999… appears to suggest a value that is infinitely close to 1, yet somehow falls short of it; in other words, 1 minus an infinitesimal.
 * However, this is not the case. The equality is based on the definition of a repeating decimal as a limit of a sequence of real numbers, which must be a real number, not an infinitesimal. 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.
 * Thoughts? --Zarel (talk) 01:05, 28 March 2009 (UTC)


 * I like the last part about "limit of a sequence", which also provides a nice lead-in to the mention that "the same phenomenon occurs in all integer bases". — Loadmaster (talk) 18:30, 30 March 2009 (UTC)


 * I like it. It covers all the key topics: notation, equality, identity, frequent disbelief in the proposition and the often-associated belief in infinitesimals, introducing limits and preparing the way for the proofs. There's just one bit that needs fixing: the words "which must be a real number, not an infinitesimal" should be replaced something like ""which must be a real number, not an expression involving infinitesimals". -- The Anome (talk) 21:17, 30 March 2009 (UTC)


 * I would just throw in here that the "nonexistence of infinitesimals" thing is a little bit of a red herring. In most contexts in which you do have infinitesimals, you still can't make sense of 0.999... being infinitesimally less than 1.  For example this notion doesn't make sense in a nonstandard model of arithmetic (or analysis), no matter whether the "omega many nines" part is interpreted internally or externally. --Trovatore (talk) 09:36, 3 April 2009 (UTC)


 * It's progress, but we have lost the explanation as to why the real set is the appropriate set in which to discuss .999... Algr (talk) 06:54, 5 April 2009 (UTC)
 * "Best known for debates between students and teachers" is ridiculously silly " Zarel may think this is silly, but it is inescapably true.  No one has found a single reference using .999... for any reason other then to refer to the =1 debate.  Algr (talk) 21:43, 5 April 2009 (UTC)
 * I said "silly", not "incorrect". It's correct that it's best known for debates between students and teachers; I just don't think that's a very encyclopedic way to phrase it. I merely prefer my phrasing: The fact that 0.999… equals 1 is well known for being counterintuitive, and students who are taught this equation often disbelieve it. --Zarel (talk) 04:55, 6 April 2009 (UTC)

OK: putting the bits together, try this:


 * In mathematics, the repeating decimal 0.999… (which may also be written as $$0.\bar{9}, 0.\dot{9}$$ or $$ 0.(9)\,\!$$) denotes a real number equal to one. In other words: the notations 0.999… and 1 represent the same real number. The fact that 0.999… equals 1 is well known for being counterintuitive. This is because at first glance, 0.999… appears to suggest a value that is infinitely close to 1, yet somehow falls short of it; in other words, 1 minus an infinitesimal.
 * However, this is not the case. The equality is based on the definition of a repeating decimal as a limit of a sequence of real numbers, which is itself a real number, without any need to consider infinitesimal quantities. 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 Anome (talk) 15:51, 7 April 2009 (UTC)


 * That looks exactly like my proposed lead section, minus the sentence: "and students who are taught this equation often disbelieve it." What was wrong with that sentence? --Zarel (talk) 20:14, 12 April 2009 (UTC)

strong proposal for easy short intuitive proof
In light of the great difficulties the general reader has with this notion I very strongly propose that a very common intuitive proof -- I added one here -- be added in the introduction itself. The sentences I added were: An easy and common intuitive proof is setting x to this number and proceeding to subtract it from 10x arithematically - the latter will be 9.999… and so on forever, and subtracting the former in the traditional school way will find 9.999… on top of 0.999…, with a 9 in the decimal of the subtrahend for every 9 in the decimal of the minuend -- clearly leaving all zeros in the decimal portion of the difference -- ie 9.00… exactly; in particular without any intuitively lingering .000…1, there being no 8 below a 9 or any other source for a digit other than 0 in the decimal portion of the difference. Thus 10x-x (ie 9x) is 9, and if 9x=9, then x=1

However this revision was not complete as I was asked to discuss it here first instead.

In particular the reason for the importance of this proof in the introduction is the high amoount of vandalism from people having difficulty understanding the identity intuitively, which ought to be (when it is so easy) cleared up right in the introduction. Remember, wikipedia is not for mathematicians but the general reader. See intro paragraph (lead section) policy: The lead should be able to stand alone as a concise overview of the article. It should establish context, explain why the subject is interesting or notable, and summarize the most important points—including any notable controversies. The emphasis given to material in the lead should roughly reflect its importance to the topic, according to reliable, published sources, and the notability of the article's subject should be established in the first sentence of the lead. While consideration should be given to creating interest in reading more of the article, the lead nonetheless should not "tease" the reader by hinting at—but not explaining—important facts that will appear later in the article. (bold mine)

For the reader (e.g. high school student about to vandalize the page) who is looking for refutation or proof, this is the overview of the article. They might not read any farther and be happy they learned a method to overcome their intuitive objections. It is explicitly in the policy (in the part I bolded): the lead should NOT hint at without explaining important facts! The proofs should NOT be "hinted at" where a simple one could be given!

I strongly believe in light of the above intro policy that a common, easy, and intuitive proof ought to figure as a parenthetical sentence in the lead summary.

Thank you for your consideration. 79.122.103.33 (talk) 17:05, 11 April 2009 (UTC)


 * I don't agree that such a "proof" belongs in the lead. Two sections down is the "Proofs" section, with a number of different proofs, starting with the most "intuitive", and progressing to more and more complicated (required real analysis and such).  If your hypothetical reader can't scroll down half a page, then there's really not a lot we can do!


 * I respectfully very strongly disagree, and what's more the intro lead policy clearly states that this should not be done where possible. (Also, to get it out of the way: these are not hypothetical readers!  A glance at the edit history will tell you that.)   But to get to what matters: there is a guideline for how to write intro paragraphs and we should follow it.  It explicitly says that the reader should not be "teased" about the facts hinted at as being in the rest of the article!  (it is the part I bolded in the above-reproduced intro section policy).  I appreciate your position in cases that are very difficult, but in this case the subject is about a very specific number.  It is not about 0.88.. (which is 8/9) or 0.77.. (which is 7/9) but specifically and only about 0.99.., where all of the misconception concentrates.  For Pete's sake there is a FAQ at the top of this very talk page!  That is proof positive that there is an overriding nead in the lead section not to only hint at the upcoming facts presented.  To give you an analogy: for a biographical article, the intro paragraph would not HINT at when the subject was born, since it is to be discussed in the body of the article (just one section down!) but in all cases where easily possible actually give you this information.  If this doesn't convince you I will think of closer analogies, from medicine, physics, chemistry, art history, philosophy, juggling, etc etc.  But I hope you see my point of view and agree that if we can keep the proof quite short and completely intuitive it belongs in the intro for the reasons above.  Thank you. 79.122.103.33 (talk) 19:30, 11 April 2009 (UTC)


 * Your example proof above is the prosaic version of the "Digit manipulation" proof already given in the article. I really don't think that the prose helps here, as it's more bulky, and breaks the flow of the lead section.  What's more, it explicitly introduces the "problem" of what 0.000...1 might mean!  Oli Filth(talk 17:26, 11 April 2009 (UTC)
 * I debated adding "(mathematically meaningless)" after the intuitive "suspicion" -- I believe it should be added, with a note that it is explained below. 79.122.103.33 (talk) 19:52, 11 April 2009 (UTC)
 * however, I agree with you both on the fact that it is more bulky and that it breaks the flow of the lead section. I wanted to work on it further but you had me move the discussion here instead.  I also realize it reproduces material from later, however this is the lead paragraph's job! (see above).  The "problem" of what 0.00...1 might mean is not a problem at all - the only problem is that htere is a psychological ("intuitive") idea among the untrained -- for whom the encyclopedia is written!  this is not mathworld - that there is a lingering 0.00..1 at the end.  There is not.  Anyway the proof given wouold ideally be just one lean sentence that weaves easily into the intro and is easily understood by the people who come to this article.  (not mathematicians you know!).  79.122.103.33 (talk) 19:52, 11 April 2009 (UTC)


 * I think we'll have to agree to disagree (to use a cliché). The intro should be a concise summary; a proof is a detail which doesn't belong.  One doesn't have to place huge chunks of body text in the intro in order to fulfil the "no teasing" requirement.  Nevertheless, let's see what other editors think (shouldn't take too long, this talk page is fairly active).  Oli Filth(talk 20:22, 11 April 2009 (UTC)


 * In fact, the lead already summarises the proofs already. "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."  I really don't believe anything more than that need be said.  Oli Filth(talk 20:27, 11 April 2009 (UTC)


 * I believe "proofs have been formulated" -- when some of these are in the rest of the article -- is a textbook example of "hinting" to the reader about the contents of the article, without mentioning them. Plese remember who visits this page! 79.122.103.33 (talk) 21:11, 11 April 2009 (UTC)


 * "Proofs have been formulated" is not talking about the proofs that have been included in the article, it's stating that "0.999... = 1" has been established in many ways as a mathematical truth. That the article contains some of these proofs is a bonus, but that is not the crux of what the lead is trying to say.


 * As as analogy, I wouldn't expect to see a proof of the irrationality of the square root of 2 in that article's lead, and indeed there isn't one, even though the article body contains several. Similarly for Proof that π is irrational.  Oli Filth(talk 21:25, 11 April 2009 (UTC)


 * I will answer Oli's call for other editors.
 * I absolutely agree that proofs should not be included in the lead. However, a nice compromise, I feel, may be to add to "Proofs have been formulated", as follows: "Proofs have been formulated (see below for examples)". That way, the reader can clearly see that there are proofs readily available in the article. --69.91.95.139 (talk) 21:35, 11 April 2009 (UTC)


 * (I started this thread) ... I support the "Proofs have been formulated (see below for examples)" solution. It would be nice if on clicking it the easiest and most intuitive example is in the first sentence, so that just clicking upon seeing that and reading on immediately nullifies the incessant vandalism of this page!  It would be a nice compromise in my opinion.  79.122.103.33 (talk) 22:39, 11 April 2009 (UTC)


 * It seem a relatively harmless compromise. Oli Filth(talk 22:42, 11 April 2009 (UTC)


 * i think that's settled then. i'm glad we could reach an agreement 79.122.103.33 (talk) 22:45, 11 April 2009 (UTC)


 * Wait, wait, no! "Proofs have been formulated (see below)" is the textbook example of hinting - i.e. what we have established as "What not to do"! What exactly is wrong with leaving it at "Proofs have been formulated"?! --Zarel (talk) 00:10, 12 April 2009 (UTC)


 * I'm not sure that's "hinting" exactly, but perhaps you're right. By all means remove the link; at the time it seemed the simplest compromise to the "problem" that 79.122.103.33 identified above.  Oli Filth(talk 00:18, 12 April 2009 (UTC)


 * Wikipedia is not paper and a link inline is as good as there. 79.122.103.33 (talk) 00:26, 12 April 2009 (UTC)


 * I don't understand what you mean. "See below" as a link is equivalent to "Click here", and not only does WP:MOS discourage it, but so does the W3C. --Zarel (talk) 00:31, 12 April 2009 (UTC)
 * are you referring strictly to the difference between "see below" and "click here to see below"? i don't think anyone insists on the altter, it is just about the link itself that we're debating... 94.27.231.28 (talk) 09:32, 12 April 2009 (UTC)


 * If this isn't "hinting", what is? I've always thought of hinting as doing things such as "Proofs have been formulated (see below)" or "Read on to section 3 for examples". On the other hand, "Proofs have been formulated", then describing them in greater detail in a later section, seems like the right way to do things. --Zarel (talk) 00:31, 12 April 2009 (UTC)


 * I've never been entirely clear on what WP:LS means by "hinting", but I'm not sure I see how "Proofs have been formulated (see below)" is hinting more than "Proofs have been formulated". Oli Filth(talk 00:38, 12 April 2009 (UTC)

(outdent) I don't think the "(see below)" link constitutes "hinting", but on the other hand I also don't think it's a significant improvement of the current state. The TOC lists "proofs" prominently anyway. Huon (talk) 10:53, 12 April 2009 (UTC)
 * if you need to see why it is so important to include it, 1) read the intro para guideline above which basically says to, 2) remember that Wikipedia is not mathworld and not for mathematicians but general readers, who... 3) have huge problems getting that far into the article before deciding to vandalize it. This article is not about 0.1111... through 0.999... all of which have the same mathematical property (equalling the appropriate number of ninth's) but specifically the one number with which the general reader (for whom the encyclopedia is written) has trouble grasping the very "facts" hinted at. It would be preferable if a complete intuitive proof were given paranthetically, linking to the appropriate section is a compromise, however having the proofs only in the body (and the TOC) is inappropriate for the reasons cited above.  Morever as just mentioned this is not a theoretical or hypothetical quibble -- there is huge vandalism arising from the general reader's concern not being spoken to early enough.  It is akin to the Shakespeare_authorship_question article giving no source for the doubt. Instead it very properly includes, right in the intro paragraph, the claims "Very little biographical information exists about him and, although much has been inferred from his writings, this lack of solid information leaves an enigmatic figure"...and "A further argument against the mainstream view is the erudition of Shakespeare's works, including an enormous vocabulary of approximately 29,000 different words."  Likewise the answers to these, the most common, claims is also given.  None of these are merely hinted at.  Therefore by direct analogy the proof (as for the shakespearean authorship question) should not be only in the body of the article, but a tiny amount (as in the shakespearean authorship article) should be right in the lead section, where per policy it belongs.  Thanks for your understanding.  94.27.231.28 (talk) 12:06, 12 April 2009 (UTC)


 * Giving a parenthetical link isn't going to make your vandals any more likely to read the proof. By your same analogy, the Shakespearean authorship article includes part of the proof in the lead section, but does not include a "See below" link or message. This is acceptable. A "see below" link is completely unacceptable (notice further that no other Wikipedia article has one because it is against WP:MOS).
 * I am not opposed to including a proof in the lead, although I would not suggest using yours - with all due respect, it's rather poorly written.
 * My proposed new lead section contains a brief definition-based argument that 0.999... = 1. It's not a complete proof, but it's better than nothing. Do you have anything against that one? --Zarel (talk) 20:02, 12 April 2009 (UTC)


 * First of all, it's not only about vandals -- vandals are just an indication that we are failing. For example, look at this article  (Bayes Theorem), which is not particularlyl subject vandalism.  It says "Bayes' theorem can be used to compute the probability that a proposed diagnosis is correct, given that observation. (See example 2)".  The reason it says that is so as not to tease the reader (if it did not say "see example 2").  A direct analogy is saying "there is a proof" without sending the reader to some (or including one).
 * As for being poorly written. I know it's poorly written and was working on it but was asked to stop.  I don't care what proof is given, but it should be prosaic and intuitive and very short in my opinion.  I don't think a definition satisfies the needs of the reader but a less than completely rigorous proof would.  Can you formulate one very concisely?  By the way the general reader doesn't find the infinitude of primes difficult to grasp ("controversial" though not in our eyes) or feel the need to vandalize the page.  But if they DID I would absolutely argue that it would show the infinitude of primes article ought to include a non-rigourous proof (--if there were only finite primes, you could multiply them all together, add 1, and the new number, not in the finite list of primes hence "not a prime", would have no prime factors -- a contradiction)  or something like that.  But that article doesn't need it as it is not the source of particular confusion, and this article does need it. I believe a glance at the FAQ at this very page (at top) will show that quite clearly.  Thank you. 94.27.151.13 (talk) 11:41, 13 April 2009 (UTC)


 * Giving a parenthetical link isn't going to make your vandals any more likely to read the proof. By your same analogy, the Bayes' theorem article could include part of the proof in the lead section, but would not include a "See below" link or message. This is acceptable. A "see below" link is completely unacceptable (notice further that no other Wikipedia article has one because it is against WP:MOS).
 * You'll notice what I said above is exactly what I said last time. Perhaps if you actually read what I wrote, I wouldn't need to repeat myself.
 * Here, I'll make the emphasis clearer: I am not opposed to a proof in the lead section. Why the *&@# are you still arguing about why you think there should be a proof in the lead section?
 * As for being poorly written, that is what the talk page is for. The rest of us are using it for improvement of the lead section, too.
 * Here is my definition-based argument in the new proposed lead section: "The equality is based on the definition of a repeating decimal as a limit of a sequence of real numbers, which is itself a real number, without any need to consider infinitesimal quantities." Is that enough? --Zarel (talk) 17:11, 13 April 2009 (UTC)

(outdent) "Here is my definition-based argument in the new proposed lead section: "The equality is based on the definition of a repeating decimal as a limit of a sequence of real numbers, which is itself a real number, without any need to consider infinitesimal quantities." Is that enough?"
 * No way, if I were a teenager (and I was at one point) or general reader, that wold be just handwaving to me. I believe we could do much simpler... Again, this isn't mathworld, and we are writing for the general, non-mathematician reader. 94.27.151.13 (talk) 18:39, 13 April 2009 (UTC)

A new digit manipulation proof

 * Note: Moved here from arguments page -- The Anome (talk) 10:50, 14 April 2009 (UTC)

(9/10)*0.999... = (9/10)*0.999... (Identity)

= (10/10 - 1/10)*0.999... (Rewriting fraction as equivalent difference of fractions)

= (10/10)*0.999... - (1/10)*0.999... (Distributing 0.999... over the difference)

= 0.999... - (1/10)*0.999... (Multiplication of (10/10)*0.999...)

= 0.999... - 0.0999... (Multiplication of (1/10)*0.999...)

= 0.9 (Subtraction)

(9/10)*0.999... = 9/10 (Converting finite decimal to rational number)

(9/10)*0.999... = (9/10)*1 (Applying multiplicative identity)

It's another form of the digit manipulation proof, but with two advantages:

1. The left side of the equation never changes, so the proof is formal (or may easily be made formal). The original digit manipulation modifies both sides of the equation and thus loses formality.

2. The original digit manipulation uses 10*0.999... = 9.999..., and doubters often claim that there should be a zero somewhere at the end of the endless 9's; on the other hand, multiplying 0.999... by 1/10 yields a much less objectionable result.

Please share your thoughts, comments, and suggestions. Gustave the Steel (talk) 18:09, 8 April 2009 (UTC)


 * You claim that those advantages are gained, but I disagree.
 * 1. There is no reason a proof which manipulates two sides of an equation cannot be formal. The main reason the original proofs are not commonly accepted is because they assume that certain arithmetic properties hold for infinite decimals. The same assumptions are being made here.
 * 2. The digit manipulation divides by 10; anyone who argues that multiplying by 10 means there will be one less 9 after the decimal than before you multiplied by 10 will have no trouble arguing that dividing by 10 means there will be one more 9 after the decimal than before you divided by 10.
 * My observation is that any alternative proof someone comes up with (something I, myself, have done) is generally nothing more than an attempt to bury the issues, and it won't work. The opponents will always figure out where exactly you buried them, and then you will have made absolutely no progress with them. --72.177.97.222 (talk) 21:54, 8 April 2009 (UTC)
 * Are you sure about your second point? I know that, to us, the idea of having "one less nine" after 10*0.999... is no more logical than the idea of "one more nine" after 0.999.../10.  However, the person who objects to 10*0.999... is relying on his/her intuition that 0.999... is missing something that keeps it from being 1.  Multiplying 0.999... by a number (greater than 1) makes the difference greater, in their minds.  I honestly do not believe that 0.999.../10 = 0.099... will contradict their intuition; if anything, adding 0.9 to 0.999.../10 would make the distance from 1 even smaller than that of 0.999..., which contradicts their very idea of 0.999... as the "closest number to 1".
 * As to your first point, well, all the formal proofs I've seen only involve one column changing. Can you show me a counterexample?  I would like to know if formal proofs can change multiple columns; there's nothing intuitively unmathematical about it, and I'd sure like to divide both sides by 9/10 at the end.  Gustave the Steel (talk) 14:45, 9 April 2009 (UTC)

"The main reason the original proofs are not commonly accepted is because they assume that certain arithmetic properties hold for infinite decimals." - Anon222 You and Gustave are close. Actually this proof assumes that certain arithmetic properties DON'T hold for infinite decimals. Namely, when a finite number is multiplied or divided by 10, (or the current base,) the number of significant digits never changes, they simply move left or right. But here step 5 requires that 0.999... and 0.0999... have different numbers of significant digits, with no explanation as to how this is possible, or why we can know that the difference between two infinite digit counts is exactly 1 and not some other number. So really this is just a more complex version of the digit manipulation 'proof' that appears in the article - It has the same problem and no advantage. Honestly, it was this and the .333...x 3 proof that first convinced me that there was something fundamentally wrong with how mainstream math looks at .999... . Algr (talk) 07:36, 15 April 2009 (UTC)


 * I believe we can accept Algr's reaction as showing that turning the digit manipulation proof upside down won't make it significantly more convincing. Thus, we can just as well keep the current one in the article. Huon (talk) 13:04, 15 April 2009 (UTC)

Heuristic Arguments
Perhaps a section on heuristic arguments (ie not relying on Mathematical rigor, but useful for understanding) might be helpful. It seems a lot of people just don't want to be convinced. Off the top of my head, you can treat the question philosphically as in "If I put my finger infinitely close to the wall, will it touch the wall? If you say no, surely you are putting a limit on how close I can get to the wall, there is some finite space between, contradicting the definition of infinity (without limit)" Obviously it is completely non mathematical, but many people are convinced of the monty hall problem's unintuitive solution by unrigorous arguments. So the challenge is, if they are separate numbers, find a number between them... Also, the fact that every single professor of mathematics will tell you that 0.999... is equal to 1. It has no bearing on the truth value of the statement, but it might go some way towards convincing "unbelievers" of their arrogance. Triangl (talk) 22:22, 5 May 2009 (UTC)


 * If you try an introduce real world examples like that you are asking for trouble - mathematics is useful for modelling the real world, but those models are rarely perfect. Modelling the separation between your finger and the wall as a real number works very well for reasonably large separations, but once you get onto an atomic scale quantum mechanics kicks in and you can no longer meaningfully assign the separation a real number. --Tango (talk) 22:39, 5 May 2009 (UTC)


 * Quite right, in fact ignore that example because someone already used it for the opposite claim in the arguments page! And to explain its indequacy requires either quantum mechanics or electromagnetic fields, and hence the same arguments apply to my own example. Triangl (talk) 23:39, 5 May 2009 (UTC)

Wow
Do people really seriously debate this?

Like, seriously? --COVIZAPIBETEFOKY (talk) 16:39, 10 May 2009 (UTC)
 * Indeed.--Noe (talk) 19:57, 10 May 2009 (UTC)


 * Hehe, ONLY on Wikipedia... -- OlEnglish (Talk) 06:44, 19 May 2009 (UTC)


 * Check out the sci.math newsgroup. There's always a thread from some crank claiming that 0.999... is something other than 1. I believe it's mentioned in a FAQ that's posted there every month or so. — Loadmaster (talk) 16:28, 19 May 2009 (UTC)

The Cyclical Paradox of the First Proof

 * → Discussion moved to the arguments page, as per the page heading instructions. || Loadmaster (talk) 14:20, 22 May 2009 (UTC)

Edits
So after posting a change on the main ".999..." page TWICE, and having it removed TWICE - I give up. If you disagree with someone you can't just dismiss them out of hand. You must provide reason. Which I tried to do but was instead deleted. Twice. —Preceding unsigned comment added by 68.114.70.149 (talk) 02:49, 30 June 2009 (UTC)

Changes to the page should be discussed here before being posted to the article. If you'd post your proposed paragraphs here, we'd be able to tell you if we agreed or disagreed on their inclusion, and suggest changes. Gustave the Steel (talk) 04:42, 30 June 2009 (UTC)

1/3 = .33333
By saying that 1/3 is equal to .333, that is already assuming .9999 is equal to 1. Therefore, I see only the 10x proof to be viable. —Preceding unsigned comment added by Siddharth9200 (talk • contribs) 17:26, June 26, 2009
 * You don't need to assume .999...=1 to prove that 1/3=.333..., just use long division. --Tango (talk) 23:30, 26 June 2009 (UTC)
 * So what is 1/3, then, if not 0.333...? --Zarel (talk) 04:07, 27 June 2009 (UTC)
 * Go ahead and try that Tango, and tell us when you get rid of the remainder. In attempting to define .333... in that way you are performing an infinite number of tasks, but how do you define what it means to do that? I would say that 1/3 can't truly be expressed in decimal notation any more then π can.   Algr (talk) 08:12, 30 June 2009 (UTC)


 * π can be expressed in decimal notation. --COVIZAPIBETEFOKY (talk) 13:36, 30 June 2009 (UTC)
 * Ok, you can do it just using long division and the ability to spot a very obvious pattern. The algorithm gets stuck in a loop with nothing changing except which decimal place you are calculating, so you can safely conclude that all the decimal places are the same (3). And that is precisely what we mean by the phrase "3 recurring". --Tango (talk) 15:16, 30 June 2009 (UTC)
 * "tell us when you get rid of the remainder"? You can't get rid of the remainder. That's the whole point and the very meaning of the ... notation. 143.167.37.22 (talk) 15:23, 30 June 2009 (UTC)
 * Exactly my point, anon. We are making assumptions about the conclusion of something that never concludes.  What kind of proof is that?  Algr (talk) 17:40, 30 June 2009 (UTC)
 * We're not making any assumptions, we are explicitly saying it never concludes. That's what "..." means. --Tango (talk) 17:57, 30 June 2009 (UTC)
 * But as soon as you ascribe any fixed value to .xxx... you are stating that it DID conclude, by stating a concluding point. (And even implying a final digit.)  If .xxx... never concludes then it is like x=x+1.  Algr (talk) 18:34, 1 July 2009 (UTC)
 * The decimal expansion never concludes, the real number represented by it is not a process so it is nonsense to talk about it "concluding". It is, in fact, the limit of that process. --Tango (talk) 18:57, 1 July 2009 (UTC)
 * By saying that it "concludes" you're stopping the number in your mind. That's the problem a lot of people have with 0.9999... = 1.  They consider it a process and arbitrarily stop that process in their mind and conclude that it doesn't equal 1.  But it's not a process.  0.9999... = 1.  0.3333... = 1/3.  And if you just can't accept that, then think about it for a second.  It's a symbol.  It works the way mathematicians want it to work.  If, working in the decimal system, the results arbitrarily changed because there was an infinite repeater in there, and you arbitrarily stopped it to make it non-1, it would be mathematically absurd.  Why is it equal to 1?  Because that's the only rational value for it, and that's what mathematicians want it to be.68.222.96.245 (talk) 01:50, 17 July 2009 (UTC)
 * How does ascribing a fixed value to, say, 0.333... state that it concludes? I shall ascribe a fixed value equal to 1/3. If it concludes, then what is its concluding point? There is none. That is the whole point. --Zarel (talk) 03:52, 3 July 2009 (UTC)

Running commentary on context of proofs?
I seem to recall several editors saying of the first two digit manipulation proofs, that they suck as proofs and should be de-emphasized. There has also been rather a lot of anonymous bile directed at them. I for one still think that they're useful and important, but perhaps the article fails at explaining how.

There's a brief but attractive treatment of the explanatory role of the multiply-by-10 proof here: Perspectives on mathematical practices - Google Books. It makes me think it should be possible to provide a well-sourced running commentary on the proofs. We have proven that X follows from assuming Y, which explains Z, but does not really explain W, for which keep reading.

Are there any strong opinions on this direction? Any suckers volunteers? Melchoir (talk) 04:45, 30 June 2009 (UTC)


 * As I mentioned before, thinking about those proofs is what convinced me that the equality was wrong. (Before encountering this article, I had thought about .999... but never quite knew what to make of it.)  I don't see anything new in that reference - it is the same problem of assuming the conclusion within the process.  Algr (talk) 08:22, 30 June 2009 (UTC)


 * Of course, you never bothered to read the more formal proofs. That would take work, and we can't have that, can we? --COVIZAPIBETEFOKY (talk) 13:39, 30 June 2009 (UTC)
 * AGF, Coviz. I've read the more formal proofs and they just hide the same assumptions more deeply. If you want work, why not show us how to describe π unambiguously in decimal notation?  (As you said you could above?) Algr (talk) 17:53, 30 June 2009 (UTC)
 * Sure. 3.141592653589793238462643383279 and so on. That clearly isn't all of it, but it goes on infinitely.
 * Hey, you never said it had to be described unambiguously using a finite number of digits. It can be described using countably many digits; that's close enough anyway. ;) --Zarel (talk) 20:51, 30 June 2009 (UTC)
 * I'm not sure where you get the idea that the decimal expansion of π might be 'ambiguous'. Every digit is well-determined, so there's no ambiguity. That is to say, the nth digit of π is always, at least theoretically, accessible by an algorithm. You can take a look at the Pi article for some examples of such algorithms; there are many of them.
 * Furthermore, I have no need to assume anything, good faith or bad. Assuming implies a lack of knowledge. --COVIZAPIBETEFOKY (talk) 11:16, 1 July 2009 (UTC)
 * Since when is mathematics about being "close enough"? Zarel, all you have done is imply π by stating a number nearby.  That is no better then me defining X=the proof that .999...≠1 in the real set. and expecting you to accept it.  Also, how do you know that the number of digits in π is countable?  If I append an uncountable number of random digits at the end of π would the result still be π?  (If this keeps up, we might want to move it to arguments.) Algr (talk) 18:46, 1 July 2009 (UTC)
 * Pi is, by definition, a real number and the real numbers are, by definition, a completion of the rational numbers. It isn't much work to show that that definition is equivalent to defining real numbers as everything that can be represented by a decimal expansion (which is, by definition, a countable sequence of digits). That means that Pi must have a decimal expansion, which must, by definition, be countable. --Tango (talk) 18:56, 1 July 2009 (UTC)
 * I've stated the exact number pi. This isn't just approximately pi, it's exactly pi. The nth digit is floor(pi*10^n-floor(pi*10^(n-1)*10)), for all positive integers n. If you disagree, feel free to prove exactly how my number differs from pi. Maybe once you fail, you'll learn why 0.999... = 1.
 * Also, as Tango says, the number of digits of a real number is always countable. It's impossible to append an uncountable number of digits at the end of pi, firstly because there is no end of pi, and secondly because you can't "append" an uncountable number of digits anywhere. You can keep on appending them for eternity and you'll still only have appended a countable number of them. It might help for you to learn what countable means in the first place:
 * "A set is countable if there's an injective function from S to the natural numbers."
 * And there definitely is. Heck, there's a bijection: I can always map the nth digit to n. --Zarel (talk) 20:21, 1 July 2009 (UTC)
 * There is nothing stopping one extending the definition of a decimal expansion to include uncountably long sequences - there are uncountable ordinal numbers. You just wouldn't have a real number any more. --Tango (talk) 21:55, 1 July 2009 (UTC)
 * You also wouldn't have a field. Such a construction suffers much the same problems as Algr's 'base infinity'. --COVIZAPIBETEFOKY (talk) 18:41, 2 July 2009 (UTC)
 * I think you could make it a field if you allowed the expansions to go on to (some uncountable) infinity to the left as well as the right. The definitions of addition, subtraction, multiplication and division would extend fairly trivially and would still be well-behaved, so I think you would have a field. There would be all kinds of other problems, though, not least the fact that it wouldn't be useful for many things. --Tango (talk) 22:11, 2 July 2009 (UTC)
 * If we define X as having 1 at the -ω'th digit and 0 for all other digits (so X is an infinite number), then closure of multiplication demands the existence of a Y satisfying Y = .5*X. I may be missing something, but I don't think such a Y exists.
 * Of course, there may be more clever workarounds to make it into a field, but if we're talking about straightforward decimal expansions, with digits limited to the integers 0 through 9, and the indices for those digits ranging over some set of ordinals (including at least ω) and their negatives, I don't think it could be a field. --COVIZAPIBETEFOKY (talk) 01:21, 3 July 2009 (UTC)
 * If we define Y as having 5 at the -(ω-1)'th digit and 0 for all other digits, then I think that counts as a Y satisfying Y = .5*X. That's the fun thing about how ω works. Also, keep in mind ω is still countable. --Zarel (talk) 03:29, 3 July 2009 (UTC)
 * Ok, we're not working with just ordinal numbers and their negatives as indeces. Then what are we working with, exactly? Their continuation by closure of addition, perhaps? Is there a name for that set of numbers?
 * And yes, I realize we haven't even gotten to the uncountable ordinals. --COVIZAPIBETEFOKY (talk) 11:39, 3 July 2009 (UTC)
 * Ordinals are closed under addition, but not subtraction. -(ω-1) doesn't make sense since ω is a limit ordinal so it doesn't have a predecessor. I think that example is a problem, indeed. There are similar problems with things like 10*0.000...1 (where the 1 is in any position labelled by a limit ordinal). --Tango (talk) 18:55, 3 July 2009 (UTC)


 * This is both a violation of WP:NPA and beside the point. If the non-formal proofs actually convince people that the equality is wrong they probably need improvement. If I'm not mistaken Algr came to accept that within the real numbers the equality holds; his doubts seem to focus on whether the real numbers are the appropriate number set for this context. Maybe Algr could explain his problems with these proofs more thoroughly so we can increase the article's clarity (within the real numbers, of course). On the other hand, these proofs are designed not to be rigorous, but to appeal to intuition - such as "multiplying a number by ten shifts the digits" without further justification - and to be understandable with a bare minimum of mathematical knowledge, and if these non-rigorous steps are the problem, it probably can't be overcome without the proofs becoming much more technical, contrary to their purpose. Huon (talk) 14:14, 30 June 2009 (UTC)
 * I agree, and that's not what I have in mind... I'll try to demonstrate when I get the time. Melchoir (talk) 03:58, 1 July 2009 (UTC)


 * I think the best plan would be to rename them. They aren't proofs, they are demonstrations (there is nothing wrong about them, but they make a lot of (true) assumptions without proving them). For rigorous proofs you need to read the rest of the article. --Tango (talk) 15:14, 30 June 2009 (UTC)
 * This is an excellent idea. I support this fully. --Zarel (talk) 20:51, 30 June 2009 (UTC)
 * I don't. Tango, I'm not sure what you're trying to say here; all proofs make assumptions without proving them. Melchoir (talk) 04:00, 1 July 2009 (UTC)
 * I'm not talking about the premises of the result, I'm talking about other results used in the proof (such as that you can meaningfully perform digit manipulations on non-terminating decimal expansions). While it is very common to omit the proofs of results that will be well known to your intended audience, the intended audience of this article is not likely to have taken a first-year analysis course at university. --Tango (talk) 04:42, 1 July 2009 (UTC)
 * Okay, this is all true. But you seem to be proposing that a section of prose that would qualify as a proof in an analysis text is suddenly not a proof if shown to the wrong audience? Melchoir (talk) 05:38, 1 July 2009 (UTC)
 * Yes, I suppose I am. --Tango (talk) 05:44, 1 July 2009 (UTC)
 * Hmm. Well, there's certainly a relevant distinction being made, but I don't think it justifies calling an argument not-a-proof in the article. The negative connotations of substantive fallacy and illegality would be hard to avoid. It's better to call it a proof and write "the preceding proof assumes X". Melchoir (talk) 06:01, 1 July 2009 (UTC)
 * ...or, come to think of it, "the following proof assumes X". But that might break up the flow too much. Melchoir (talk) 06:02, 1 July 2009 (UTC)
 * If you are going to do that you need to make it clear that the assumption can be proven but is just too complicated for the article. --Tango (talk) 06:11, 1 July 2009 (UTC)
 * Sounds about right, will do! Melchoir (talk) 06:19, 1 July 2009 (UTC)