Talk:0.999.../Archive 19

Request for comment: Which version neutrally summarizes the cited sources with appropriate weight?
For the discussion section of the "Algebraic proofs" given in the article, which of the following pieces of text more accurately reflects the opinions expressed by the cited sources, and represents established scholarship with appropriately due weight: Sławomir Biały  (talk) 16:20, 21 July 2017 (UTC)

Version A:

Version B:


 * Version B. The cited sources do not present the algebraic arguments as convincing demonstrations (as evidenced, for example, by Byers placing scarequotes around "proof").  On the contrary, in the Byers source, these arguments are presented as convincing but fallacious arguments to illustrate that students can become convinced of the identity of 0.999... and 1 without understanding that identity, and he goes to great lengths to distinguish between process and object.  This context therefore carries a significant caveat that is completely lost in version A.  Byers' view is suppressed, being relegated to a meaningless tweet in a footnote.  Furthermore, version A also fails to capture the full context of the view of Peressini and Peressini, who say "Such an explanation would probably involve considerably more, e.g., explaining the distinction between rational numbers themselves and a decimal representation of them, how the decimal representation is related too a (potentially) infinite series, and also the Cauchy-Weierstrass property (or an equivalent one)."  They do add that "This simple proof may actually, in certain less obvious contexts, have explanatory power", but these "less obvious contexts" (such as why $$0.33... < 0.34$$) are unexplained and have little bearing on the subject of this article.  Finally, it is also a fact that significantly more is required to prove that 0.999...=1 than what has been offered in the "Algebraic proofs" section.  The identity requires the completeness property of the real number system.  Version A (as well as the earlier version of the lead-in to the section on the proofs) attempts to minimize this aspect of the issue, when in fact it is at the very heart of the matter.  A reader could easily walk away from the article believing that the identity follows from some trivial algebra, apart from one or two finer points of rigor.  This is directly undercut by the quote to Peressini and Peressini given above, and also by Byers', for instance (p.41): "understanding involves the realization that there is 'one single idea' that can be expressed as 1 or as .999..., that can be understood as the process of summing an infinite series or an endless process of successive approximation as well as a concrete object, a number."  However, fruitless discussions on the talk page have lead me to the conclusion that actually some routine editors of the article and its talk page seem not to appreciate the importance of these nuances that are amply evidenced in reliable sources, and wish to place the article's accessibility ahead of the need to represent sources in accordance with the demands of due weight and accuracy.  (This is evidenced in particular by User:Calbaer's remarkable assertions that "Sławomir's concern should not be 'representation' of any 'viewpoint.' Instead, the overall concern should be explaining the matter without sacrificing either accuracy and comprehensibility." and "It also might be that it's not at all clear how the [NPOV and WEIGHT] policies cited have anything to do with the article in the first place... Those policies are about making sure that viewpoints are properly represented, but method of explanation is not viewpoint, so the policy is not applicable to what is being discussed here.)   Sławomir Biały  (talk) 16:20, 21 July 2017 (UTC)


 * Subcomment: Proportionate representation. I see that certain editors here are entertaining the possibility of shortening the added material.  I do not believe that is consistent with the WP:WEIGHT policy, which requires that we cover topics proportionate to their coverage in reliable sources.  The Byers source, in particular, spends less than 10% of the text on the proof, and more than 90% of the text discussing the issues that have been condensed into several short paragraphs.  Less than a quarter of the total character count in the coverage in the Peressini and Peressini is the actual proof itself.  My argument is, and always has been, that we include the full context of each source that we use.  I submit that to do otherwise would specifically violate the proportionate treatment aspect of the neutral point of view policy.  I have no objection to editors cleaning up the treatment to make it more palatable, but the whole summary of the sources must be there if we are to include the proofs at all.  Policy is absolutely crystal clear in the matter.  Content doesn't get a pass if it's just for pedagogical or educational purposes.  All content is subject to the neutral point of view policy, no exceptions.   Sławomir Biały  (talk) 18:38, 22 July 2017 (UTC)


 * Version B.It would be nice if the problem could be dismissed easily but the example of Hackenbush game theoretic values shows it is not altogether straightforward. Version A would contradict the lead and the lead is correct - it is true within the standard real number system but can be false in other systems. There should be citations for the algebraic proofs. Citation 1 in the discussion is rather cryptic and should be expanded to reference the actual publications - it might help if Harvard citation templates were used as the artcle does tend to that style. Dmcq (talk) 17:03, 21 July 2017 (UTC)
 * Version B. seems to reflect the situation best.Slatersteven (talk) 17:19, 21 July 2017 (UTC)
 * Version B. Version A comes across to me as too facile, too sloppy about the distinction between the definition of real numbers vs the definition of decimal notation and whether there is even a single universal definition for either, too condescending, and also problematic from the point of view of egg submarines. —David Eppstein (talk) 17:44, 21 July 2017 (UTC)
 * Version B, provided the section title Algebraic proofs is amended to Algebraic motivations. - DVdm (talk) 18:21, 21 July 2017 (UTC)


 * Neither: What is being suggested here is replacing "Although these proofs demonstrate that 0.999… = 1" with "Although these 'proofs' purport to demonstrate that 0.999… = 1". But the lead of the article clearly says "In mathematics, the repeating decimal 0.999… denotes a real number that can be shown to be the number one. In other words, the symbols "0.999…" and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor..." Version B looks like a blatant attempt to give the fringe theory that 0.999… does not equal 1" undue weight. This is not to say that version A cannot be improved -- those who have pointed out the flaws in it have a point -- but we should not "fix" the problem by supporting fringe science. --Guy Macon (talk) 19:16, 21 July 2017 (UTC)


 * : It is not the intention of the section to advance a fringe position. In fact, these supposed "proofs" are presented in the very sources that we cite as examples of fallacious arguments.  A fallacious argument is a fallacious argument regardless of the truth value of its conclusion, and it is not fringe to point that out.  The article does contain several more rigorous proofs.  These still need work, because the present article fails to define clearly what is meant by the notation "0.999..."  But if we are going to present flawed proofs, then it is very important that the article point out that these proofs are flawed.  A failure to do so, apart from violating the neutral point of view, is just fodder for the usual bunch of fringe theorists who will attempt to undercut those supposed "proofs"; ironically, in this case, they would have a point.  Perhaps you are leaning towards a third option, namely: why should the article present false or misleading proofs at all?  That might be worth discussing.  There seems to be something inherently dishonest about using these arguments as "explanations" of the concept, when they are explicitly denounced by reliable sources as having little explanatory value.  On the other hand, mathematical explanations often benefit by having both examples and non-examples.  If we present them as non-examples, then we must be upfront that this is what they are.  This is the reason version B is written in the way that it is.   Sławomir Biały  (talk) 19:39, 21 July 2017 (UTC)


 * Please don't ping me. When I make a comment I watch for replies. I have no problem if the article presents false or misleading proofs, but they should be clearly labeled as such. Changing proofs to "proofs" and changing ...demonstrate that.. with ...purport to demonstrate that... does not make it clear that the argument is false or misleading.


 * In other words, this RfC is an example of "A is wrong. Something must be done. B is something. Therefore, B must be done" See False dilemma. --Guy Macon (talk) 19:53, 21 July 2017 (UTC)


 * I object strongly to the characterization that version B comes from version A by simply "changing proofs to 'proofs' and changing ...demonstrate that... with ...purport to demonstrate that...". In any case, a full context can be seen at this revision (diff), where the first paragraph of the section includes an explicit indication that they fall short of being valid mathematical demonstrations and why, and also the lead section of the article contains a definition of the subject (which remains absent from the status quo revision).  Perhaps your concerns are assuaged by that revision, seen in full?
 * I am aware that option A/option B RfC's can often miss nuances, but I wanted to avoid going out into the weeds regarding revisions to other parts of the section. I do offer my apologies for making you decide in a binary fashion like this, if you feel that both versions have serious shortcomings.  Since you seem to have identified just the first sentence of Version B as problematic, I'd like to invited you to attempt to rewrite it so that both of our concerns are satisfactorily addressed.  Also, sorry for the ping.  I won't do it again.  Thanks,  Sławomir Biały  (talk) 20:11, 21 July 2017 (UTC)


 * Use


 * "Although these proofs demonstrate that 0.999… = 1, they fall short of being valid mathematical proofs."


 * instead of


 * "Although these "proofs" purport to demonstrate that 0.999… = 1, because they do not actually rely on the relationship between decimals and the numbers they represent, they fall short of being valid mathematical proofs."


 * The rest seems fine. --Guy Macon (talk) 21:49, 21 July 2017 (UTC)


 * I heartily endorse this suggestion, with one small emendation: "Although these arguments demonstrate that 0.999… = 1, they fall short of being valid mathematical proofs." Something that is not a mathematically valid proof should not be called a proof.  I will make the change.  I do not anticipate any objections to making this change in "Version B", so I have gone ahead and done it.  Anyone may feel free to revert me if they object.  Thanks,  Sławomir Biały  (talk) 21:57, 21 July 2017 (UTC)


 * Version B is the better of the two, but this of course does not rule out the possibility that a still-better version could be written. XOR&#39;easter (talk) 20:06, 21 July 2017 (UTC)


 * Version B, for the reasons that I have given in my previous post in, and with changing the heading Algebraic proofs to Algebraic motivations. D.Lazard (talk) 20:11, 21 July 2017 (UTC)


 * Version A: What Guy said. Neither is perfect, but B is verbose and the fact that a reader could come away from it thinking that it supports 0.999... not equal to 1 means that it is deceptive.  Better to be retain the current material than to replace it with something that is deceptive and more difficult to read, and gets us further away from the desired state.  Of course, ideally, someone would be present an alternative that is both clear and precise.  But for now, Guy nailed it: classic false dilemma.  I'd also add loaded question; the idea that due weight should be the only criteria in judging which text is better for the article is false. Calbaer (talk) 22:48, 21 July 2017 (UTC)
 * Version C: "Although these arguments demonstrate that 0.999… = 1, they are not rigorous proofs. They are useful for the sake of pedagogy, as rigorous proofs might be inaccessible to those without knowledge of higher math.  However, they do not prove why the intuitive mathematical steps within them work on these repeating decimal representations [1] .  As such, formal definition of the decimal representation scheme and use of real analysis are necessary for formal proofs of 0.999… = 1."  Short, sweet, and to the point.  Don't get ahead of yourself by introducing advanced math that we don't even use, let alone expect the reader to know.  The important thing is that the initial "proofs" are not rigorous, but can be used to give intuition and think about how a formal proof might work. Calbaer (talk) 00:08, 22 July 2017 (UTC)
 * I am not sure what part of it you feel would lead a reader to "could come away from it thinking that it supports 0.999... not equal to 1". Not misleading the reader on the matter of whether $$0.999...$$ is equal to $$1$$ is actually very easy to accomplish.  Apart from being telegraphed in the very first sentence of the article and the paragraph preceding the arguments in question, the very last paragraph of Version B says, quite explicitly: "Once the real number system has been formally defined, its properties can be used to establish the decimal representation of real numbers. The properties of the decimal representation can then be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number. This is done below."  This is, actually, considerably more detailed in its description of how the proofs can be made correct, than version A.  The only caveat is that they indeed require a detailed analysis of decimal representations.  Without the added indication of what is wrong with these proofs, however, the reader is far more likely to be convinced that the proofs are actually correct without modifying in a deep way their understanding of concept of "decimal representation".  This is the chief danger for readers of the section in question.  Furthermore, we don't even need to be hypothetical about this being the problem.  The research literature, including the sources that we cite, tell us explicitly that this is the problem with reader understanding, in an extremely detailed way.  I happen to think the published, peer reviewed, assessments of what readers do and do not find confusing about the subject of the article should be given considerable weight in our assessments of what we think readers will and will not find confusing.  If indeed the true aim is pedagogy, as you've repeatedly raised, then the opinions of highly qualified published experts on mathematics education should at the very least inform our own approach to the question of presentation, and certainly not be tossed out just because we think we know better than they do what will help readers understand the problem.   Sławomir Biały  (talk) 00:17, 22 July 2017 (UTC)
 * Regarding "I am not sure what part of it you feel would lead a reader to 'could come away from it thinking that it supports 0.999... not equal to 1,'" I am referring to Guy's statement that, "Version B looks like a blatant attempt to give the fringe theory that 0.999… does not equal 1 undue weight." Regarding "This is, actually, considerably more detailed in its description of how the proofs can be made correct, than version A," first of all, it doesn't describe how they can be made correct.  Moreover, something that includes unnecessary details will only confuse the reader, which is a point I've repeated made.  So you shouldn't argue "considerably more detailed in its description" as a point in that description's favor.  I'll gladly toss out any details that fail to improve the article. Calbaer (talk) 00:44, 22 July 2017 (UTC)
 * Firstly, "Guy's statement" is not an answer to "what part of it you feel would lead a reader to..." (etc).  In any case, Guy's objection has now been settled, and so this is a moot point entirely.  Secondly, from Version B: "Once the real number system has been formally defined, its properties can be used to establish the decimal representation of real numbers. The properties of the decimal representation can then be used to justify the rules of decimal arithmetic used in the above proofs."  This says exactly how the proofs can be made rigorous.   Sławomir Biały  (talk) 01:10, 22 July 2017 (UTC)
 * I strongly object to the claim that "Guy's objection has now been settled" without any attempt to ask Guy whether he considers it to be settled. I haven't even read the comment that supposedly settled this (busy with real life) and probably won't find time until Monday or Tuesday. --Guy Macon (talk) 09:25, 22 July 2017 (UTC)
 * Did I misunderstand your assertion "The rest seems fine"?  Sławomir Biały  (talk) 11:03, 22 July 2017 (UTC)
 * I had not noticed that you changed one of the versions. Doing that causes problems with same !votes being cast for the old version and some for the new.
 * Here is the best way to do that: let's say someone writes
 * "It is a little known fact that automobiles are the masters and humans exist on to serve their needs" --Example (talk) 01:11, 01 January 2000 (UTC)"
 * A few people !vote on it, then they realize that they meant to say something else. Rather than just changing it, they should do it like this:
 * "It is a little known fact that automobiles cats are the masters and humans exist on to serve their needs" --Example (talk) 01:11, 01 January 2000 (UTC) Modified 02:22, 01 January 2000 "'
 * (Five "~" characters instead of four gets you just the date). That way it is clear what was changed and when it was changed.
 * The removal of the scare quotes and the "purportedly" does indeed settle my specific objection about promoting a fringe view. I neither support nor oppose the other proposed charges at this time. --Guy Macon (talk) 12:58, 27 July 2017 (UTC)
 * Yes, such violations of WP:REDACT make it very difficult to hold a coherent discussion. The user here has repeatedly edited his own comments after significant amount of time - days in the aforementioned example - without any indication (strikethrough, updated dates) of having done so.  The guideline states, "So long as no one has yet responded to your comment, it's accepted and common practice that you may continue to edit your remarks for a short while to correct mistakes, add links or otherwise improve them....  Once others have replied, or even if no one's replied but it's been more than a short while, if you wish to change or delete your comment, it is commonly best practice to indicate your changes."  While I'll admit once accidentally changing a comment which was replied to between the time I hit "edit" and "save," at least I'd updated the timestamp, so what happened was obvious to all.  (The follow-up in question fortunately did not regard what I had edited, so it didn't cause a problem.)  Granted, this is a guideline, not a policy, but changing a paragraph in place after dozens or hundreds of comments on it makes this confusing, strains the ability of others to assume good faith, and brings us further from consensus, as we can't even agree what we're discussing.  Here's the diff between the original and current; search for "Version B" to see the change.  I would encourage the editor to restore the original and place the change in a place which makes it obvious there has been a change (e.g., strikethrough and bold with a timestamp). Calbaer (talk) 17:30, 27 July 2017 (UTC)
 * I have noted the substantive diff in a separate section below. If anyone wishes to annotate these changes in a way that they would find more satisfactory, they are of course welcome to do so.  I personally do not see the necessity of doing that, at the risk of making an otherwise fairly simple RfC more complex, and think that dwelling on the details of RfC form as opposed to discussing the content improvements of what clearly is the consensus option at this point, is not likely to lead to a different outcome.  I will also note that, should this turn into an argument in support of Version A, that it be based on a genetic fallacy.  Sławomir Biały  (talk) 18:08, 27 July 2017 (UTC)
 * No consensus exists at present. From Consensus: "Consensus on Wikipedia does not mean unanimity (which, although an ideal result, is not always achievable); nor is it the result of a vote. Decision-making involves an effort to incorporate all editors' legitimate concerns, while respecting Wikipedia's policies and guidelines."  Unless you dismiss all concerns you don't like as "illegitimate," I fail to see how anyone could think of the current state as a consensus.   In particular, concerns about comprehensibility have not been sufficiently addressed, meaning that the desired changes will be contrary to the primary mission of Wikipedia: to develop educational content under a free license and to disseminate it effectively.  Furthermore, I still think that this is a false choice presented as a loaded question, and contrary to WP:RFC, which states, "Keep the RfC statement short and simple."  Changes this long and complex are better suited to collaborative development, not yay and nay opinions. Calbaer (talk) 18:46, 27 July 2017 (UTC)
 * I would say that consensus plainly does exist, for version B. This is true both in terms of number of votes and strength of the arguments.  One side, notably, fails to mention any policies whatsoever in support of its position, nor are the policy points of the other side addressed, despite multiple friendly entreaties and reminders.  But that is not for you or I to decide.  And I certainly consider this apparent effort to undermine the validity of the RfC as highly questionable in light of the good faith effort I have made below to clarify what changes were made in response to what comments in the request for comment.   Indeed, most of your efforts here appear to be aimed at drawing the RfC into question, or otherwise undermining the changes that you don't happen to like for ad hominem and genetic reasons, rather than focusing on the substance of the proposed changes.  But this should be discussed below, in a separate section, if anywhere.  And I strongly object to the insinuation that the changes I have made have not been in good faith.  Such insinuations have no place in this discussion.   Sławomir Biały  (talk) 19:17, 27 July 2017 (UTC)
 * If you are referring to my agreement with another editor that modifying Version B makes this discussion more difficult and ambiguous, I am advising people in general - not just you - not to furtively alter additions to the talk page after hours, days, or weeks. That tests good faith, no matter who does it; in fact, my own edit after a few minutes caused another editor to question my good faith, so even such good faith edits are best avoided.
 * As for the "consensus" on the RfC, you have long been assuming that this is a matter of the merits of Version A versus Version B. In fact, giving alternative text makes it a choice between making a specific change versus not doing so, which is different.  Many if not most of those who "voted" for B expressed reservations about its clarity, content, and/or accuracy.  It would be wrong to treat their opinion as evidence of consensus about the exact content (the actual text) rather than general direction (going into precise detail).  If you say there is consensus on the latter, then that's a bit more understandable; though I do not support Version B, I'm all for more precision if readability is not unnecessarily sacrificed. Calbaer (talk) 20:39, 27 July 2017 (UTC)
 * So what? There is clear consensus that "Version B" is either better as it is, or is a better starting point for future article development, than "Version A".  If B can be improved, then please discuss those improvements clearly.  I have already responded to your proposed "Version C", several times.  And I have been systematically engaging with others on improving the proposed content.  That is fully in line with building consensus.  You can be a part of that process or you can choose not to be.  So far, I see you leaning towards the latter option.   Sławomir Biały  (talk) 21:12, 27 July 2017 (UTC)
 * I have given plenty of suggestions in attempts to address your concerns; it's just that they're suggestions that you have decided to oppose rather than use. Editors who disagree with you are not obstacles to overcome via attrition, but people who honestly want to serve the aforementioned mission of Wikipedia.  (I personally believe you want to improve this article, but I am not confident in your ability to do so, due to your desired suggestions being at odds with comprehensibility.)  As for RfC, you believe there is consensus, but I disagree.  I believe the its length alone indicates serious problems that began when it was formulated in the precisely the opposite manner of how RfCs are recommended to be formulated, being neither brief nor neutral.  That has made this a far more confused process than it needed to be, but it's what we have, unfortunately.  Calbaer (talk) 21:46, 27 July 2017 (UTC)
 * I'd be happy to engage with you on any specific discussion about article content. I replied to your "Version C", with a query for more information as to how failing to summarize the context of the cited sources is compatible with the neutral point of view policy.  You appear to have provided no such justification, but instead falsely assert that I "have decided to oppose rather than use" your suggestions.  If you can strongly make the case that your proposal (or Version A, or something else) is policy-compliant, then go for it.  But you can't punt on policy and then claim that I'm opposing because WP:IDONTLIKEIT.  That's not how this works.  Furthermore, I object to referring to this as "attrition".  You could have addressed the content directly rather than continuing the personal assault against me, in far fewer words.  If you find it exhausting to make useless and unproductive arguments, you should stop making them.   Sławomir Biały  (talk) 21:58, 27 July 2017 (UTC)
 * WP:AOBF Assume the assumption of good faith is applicable here. Calbaer (talk) 20:44, 27 July 2017 (UTC)
 * I strongly oppose to the argumentation for Version C. Either something is a (sketch of a) proof, lacking details, necessary for the not fully initiated to follow, maybe even accessible to strict formalization, or it is detrimental -and not useful- to achieving a sound mathematical education, when promulgated as mathematical reasoning (I do not want to deny occasional inspirational potential). All these numberphile-isms, even when viral, striving for acceptance by as many as possible, disregarding their mathematical competence, should be refuted within a serious encyclopedia, rather than constituting "content". The provided line of thoughts just demonstrates wishful thinking -wouldn't it be nice if it worked like this?-, but provides no fruitful intuition. Purgy (talk) 08:30, 22 July 2017 (UTC)
 * Also, the only reliable peer-reviewed sources that we have for these proofs even tell us that they provide no fruitful intuition.  Sławomir Biały  (talk) 11:12, 22 July 2017 (UTC)
 * What peer-reviewed sources? Both references are books (here and here), which are not peer reviewed.  (How can you try to discredit other editors on an appeal-to-authority basis &mdash; "those who I rate as non-mathematicians" &mdash; and not know that?) Calbaer (talk) 15:33, 22 July 2017 (UTC)
 * Peressini and Peressini is published in the peer-reviewed Springer series Perspectives on Mathematical Practices. Byers is published by the academic publisher Princeton University Press, which certainly does conduct peer review.  But a bigger question is, why are you trying to question obviously reliable sources, and also dismissing adherence to sources as "appeal authority"? You aren't new to Wikipedia, and should know better.  In any case, citing expert sources in support of a carefully argued position is not an appeal to authority.  It is the hallmark of all scholastic discouse.  But I suppose you wouldn't know that, would you?  Sławomir Biały  (talk) 16:14, 22 July 2017 (UTC)
 * Actually, it can be questioned whether the authors and publishers are experts in mathematical pedagogy. They are experts in mathematics, but all the arguments can be made rigorous, and they may not be experts in which arguments are most questionable.  — Arthur Rubin  (talk) 18:23, 22 July 2017 (UTC)
 * I don't see why being an expert in mathematics is mutually exclusive with being an expert in math pedagogy. On the contrary, I would say that the two attributes exhibit a strong positive correlation.  But that's beside the point: it seems like pedagogy they're writing to me.  I mean, it seems like our criterion for someone being an "expert in pedagogy" here, would be someone who writes peer-reviewed pedagogy research.  And we have that.  So I don't see what the problem is.   Sławomir Biały  (talk) 19:05, 22 July 2017 (UTC)
 * Please show me where it says the book is "peer reviewed." It's edited.  That's a completely different process than peer review.  Don't get me wrong, non-peer-reviewed sources are fine for Wikipedia.  But don't claim that you're a mathematician struggling against us less qualified "non-mathematicians" when you don't even know what peer review is. Calbaer (talk) 19:48, 22 July 2017 (UTC)
 * This is, sadly, becoming rather less about improving the encyclopedia. Does it matter if the content in the reliable sources meets some criterion for peer review?  Both are serious academic sources.  If you think that the sources are unreliable, then this is a fruitful line of inquiry.  If I used "peer-review" in an off-the-cuff way to refer to sources about which there is no real dispute, then I don't see that resolving this irrelevant ambiguity is constructive.  If you wish to discuss your own qualifications, showing a knowledge of the subject of the article would be a good start, or at least an eagerness to acquire the necessary background.  But attacking those editors in this process who are the most qualified to improve the article, seems like a waste of everyone's time, and not beneficial to the encyclopedia.   Sławomir Biały  (talk) 20:07, 22 July 2017 (UTC)
 * You are the one who appointed yourself most qualified to improve this article; no one else did. I am merely questioning your qualifications, given that you don't know one of the most basic aspects of mathematical and scientific research.  I had hoped this observation might convince you to stop throwing stones from your glass house (i.e., writing ad hominem attacks against the qualifications of others).  If this results in less elitism and hostility on your end, then, yes, it was indeed constructive, since these are hampering efforts to improve the article. Calbaer (talk) 03:08, 23 July 2017 (UTC)
 * From academic publishing: "many academic and scholarly books, though not all, are based on some form of peer review or editorial refereeing to qualify texts for publication." I have little doubt that both sources were subject to some form of peer review.  But that is not relevant to improving the article, and the matter is not worth further discussion.  Regarding glass houses, let me remind you that (from Credentials matter): "Amateurs may not have the experience or education necessary to evaluate sources adequately, or may not understand the material well enough to organize it into a coherent whole. And they may not be aware of how poor their understanding might be (the Dunning–Kruger effect). Experts are not perfect, but amateurs are on the whole less perfect, and especially in their judgement of the work of experts."    Sławomir Biały  (talk) 10:37, 23 July 2017 (UTC)
 * The first words of Credentials matter are: "This page is an essay. It contains the advice or opinions of one or more Wikipedia contributors. Essays are not Wikipedia policies or guidelines. Some essays represent widespread norms; others only represent minority viewpoints." Not a policy, not a guideline, just an editor's opinion.  As for Dunning–Kruger, I'd point you to the transcript this interview with Dunning: "We all have our specific pockets of incompetence, and we know some of them. But there are a lot of them we simply don't know."  When challenged on teaching expertise - given that the point of the article is to educate - you replied that expertise in math should be sufficient.  It's clearly not (assuming you have credentials in the first place; for all your talk of their importance and how they make your arguments superior, you curiously never state yours).  Anyway, please don't cite opinion as though it were policy, and please don't misrepresent sources. Calbaer (talk) 13:40, 23 July 2017 (UTC)
 * The argument that I feel that expertise in math is sufficient for math pedagogy is incorrect, and I would point out that the sources under discussion clearly concern math prdagogy, not research mathematics. But we could focus the discussion on mathematics teaching expersience.  How many mathematics courses have you taught, in which the subject of this article was discussed, for example?  I heartily agree the we should not misrepresent sources, and am grateful that you finally acknowledge this.  It seems like we should be discussing how to do that.  Do you agree that the best way not to misrepresent dources is by summarizing what those sources Ave to say, supported by in-text attribution and direct quotation?  If not, how to you propose this non-misrepresentation be achieved?   Sławomir Biały  (talk) 13:52, 23 July 2017 (UTC)
 * Appeal to probability (many do, ergo this one does) is a fallacy. And it's silly to expect me to answer questions you refuse to answer yourself, such as those on qualifications.  My point is that you should drop the "you're unqualified" attacks, because they're counterproductive and you don't have a leg to stand on there anyway. Calbaer (talk) 14:23, 23 July 2017 (UTC)
 * Not sure what "appeal to probability" you're referring to: Experts in mathematics are "probably" experts in pedagpgy, and we're discussing sources on pedagogy. You seem to be saying that if something is "probably true", then it must be false, like your earlier apparent belief that because authorities say something, we should (apparently) disregard it, regardless of what other reasons are given in support.  This same logic pervades your denunciation of the supposedly "loaded question" that you object to: simply saying that "the idea that due weight should be the only criteria in judging which text is better for the article is false" does not absolve one of the responsibility to address what is and is not due weight.  Nor does dismissing the credentials of other "experts" or "supposed experts" or whatever, absolve you of the responsibility of responding to what they write in a substantive manner (if you respond at all: no one can oblige you to have an opinion on the matter one way or the other).
 * And no one besides you seems to be keeping the issue of credentialism alive. Nothing you have said in this discussion appears to relate in a substantive way either to the specific policy points (which have nothing to do with qualifications), or the specific sources themselves, aside from question the extent to which "peer-review" is applicable.  You don't have to have formal qualifications to edit Wikipedia, but to continue productively to a discussion about mathematics does require a certain competence that you're not demonstrating.  The only mathematical thing you mentioned was an apparently mistaken belief that the proofs under discussion follow from the axioms of mathematics, which is just wrong, while insinuating that it is my own understanding of the subject that is flawed .  I hope you can see how this comes across, and how by doing this you did make qualifications relevant to the discussion.   If you do not wish qualifications to be relevant, you should stop discussing them.  And, in particular, stop trying to attack the qualifications of others who, plainly, have far more substantive things to say about the article than you do.  That is not constructive, unless your goal is to prevent those who know what they're talking about from contributing.  Is it?  I'm beginning to wonder.   Sławomir Biały  (talk) 15:00, 23 July 2017 (UTC)
 * The appeal to probability ("some therefore all") is your conclusion that because some books are peer reviewed, the Springer one is, in spite of there being there being no evidence that this is true. Note that the link above shows you confusing language, not mathematics.  Language, not math, is your primary difficulty in making a positive contribution here.  A secondary difficulty is hostility.  On that point, you are the one keeping credentialism alive.  Editors such as Huon, MjolnirPants, and myself are trying to dissuade you from doing so, to consider ideas on their own merits rather than attacking the qualifications of those presenting them.  If you believe this line of argument is useless, then just stop starting such arguments. Calbaer (talk) 19:46, 24 July 2017 (UTC)
 * Calbaer, I have been doing nothing but discussing "ideas on their own merits" with other editors. I don't care what your (or anyone's) credentials are, in truth, and dismissing others as "non-mathematicians" was worded in a less-than-ideal manner.  Indeed, I feel strongly that contributing productively to a discussion about a mathematical topic does require a willingness to admit that there is a great deal that one does not know.  But your behavior throughout this discussion, beginning with this post, your dismissal of what I have to say as a Gish gallop (while perversely at the same time saying that I am ignoring you), your reverts to the edits of the article, and finally this current inquest, seem to be intended to shut down discussion about ideas and their merits.  They are just meta-discussions rather than proper discussions, observations of form rather than substance.  If you want me to discuss with you "ideas on their own merits", give me some ideas with merit to discuss with you and I would be happy to do so.  Or you could discuss my ideas on their merits.  It's up to you.   Sławomir Biały  (talk) 09:27, 25 July 2017 (UTC)
 * I gave alternatives to your suggestions several times, so you can't credibly accuse me of not producing any ideas or not discussing yours. And your attacks and mistakes might be matters of unfortunate wording to you, but all we have are your words, so that's what we have to go by.  If you have difficulty conveying your thoughts in print, then presenting your ideas on talk pages for refinement before inclusion in articles might be preferable to editing first, only to find your contributions reverted by multiple editors. Calbaer (talk) 18:58, 25 July 2017 (UTC)
 * I responded to your suggestions. If I did not, please tell me which suggestions you would like my input on.  You are certainly welcome to respond substantively to any of my posts as well (or not).  But please don't make excuses like that I "have difficulty conveying [my] thoughts in print".  If something is unclear to you, try reading it.  If it's still unclear, try reading it a second time.  If it's still unclear, you can always ask for clarification.  There are people here from whom you can actually learn something, if you're not so busy trying to shoot them down.   Sławomir Biały  (talk) 19:46, 25 July 2017 (UTC)
 * "[D]ifficulty conveying your thoughts in print," is indicated by your statements that it was your wording to blame for the appearances that you were lodging ad hominem attacks ("worded in a less-than-ideal manner") and didn't know the definition of "peer review" (which you characterized as "an off-the-cuff way to refer to sources about which there is no real dispute"), rather than credentialism and ignorance, respectively. Unfortunately, an encyclopedia article is not something from which a reader will be able to "ask for clarification," so we need to be careful about wording there. Calbaer (talk) 20:11, 25 July 2017 (UTC)
 * Fine. Point taken.   Sławomir Biały  (talk) 20:14, 25 July 2017 (UTC)
 * Purgy, that's a problem with A, B, and C, not just C. All present the above material as not complete enough to constitute sufficient formal proof, not as "bad" material that the reader should avoid being fooled by.  Even B's proponent has claimed that B showed "how the proofs can be made rigorous."  Although I don't buy that, that indicates a desire to keep the material and present it as incomplete, not as a cautionary tale.  B would just muddy the waters, trying to have it both ways.  If you dislike C, that's fine; I just want something that resolves the objection to accuracy while retaining readability, criteria B fails to satisfy. Calbaer (talk) 15:33, 22 July 2017 (UTC)
 * Calbaer, thank you for responding in very calm words (quite rare these times) to my rather harsh accusations. Nevertheless, I disagree with holding all three variants equal wrt their distance to the non-proofs. While I perceive version A as, blatantly and fully intentionally, distracting from the problems at the heart of "infinitely long" division and "infinite" multiply through, version B constantly appeals to the higher ideas in the respective math. I agree that these approaches reflect themselves in the current readabilities, but I rather take the degraded readability of B, hoping for achievable improvement, than the dishonest soothing of everything is easy in the first section of A, suggesting that the nitpickers, best to ignore, may articulate their troubles and pipe dreams in the rest of the article, best ignored by the average reader. Imho: A (and C) blissfully fail accuracy, B is improvable, especially on readability. Purgy (talk) 12:32, 23 July 2017 (UTC)
 * I was judging based on which is better for inclusion, not which is improvable. I appreciate the concern here, but I'd personally prefer an incomplete (and thus inaccurate) but readable text to complete and accurate but unreadable text.  Thus my desire to wait for a "version C" (not necessarily mine, since I just quickly typed it out as an example).  I'd like something that is both accurate and not so opaque and meandering as to lose everyone who might actually gain knowledge from the article. Calbaer (talk) 19:52, 24 July 2017 (UTC)
 * Regarding "the idea that due weight should be the only criteria [sic] in judging which text is better for the article is false." That is true.  But do you agree that: compliance with WP:WEIGHT is a mandatory requirement of all article content?  On the assumption that the answer to this question is "yes", how is it that Version A is due weight, when it spends just a third of the time on the discussion, with two-thirds dedicated to the proof; whereas, in the Byers source, only 10% of the text is actually devoted to the proof itself?  Also, does version A accurately and neutrally summarize the viewpoints expressed in Byers, Richman, and Peressini?  Is relegating the opinions of these cited authors to a footnote, rather than the article text, consistent with the "prominence of placement" expressed in WP:WEIGHT?  Finally, it does not have to be a binary decision, as I noted in my comment to Guy.  You are certainly free to make suggestions on how Version A or B might be improved, if you feel that it can present the viewpoints expressed in the reliable cited sources in a more satisfactory manner.  Your suggested Version C, however, does not attempt to express those views at all, and so it also fails the mandatory policy requirement of NPOV.  Thus it is not an acceptable alternative.   Sławomir Biały  (talk) 17:16, 23 July 2017 (UTC)

Break

 * Version A: I am going out on a limb here, I know, but I have multiple concerns about Version B. First of all, we say you have to understand the relationship between compressed infinity, completed infinity and the Cauchy completeness property. None of which are heard from again. And the last-mentioned points to the article on Cauchy sequences where the completeness property is hard to find; perhaps the Construction of the real numbers would be a better place to send them? Then we mention "the completeness axiom of the real numbers". What is that? An an axiom? The reader might think that we've just produced a new concept, which we are asking to be accepted without proof. And it begs some more questions. First of all, the idea of the equality of 0.999... and 1 arising from the construction of the reals is a good one, but as we just pointed out in the preceding paragraphs, we haven't established that 0.999... is a real number yet. Secondly, the reals can be constructed in multiple ways, and we go on to use others below, so are we saying that we must use Cantor's construction, using the Cauchy sequence? If not, then doesn't that invalidate our argument? Is it true that 0.9999... = 1 under any construction of the real numbers, or just some? Hawkeye7 (talk) 00:00, 22 July 2017 (UTC)
 * I am happy with the change from "proofs" to "arguments" in both versions. Hawkeye7 (talk) 00:02, 22 July 2017 (UTC)
 * I'm not. They have been presented as proofs not arguments even if they are not halfway rigorous proofs. And we don't need handwavey arguments here. We should say what they are which is purported or incomplete proofs. Dmcq (talk) 12:16, 22 July 2017 (UTC)
 * The real numbers are axiomatically defined as the unique complete ordered field up to isomorphism. There are many different ways to construct this field, but there is only one of them (at least in the standard foundations of mathematics), and the model does not affect the truth of the equality $$0.999...=1$$.  This is another thing that (in my opinion), the present article gets wrong: it appears to hang the question of equality on the specific model of the reals.  This also needs fixing.  But that is a different discussion.  The further objection seems to be that many unfamiliar mathematical concepts are required to understand the subject of the article, and these unfamiliar concepts seem very sophisticated.  There is a reason for that: the subject of the article is a genuinely difficult thing to understand, even for students who have a thorough understanding of university calculus.  For many individuals, it requires a radical restructuring of the very concept of "number".  We should not present proofs that secretly rely on implicit assumptions that do not correspond to the assumptions that the target readers will have going into the proof (it would be like saying "triangle" in a proof, but really secretly meaning hyperbolic triangle).  Those implicit assumptions should be made explicit, and we should use the mathematically correct vocabulary for them.   Sławomir Biały  (talk) 01:27, 22 July 2017 (UTC)


 * Version B: Obviously, this version is amenable to improvement, too. However, it is by far mathematically better reasoned, and therefore, up to now, also prevails in the !voting by several mathematically educated editors. Attempting to achieve a fallacious understanding of a "deep" fact, not easily accesible, by pretending accessibility via simple mechanisms does not serve well unwary readers. As for the original question in the RfC, to me the obvious concerns in the sources are swept under the rug by Version A. Purgy (talk) 08:53, 22 July 2017 (UTC)


 * Weak support for Version B. Once I saw Sławomir Biały involved, I expected to side with him. And to large extent I do. However Version B is simply too long, and reads as a bit axe-grindy. I think this part is very important, and should stay: The completeness axiom of the real number system is what allows infinite decimals like $$0.111\ldots$$ and $$0.999\ldots$$ to be regarded as objects (real numbers) in their own right ...establish the decimal representation of real numbers." I think the other added bits specifically critiquing algebraic proofs should be reduced. I appreciate the subtlety of Sławomir's objections to version A, but I think he's responded with overkill. What constitutes a rigorous proof is context-dependent, and it's not clear that this is the best place to dive in to a level of detail that most readers will find to be some mix of baffling and unnecessary. I would like to Support a shorter Version B. I realize now this is bad form for an RfC, but I can't support either as-is. SemanticMantis (talk) 13:36, 22 July 2017 (UTC)
 * That is perfectly in order. An RfC is supposed to be a request for comments - not a vote. I'd be quite happy with a slightly expanded version of the top of the section on algebraic proofs and remove the discussion altogether, the recent change by D.Lazard is a good basis for that. Version A is just wrong and jangles. The sooner the article passes over the algebraic section the better. And really a lot of the interest is just in how people try proving something without really understanding what they're working from or what they're actually doing. In fact a lot of the next section on analytic proofs is the same trying to do it without referencing the definition of real numbers. I wouldn't mind the whole 'analytic proofs' section disappearing too and just go on to proofs from the construction of the real numbers. Dmcq (talk) 17:19, 22 July 2017 (UTC)


 * Version A is the better of the two. Most of the differences are wrong (irrelevant) and fringe (relevant).  The completeness axiom should be emphasized more, but potential and completed infinities should not, because potential infinity is a fringe concept, at best.  The difference between seeing 0.999... as an object and incorrectly seeing it as a process might be emphasized, but not using fringe terminology.   — Arthur Rubin  (talk) 18:23, 22 July 2017 (UTC)
 * Arthur, the Peressini source mentions "(potentially) infinite series", which I read as a reference to potential infinity, which is referred to elsewhere as well in the research literature (Katz and Katz). But if you are more comfortable, would simply changing the phrase "the relation between potential infinity and completed infinity" to "the nature of infinity" be sufficient to sway your opinion?   Sławomir Biały  (talk) 19:12, 22 July 2017 (UTC)
 * Also, on a slightly more nuanced note, I do not think it is correct to dismiss the idea of "0.999..." as a process. When we mathematicians use it, we do mean a process (or rather its result, in the limit).  We write 0.999... as a limit of partial sums.  The sequence of partial sums is the "process", which still plays a fundamental role.  Students understand this process aspect, but fail to grasp the limit aspect that also makes it an object because that is much subtler.   Sławomir Biały  (talk) 19:21, 22 July 2017 (UTC)


 * Support a shorter Version B; the last paragraph seems incorrect. There are ways to give "0.999..." and "1.000..." meaning in other number systems. If sources argue that's impossible, I'd like to know what they are. That said, much of B is indeed an improvement over A. Huon (talk) 19:38, 22 July 2017 (UTC)
 * I agree with Guy, and therefore argue for no change based on the current proposals until something better is found William M. Connolley (talk) 11:33, 23 July 2017 (UTC)
 * William, could you please clarify how you think that the material can better summarize, in a proportionate way, the cited sources? Do you feel that "Version B" does not properly summarize the views expressed in the cited sources, while "Version A" does summarize those views in a more proportionate way?  If so, could you please explain why?  Version B, in particular, is supported by more page references, in-text attribution, and quotations to the sources, than is version A, and so at least superficially seems to be the more policy-compliant of the two.  Accordingly, I feel that a view that the less detailed version lacking such attribution in fact summarizes those sources "better" requires some justification.  This is not a vote, and you will note that the phrasing of the RfC specifically concerns this question.   Sławomir Biały  (talk) 12:38, 23 July 2017 (UTC)


 * No change per Guy & William until a better alternative is presented. Keira  1996  23:53, 23 July 2017 (UTC)
 * Reminder to all participants (and the closing administrator). This is not a vote on which version we happen to like better.  The parameters of the RfC are, specifically, which version properly gives the opinions expressed by reliable sources WP:DUE weight.  It is true that Wikipedia is based on consensus, but the consensus must be based on valid policy-based arguments.  Not WP:IDONTLIKEIT non-arguments that fail to respond to requests for clarification.  (Much less now the apparent piling on to Guy's suggestion which seems to me to have been amicably resolved with a minor edit to the first sentence.) None of the most recent posts have addressed the question set forth in the RfC, so these will be of limited use in consensus building.  The only compelling policy argument advanced thus far has been this one, and it has not been challenged by any of the participants here.  I note that the Arbitration committee has repeatedly held that articles must comply with the neutral point of view policy.  For example : "Wikipedia articles are to be written from a neutral point of view and without bias... To comply with the verifiability policy, assertions of fact, particularly controversial ones, should be supported by an inline citation to a reliable source." : "All Wikipedia articles must be written from a neutral point of view. Where different scholarly viewpoints exist on a topic, those views enjoying a reasonable degree of support should be reflected in article content. An article should fairly represent the weight of authority for each such view, and should not give undue weight to views held by a relatively small minority of commentators or scholars. The neutral point of view is the guiding editorial principle of Wikipedia, and is not optional."  (And so on...)  I wish to emphasize that the closing administrator should not be counting votes, but read the reasoned policy-based arguments, knowing that the Arbitration Committee may review their decision if our policies fail to prevail.   Sławomir Biały  (talk) 00:23, 24 July 2017 (UTC)
 * You're quick to cite WP:NPOV to support your argument, but you blatantly ignore WP:UNDUE and WP:FRINGE which state that we're not required and in fact 'discouraged' from providing a platform for promoting fringe views (like the view that 0.999... isn't equal to 1. It is.). 74.70.146.1 (talk) 01:52, 24 July 2017 (UTC)
 * How about actually looking at the sources? Have you better? They like to promote mathematics as based on reason rather than faith. I hope we're not into the territory of the Australian PM who recently said 'The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia'. Nobody is saying that as real numbers they are unequal - it is just that the algebraic proofs don't state their assumptions, as it says in the lead of the section 'However, these proofs are incomplete or not rigorous, as they do not include a clear definition of 0.999… and of the operations that are allowed on such a notation.' Dmcq (talk) 07:19, 24 July 2017 (UTC)
 * My objective is not to support the view that 0.999... and 1 are anything but equal (which would be fringe). But presenting as a proof a subtly misleading argument of a true fact is not justifiable, when the cited literature presents those proofs actually as examples of misleading arguments.  Indeed, it would be better to give no such proof at all: our obligation to "the truth" stops at saying that the real numbers $$0.999...$$ and $$1$$ are equal, supported by a reliable source.  The article is under no policy obligation to convince the reader of this standard fact, just like we aren't under any obligation to convince the reader that the statement "An electron is a fundamental particle" is true by presenting them with scattering data from particle accelerators.  This having been said, if we include these proofs at all, then policy does require us firstly to cite sources appropriately, and secondly to summarize what those sources actually have to say on the matter in a proportionate way.  As I noted above, the Byers source spends 10% of the text on the "proof" (with Byers' scarequotes), and 90% of the text on the discussion.  So it's hard to see how spending more text on the proofs than the discussion is any way proportionate (and indeed, no effort is made to summarize his views in a coherent way at all).  If you can think of a way to summarize the sources better, then please suggest it.  That's why there is a request for comment.  I'm rather alarmed though that so many editors do not seem to care what sources actually say.  Needless to say, views that are not actually based on any analysis of sources should be disregarded by the closing administrator.  If they are not, then ArbCom remains a likely outcome.   Sławomir Biały  (talk) 11:15, 24 July 2017 (UTC)
 * If you argue this way, it seems to me neither version A nor version B are a solution, but strictly speaking you'd need to ask for the deletion of the article.
 * If the purpose of this article is simply to state the well known fact 0.999...= 1, then we don't need an article for that all. That fact can simply be mentioned in the article on Real numbers and/or Decimal representation and that's it (maybe instead of deletion turns this article into a redirect). Having a separate article only makes sense, if it deals explanations and backgrounds infos on that fact, which loosely speaking is kinda like "convincing the reader of that fact". Also having a (minimal) encyclopedic obligation does not block us from doing more. So while we are not obligated to do more, we very well may choose to nevertheless.--Kmhkmh (talk) 16:40, 24 July 2017 (UTC)
 * I did not mean to suggest that we do not include a proof at all. By all means, we should include a really convincing proof; several even!  What I object to is presenting a proof that absolutely should not be convincing to anyone that actually needs to be convinced, as if it were convincing.  Indeed, our sources say that students who find these proofs convincing do not understand the identity.  Yes, to someone with prior knowledge of the completeness axiom, they can be made precise and rigorous; but I've been repeatedly told that it's not that group for whim these proofs are intended.  On the contrary, the algebraic proofs are also being presented as easy to understand, which they are not.  Indeed, the algebraic proofs are every bit as hard to understand as the analytic proofs, possibly moreso.  If we aren't committed to conveying a proper understanding of these proofs, supported by reliable sources, then I do not believe that they should be included at all.  Does this make sense?   Sławomir Biały  (talk) 17:17, 24 July 2017 (UTC)
 * I'm somewhat neutral on the A versus B and I'm not going to state a preference. I can live with the inclusion of the problematic/iffy proofs as long as they are found in (reputable) literature and their potential issues are pointed out in the article. In particular if they tend to show up math books for general audiences or (high) school books, it might be a good idea to have a Mathematically proper treatment of them in WP. Here i just wanted to point out that the logical consequence of strict "just state mathematical fact"-argument would imho be a request for a deletion or redirect, as that bit of information doesn't warrant an article.--Kmhkmh (talk) 17:40, 24 July 2017 (UTC)


 * Version B: Version A is too handwavey. I'll note here that several people read the second version as questioning the fact that 0.999... = 1. That is not what it's doing: indeed it notes the fact in the first sentence. What it says is that a rigorous proof is tricky. I agree with the sentiment, therefore prefer Version B. It could perhaps be shortened (not sure how). Even though it's longer, it's a bit easier to read and has less jargon than the shorter version. Kingsindian &#9821; &#9818; 16:26, 24 July 2017 (UTC)
 * Version B: Is it perfect? No. Is it too long? No, because either the reader is equipped to follow it or not. If not s/he will pass on, skipping all of say, 200 words. Readers capable of following are condemned to read the extra 200! Biiig deal... Could B be improved on? Sure, and feel welcome, but until then B should be used because it is far more helpful than A. C did not cut the mustard. Pardon me for not participating in the wall-of-text choir, but feel welcome to rattle my cage when the new all-singing-all-dancing Version D arrives; meanwhile, sorry, things to do... JonRichfield (talk) 05:58, 25 July 2017 (UTC)
 * version B. I don't know who the intended audience of the paragraph is – but version B makes more sense to me. Maproom (talk) 08:05, 25 July 2017 (UTC)
 * Something else. Look, I'm on board with avoiding lies-to-children; Sławomir is correct that those have no place in this sort of project.  But I don't see the point of the article in the first place, if readers have to understand the real numbers rigorously.  Anyone who understands the real numbers rigorously doesn't need this article. It seems to me that the "algebraic" proofs can be saved with some caveats.  Emphasize that if there is a way of getting from infinite decimal strings to numbers, and if that way has certain properties that will seem reasonable (the behavior of multiplication or subtraction, and maybe the Archimedean property), then it follows that 0.999... equals 1. I think the Archimedean property is much easier to follow for the target audience than completeness, and we should emphasize it over completeness in the early going. Then, further down into the article, we can segue into the motivation for the reals.  The Zeno stuff goes well there &mdash; I've complained a couple of times that the real number article doesn't mention Zeno, which I think is a major oversight, but I haven't gotten around to doing anything about it.  But I think we can start with the "algebraic" arguments, provided we stress that they make assumptions that can't be justified at the level of that exposition. --Trovatore (talk) 09:37, 25 July 2017 (UTC)
 * Doesn't the definition of 0.999... already rely on completeness?  Sławomir Biały  (talk) 10:33, 25 July 2017 (UTC)
 * My proposal is that, for the introductory part of the article, we not try to prove that 0.999... denotes a number, but that we rather show that, given that it denotes a number, and given that the interpretation has certain properties that will seem reasonable to the reader, then the number it denotes must be 1. Then we don't need to mention completeness.
 * That will be enough for the naive reader, and we haven't told him/her any lies. Then the more advanced, or more curious, reader can read on to see how we justify the "given" parts of the above. --Trovatore (talk) 10:46, 25 July 2017 (UTC)
 * Yes, that could work. That discussion would ideally go before the proofs I think.  But don't the proofs still obscure the main point, which is either completeness or the Archimedean property?  I do not think that making the algebra explicit axioms that are assumed to be valid eliminates the need for a discussion such as "Version B".   Sławomir Biały  (talk) 10:59, 25 July 2017 (UTC)


 * Yep we have to mention either completeness or the Archimedean property if we're not indulging in flimflam just trying to convince people who don't know better. Probably the Archimedean property is best. To some extent it just gets rid of the problem by making an assumption but we can't get rid of it just using algebra - that would be like showing parallel lines exist without assuming Euclid's fifth postulate. Just saying real numbers doesn't cut it as people will just assume they know what is meant by that. Dmcq (talk) 11:19, 25 July 2017 (UTC)


 * I imagine we could infer that some version of the Archimedean property is a consequence of taking the algebraic properties for decimals as axioms, which may not be so bad. But I think that would probably stray into original research, and I imagine that it would make a Platonist uncomfortable.   Sławomir Biały  (talk) 12:19, 25 July 2017 (UTC)
 * I can't see how one could do that without assuming the result is true. For instance the very first line of the first algebraic proof goes wrong when it says 1/9 = 0.111... Yes one gets a sequence of 1's using the algorithm but one can't say they are equal just using algebra without assuming the Archimedean property. Dmcq (talk) 13:07, 25 July 2017 (UTC)
 * I see your point. Perhaps all sides would be more satisfied if, instead of presenting these arguments as proofs of anything, they are presented as seeming paradoxes that, instead of definitively demonstrating something specific, suggest a re-examination of the number concept?  If real numbers and decimals meant the same thing, then these arguments lead to a paradox, because the decimal "1.000..." is not the same as "0.999...".
 * As we know, algebra students nowadays are entirely reliant on their calculators for even the most basic of tasks. An algebra student perhaps would agree that $$1/9=0.111...$$ because he checks his calculator, which reports that $$1/9=0.11111111$$, and perhaps infers that the 1s actually go one forever, but are simply rounded to the number of digits of precision.  Then multiplying this through by nine, he would get $$1=0.99999999$$.  This equation is, of course, not actually true any more than the equation $$1/9=0.11111111$$ was true.  But the question is, how to interpret it?
 * I think we start with the equation $$1/9=0.111...$$ (with infinite 1s), multiply through by nine to give $$9*(1/9)=0.999...$$. By the associative law for multiplication, the left-hand side simplifies to unity.  Now, we started with a "true" equation, "multiplied" by nine, then applied a valid algebraic rule for multiplication, and arrived at an equation.  The student knows that multiplying both sides preserves the equation of the two sides.  But here we have an equation that his calculator tells him is false: he enters "1", and the calculator tells him "1".  So how can this apparent paradox be resolved?   Sławomir Biały  (talk) 13:34, 25 July 2017 (UTC)
 * The same algebraic manipulation that gives 1 + 1/9 = 1/(1-1/10) = 1.111... also gives -1/9 = 1/(1-10) = 1+10+100+.. = ...11111.0 so 0 = 1/9-1/9 = ...1111.1111... ;-) It's really got to be based on the sources. Dmcq (talk) 14:04, 25 July 2017 (UTC)
 * "It's really got to be based on the sources." I don't think anything else really needs to be said ;-)  Sławomir Biały  (talk) 14:12, 25 July 2017 (UTC)


 * Questions I see in the article an explanation of the "Archimedian Property" but no explanation of the "completeness axiom" in the proposal or in the article and how, if at all, they relate or compare. Also, why is not the "completeness axiom", what it is and what it means upfront in the proposal (and in the lede), since the "completeness axiom of the real number system is what allows" the equality? Alanscottwalker (talk) 16:07, 25 July 2017 (UTC)
 * I must confess that I too am puzzled by the article's aversion to the completeness axiom. Several knowledgeable editors have expressed the opinion that it is the Archimedean property that should be emphasized, as opposed the completeness axiom.  I might be missing something, but it seems to me that completeness is required anyway, regardless of whether the Archimedean principle is emphasized (at least, without modifying things in an exotic way).  Sławomir Biały  (talk) 16:18, 25 July 2017 (UTC)
 * Completeness axiom is the existence of least upper bounds for upper bounded sets. This is the axiom, which implies that $$\sqrt 2$$ is a real number, as being the least upper bound of the rational numbers whose square is less than two. Here, as 1 is a rational number, the Archimedean property is sufficient for proving that 1 is the least upper bound of all 0.999...9. It is what was shown in the first version of section "Motivation". I am not sure that it was a good idea of removing this from the end of the section, as the only possible definition of the notation 0.999... is to denote the least upper bound of all 0.999...9. D.Lazard (talk) 17:07, 25 July 2017 (UTC)
 * Thanks. "Completeness axiom is the existence of least upper bounds for upper bounded sets" . . . bear with me: The completeness axiom holds that for upper bounded sets there is [a] least upper bounds. All real numbers, such as 1 are upper bounded sets and 0.99 is its least upper bound. Is that close? Alanscottwalker (talk) 17:58, 25 July 2017 (UTC)
 * I've tried to explain it here. Any good?   Sławomir Biały  (talk) 21:14, 25 July 2017 (UTC)
 * Thanks. Two things: 1)"The meaning of the notation $$0.999...$$ is the first point after the sequence of finite truncations $$0.9,0.99,0.999,\ldots$$ (and similarly for the meaning of any infinite decimal)." Why cannot that thought be completed explicitly with reference to the statement one equals 0.999 . . . (thus, therefore, and, etc.) . . . close the hanging issue for the reader.  2) Avoid reference to it as a "property" and an "axiom", unless your directly explain somewhere something in the article, like, 'this property is an axiom in real number theory.'  Alanscottwalker (talk) 12:29, 26 July 2017 (UTC)
 * How about this?  Sławomir Biały  (talk) 12:54, 26 July 2017 (UTC)
 * It was not my intention to remove it, but rather to rephrase it in an intuitive way suitable for motivation (thus, $$P_\infty$$ is the first point following all $$P_n$$). But you're right that it should be made explicit that this is the completeness property that one is referring to.   Sławomir Biały  (talk) 18:53, 25 July 2017 (UTC)
 * I do appreciate you being responsive, and I do hope these questions/comments have improved the article. On the overall issue of this RFC, I am persuaded of one thing, with respect to A and B - B is sourced with RS and I am therefore inclined to it, although I do wish it could be shorter. Alanscottwalker (talk) 22:24, 26 July 2017 (UTC)
 * I take that "shorter" part a bit back, I now think in particular the introduction to thinking about a "process" and an "object" is important for better understanding the article's later sections. -Alanscottwalker (talk) 12:42, 27 July 2017 (UTC)


 * Monkey wrench — I think the first paragraph of Option B is justified by the sources, although "these arguments fail to supply a satisfactory explanation of why the equation should hold" begs the question of satisfactory to whom? But, the framing seems to imply that the equality was false, or impossible to adequately explain before the formalization of real numbers. Yet all repeating decimals are rational numbers and the mathematics of their values and the process of conversion from repeating decimals to rationals was well established before. Hence all the (scare-quote) proofs. So, I think if we're to say that the arithmetic explanation is "unsatisfactory" now, we need to indicate that it was satisfactory to (probably the vast majority) of mathematicians before the elaboration of real numbers and theorizing about the meaning of decimal representations.--Carwil (talk) 17:40, 25 July 2017 (UTC)
 * That's an interesting perspective, and I wonder if there are sources. I have serious doubts that the algebraic arguments would have been regarded as satisfactory before the introduction of the real number system.  Generally, mathematics dealing with infinity were viewed with heavy suspicion prior to the 19th century.  Archimedes famously found the area enclosed by a parabolic segment by what would now be recognized as the infinite sum of a geometric series, but he did so with the method of exhaustion: proving that the presumptive "sum" would need to be neither less than nor greater than some given value.  So I think the idea that least upper bounds were important to proving statements like this were recognized well before the real number system was properly organized in the 19th century.  Another issue with these historical kinds of questions, too, is that it is in some sense meaningless to impose our standards of what a proof meant, for someone in the past like Euler (or Archimedes).  It may have been a rigorous proof to Euler, but it would have meant something different to Euler, and so the comparison to the proof that is understood today, by an algebra or calculus student, is a very different "proof-idea".  I fear that makes such comparisons meaningless without very good historical sources.
 * In any case, I think it is simplest to forgo these questions by rewriting the sentence. An inline attribution, like "According to Peressini and Peressini, ..." or possibly even a direct quotation, would then not be something asserted in Wikipedia's voice. I have gone ahead and done this.  Sławomir Biały  (talk) 21:00, 25 July 2017 (UTC)


 * Note: It is difficult to conduct an evaluation such as this if Version B changes in-place, as observations might no longer be applicable. (In general, anything on the talk page should be added to, rather than modified in place, though I'll admit to occasionally - but rarely - modifying something a few minutes after writing it if I find an error or ambiguity in what I wrote.)  It might be more useful to add the new version to the bottom. Calbaer (talk) 21:34, 25 July 2017 (UTC)
 * I like the modification to attribute "unsatsifying." Encyclopedia Brittanica (1796) offers an explanation that references limited difference (1-0.9999… = 1/10, no 1/100, no 1/1000 etc.) but not infinitesimals or real numbers. See here.--Carwil (talk) 23:28, 25 July 2017 (UTC)
 * Thank you, that is most interesting. They say that 9/9=0.999... "signifies" 1.  Thus it seems to be a matter of defining it do be something, rather than a statement about numbers, to be proved.   Sławomir Biały  (talk) 23:35, 25 July 2017 (UTC)
 * For the record, I'm okay with the revised Option B.--Carwil (talk) 02:46, 30 July 2017 (UTC)


 * Version B: This discussion (and article) stopped being ridiculous from reading B. TVGarfield (talk) 02:04, 26 July 2017 (UTC)
 * Version B as more rigorous, albeit could be shortened a bit. Let's work on that after the RfC closes. — JFG talk 22:12, 26 July 2017 (UTC)
 * Version B (invited by the bot)   "A" sounds like an overreach  / overly categorical statement about a complex area which involves human definitions and interpretations about a human-generated system.  North8000  (talk) 15:54, 31 July 2017 (UTC)

Break 2

 * Neither Both introductions give undue weight to the idea that there is even any small miniscule amount of controversy around this.  Klaun (talk) 14:13, 28 July 2017 (UTC)
 * Neither is an introduction. Version A appears in the current article, in the "Discussion" section following the "algebraic proofs".  Version B is proposed to replace that.  I apologize if this was unclear.   Sławomir Biały  (talk) 14:30, 28 July 2017 (UTC)
 * The whole article is pretty awful. It is dominated by fringe opinions that don't represent actual Maths.  Klaun (talk) 15:42, 28 July 2017 (UTC)
 * I agree that some fringe views are present, but I disagree that they dominate the article. Many of these fringe views are, however, "actual Maths".  For example, the p-adic numbers are an actual thing that serious mathematicians study.  So are the hyperreals (although those are arguably more fringe than the p-adics).  It has long been the consensus here that such views should be present with appropriate weight in a section towards the end of the article.  While I do think the mention of the hyperreals is perhaps too specific for the lead, otherwise the focus seems reasonable (especially after User:D.Lazard's recent edits).  Thoughts?   Sławomir Biały  (talk) 15:49, 28 July 2017 (UTC)
 * I am also puzzled and cannot understand which fringe opinions Klaun thinks "dominate" the article. P-adic numbers and hyperreals are both well-established topic areas in mathematics, and so definitely not fringe theories. Gandalf61 (talk) 16:04, 28 July 2017 (UTC)
 * Perhaps an ill-considered and hasty comment. I don't think the digressions into the more esoteric areas of number theory, measure theory, etc. serve the purpose of what should be the encyclopedic thrust of the article.  What is the typical reader looking for when they search for this?  I'd posit not digressions into alternate or extended number systems.   Maybe splitting all the distraction into its own article would serve the subject better. Klaun (talk) 16:33, 28 July 2017 (UTC)
 * I think the alternative number systems content is germane, and I don't think there's any reason to think that readers would be put off by it. They will likely wish to understand what happens if you change the assumptions, and the context in which such questions have been studied.  The contents seem fairly accessible, but detailed enough that they are mathematically clear and precise.  I also think the weight and prominence of placement are about right.   Sławomir Biały  (talk) 21:01, 28 July 2017 (UTC)
 * I think the vast majority of people seeing a repeating decimal assumes it represents a rational number and consider it in terms of absolute value metric when comparing it to other numbers. The weight given to other topics in this article like alternate number systems just seems pedantic and kind of WP:COATRACK.  If people are interested in those other math topics, why not let them go and find articles about those topics?  I see value in content on a subject being concise. Klaun (talk) 21:33, 28 July 2017 (UTC)
 * I agree that the reader should assume that it represents a rational number, and that is not in question. But I think in order to understand what something is (in this case a notation like "0.999..." for denoting a rational number), it's useful to illustrate also what it's not, and these alternative number systems are supposed to show that.  I can definitely see, however, that the article fails to indicate how strange these alternative number systems are, thus lulling the reader into a sense that there is really some debate about the consensus interpretation.  So while I am skeptical that any meaningful consensus will emerge from this exchange, on the matter of the alternative number systems, I agree that this aspect of the article should be discussed at greater length at a later time.   Sławomir Biały  (talk) 11:19, 29 July 2017 (UTC)
 * That seems like a perfectly rational explanation for referring to alternate number systems and other topics peripheral to the main topic of this article and I accept that. However, I still think they are discussed too much.  If for no other reason, I think it's better to confine topics to their own articles.  If we have four paragraphs on p-adics and 5 more on infinitesimals in every article where it is appropriate to mention them, that will be a lot of content to manage.  Over time that content is bound to diverge with the end result of that divergence being that you have contradictory information about the same topic in two different articles.  This is something I find terribly annoying about Wikipedia (internal inconsistencies) and think it reduces its value to the users.  I feel like some of the explanations here are enough info to be dangerous.  For example, the 10-adic explanation shows a convergence for the "...999" infinite series, but doesn't really explain why or note that this would not be true generally for p-adics (where we are free to choose any value of p and not just a prime), nor why it converges for 10 but not all p's, makes liberal use of the negative sign without stating this denotes the additive inverse rather than a value less than zero (which doesn't exist for 10-adics), etc. Klaun (talk) 16:32, 29 July 2017 (UTC)


 * Version B is much more illuminating IMHO. Double sharp (talk) 15:20, 6 August 2017 (UTC)
 * Version B After spending some time dwelling on the options and reviewing the considerations for each version above, Version B seems like the best alternative. I understand the arguments above that suggest it may be too verbose; however, I tend to agree with the view above that the extra text (while perhaps making it a bit more clunky) makes the algebraic proofs section far more digestible to a motivated reader. I disagree with the suggestion that a reader could possibly leave mislead... The text might require careful reading, but the subject warrants it. Lizzius (talk) 19:59, 7 August 2017 (UTC)

Updates to version A/B
Since some editors have requested on form that I present a clear description of the updates to "Version B" in response to several criticisms and suggestions, I have produced a diff of where those changes may be clearly observed. The changes were mostly minor points of clarification in response to comments due to Guy Macon,, and. Other fruitful discussions, involving, myself, and (and others), lead to a rather dramatic article improvement, which is much more likely to influence the outcome of this particular RfC than such small changes to the "Version B" in the original request for comment. Anyone is free to annotate or document this change along the lines Guy Macon is suggesting if they wish.

I do note that a number of the editors siding with Macon, in their stated preference of "Version A", did so after the statement of the RfC was changed to something that  Macon found more palatable. He has subsequently withdrawn his specific objection to that sentence. Sławomir Biały (talk) 14:33, 27 July 2017 (UTC)

RfC going astray
The above RfC "almost" ended in broad agreement (e.g., the editor bringing it to WP:FTN appeared to see his misgivings scattered), when one editor deemed it necessary to proclaim a prejudical outcome of this RfC. Not only that it would result in "no consensus, clearly", but also to the effect that the status quo ante would have to be maintained, therefore.

Obviously, this required the editor, carrying the main burden of the efforts to improve the article against the intentions of a plain minority in numbers, to reply in an objecting manner, enabling an ongoing struggle in this unlucky RfC. I do not want to heap more trouble on this, I (hopefully) unambiguously stated my preference, but I ask politely to bring this RfC to a due formal end, treating all opinions, in spite of possible imperfections, in a meaningful manner.

Please, keep this mathematical(!) topic out of ignorant carrying ahead of ideologies. Purgy (talk) 08:23, 28 July 2017 (UTC)


 * I too am troubled by the systematic efforts of one editor to undermine the legitimacy of the RfC, and question whether this is in the best interests of the encyclopedia. Anyone may request closure at Administrators' noticeboard/Requests for closure, although I believe most RfCs run a course for a maximum of 30 days.  However, on the one hand, it seems to me that the discussion has not yet fully run its course, as various requests for clarification have apparently been neglected, and it seems premature to judge whether those comments have been abandoned.  On the other hand, certain other aspects of the discussion are not really worth pursuing further in my mind, and are merely likely to lead to continued fomentation of ill-will among certain parties should the RfC be allowed to continue until its statutory conclusion.  Thus it comes down to a question of, to what extent WP:SNOW may be applied to the outcome of the RfC, in the interests of sparing the community such continued hardships?  I feel that any suitably experienced uninvolved administrator should be able to make that determination, and I would support a neutral entreaty to the WP:ANRFC requesting a senior administrator either to draw the matter to a formal close, or to commit to do that at some time in the future.   Sławomir Biały  (talk) 14:14, 28 July 2017 (UTC)


 * I have had quite enough of this sort behavior. Since this RfC was posted, you, Sławomir Biały, have posted FIFTY THREE comments related to the question asked in the RfC. It is difficult to find a single comment disagreeing with your desired conclusion that you failed to challenge, sometimes repeatedly. And now you have decided to make an accusation ("the systematic efforts of one editor to undermine the legitimacy of the RfC, and question whether this is in the best interests of the encyclopedia") of bad faith, without naming to individual you are accusing? You are clearly WP:BLUDGEONING the process. No, it is not true that you and you alone are working in the "best interests of the encyclopedia" and it is not true that those who fail to agree with you are engaging in "systematic efforts of one editor to undermine the legitimacy of the RfC". You really need to apologize, and then you need to back of and let someone else have a say for a change. --Guy Macon (talk) 14:52, 28 July 2017 (UTC)


 * I did not mean to implicate you in the above, as you made constructive and valuable suggestions, for which I am appreciative. And, for the record, I do apologize to you.  It was never my intention to cause offence or upset.  It was an unacceptable presumption on my part to infer a withdrawal of your objection to "Version B" from your statement that "The rest seems fine" above, and I should have waited for a clearer assent on your part.  I should have issued a personal apology earlier on your user talk page as well.  However, I began to see other editors trying to ride your coattails shortly following what was (to my mind) the resolution of our dispute.  While of course they bear responsibility for that themselves, I cannot escape the feeling that they are also worthy of your admonishment.
 * I further apologize if my style sometimes chafes the norms of such processes (see WikiDragon for a much-needed light-hearted interpretation of such behavior). However, I have responded to many comments, which involved productive discussion related both the the material under discussion, and indeed other parts of the article as a result of this RfC.  I have attempted to take the opinions of all editors respectfully into account, and even attempted to improve the proposal based on those opinions (although perhaps in a manner that you object to).  I am not aware of any statutory limits on the number of comments that may be posted, including genuine requests for clarification.  Since the discussions have actually lead to measurable and substantial article improvements, it seems like that should be taken as evidence that the massive volumes of discussion have actually been useful and productive, rather than unproductive argument-for-its-own-sake that would suggest WP:BLUDGEONing behavior.
 * However, I agree with you in part that I should step back at least from certain less productive venues of discussion that have become unnecessarily heated, and will refrain from such interactions in the future.  Sławomir Biały  (talk) 15:18, 28 July 2017 (UTC)
 * I never imagined that it was me that you were referring to. Also you can have good-faith, productive conversations, but it is still bludgeoning if you start one every time someone cast a !vote on an RfC. Making disruptive comments is an entirely different concept, which I do not see anyone here doing. It's not what you write that is the problem -- each individual comment and the replies to it have been productive. The only real problem is that there are just too many of them. Let's just drop this and get back to discussing improvements to the article, not user behavior, OK? --Guy Macon (talk) 20:12, 28 July 2017 (UTC)
 * Sounds good. Thanks,  Sławomir Biały  (talk) 20:23, 28 July 2017 (UTC)


 * Comment. I strongly contest the edit summary of User:Calbaer: "rv. to last Purgy 794192130 version (editor attempting to impose desired outcome via edit war that is contrary to RfC he admitted to have lost)", as well as the recent edit war that he is instigating:, .  The tendentious discussions below, to justify an entirely unsourced section (with a perverse request for citations) as preferable to a fully cited section that is mostly supported by direct quotations and close paraphrases.  I think it is time to seek administrative intervention.   S ławomir  Biały  14:59, 8 August 2017 (UTC)

Reverted sizable unsourced edits, seem like WP:OR
Reverted lots of edits that did not cite any references, seemed WP:OR. Perhaps some consensus in talk can be built before making big content changes in the article. Although some of the content added reads like a debate of the article subject rather than encyclopedia content. Klaun (talk) 14:57, 7 August 2017 (UTC)


 * There is a discussion in the previous thread. Please comment there rather than starting a new section.    Sławomir Biały  (talk) 15:12, 7 August 2017 (UTC)


 * That seems to be a discussion about the subject of the article, not the content. If the article is missing explicit links between content (or proposed content) and the sources, then it needs more explicit links between that content and RSes.  Klaun (talk) 15:39, 7 August 2017 (UTC)


 * No, the above discussion specifically concerns the section on algebraic arguments, and in particular concerns the extent to which they "show" anything. What my edit was intended to do was to clarify what these arguments actually do show, namely that 0.999...=1 holds provided certain assumptions are valid.  Furthermore, that edit is at least as explicitly linked to the RSs that are cited at the bottom of the section as the existing content.  As I already said anove, those sources (Buers, Peressini, and Richman) discuss the fallacies of these arguments.  But since folks seem not to want to represent those sources faithfull, I have removed the section until a version is drafted that complies with our policies.   Sławomir Biały  (talk) 15:52, 7 August 2017 (UTC)


 * Strange revert. None of the stuff which was reverted to had any citations either which I would have thought would be the prime requirement of a revert based on the rounds given. Dmcq (talk) 16:24, 7 August 2017 (UTC)


 * Well, the supposed proofs are gone now. I think it's probably best simply to start over from sources, being careful to preserve the context of these arguments.  They are not usually presented as proofs, but as illustrations of the limitations of arguments like these, so I've been very reluctant to include them at all, without significant caveats.  But yeah, reverting to other at least equally bad versions doesn't seem like a way to make progress.   Sławomir Biały  (talk) 16:31, 7 August 2017 (UTC)


 * I've restored to the last version by Purgy (though, as noted above, I'm hoping to alter the wording). We've been discussing and refining this for weeks now, so just wiping it all out is an approach which explicitly defies the consensus-building approach we've been working hard on in all this time.  That's especially true when the section has been here since the article made FA about a decade ago.  If we want to rebuild it from scratch, the appropriate way of doing so would be on the talk page or in some other sandbox.  Otherwise, we should alter it via edits that don't remove everything before introducing a replacement.  If we want to add references, there is the the aforementioned Encyclopedia Britannica entry, which is exactly the same as the reasoning for "Fractions and long division." This is not "original research," by any stretch, just information without the level of footnoting we'd ideally want (and information that is controversial since it has no rigorous proofs, and thus some editors feel like it leads readers in the wrong direction). Calbaer (talk) 19:12, 7 August 2017 (UTC)
 * You're citing an 18th century book, that isn't even a mathematics book. That's hardly a reliable source for mathematics.   Sławomir Biały  (talk) 19:28, 7 August 2017 (UTC)
 * This is an article mostly about mathematics education, not strictly about mathematics; otherwise, we could just end the article after the first section. The "algebraic" arguments appear over and over again as way to illustrate the property.  That's the whole reason some sources (according to what you say, which I believe but haven't checked) give them as flawed examples.
 * Advocating for removing this section - rather than modifying and/or moving it - won't get us anywhere near agreement on this. Insisting that it be constructed solely from universally agreed-upon sources, though ideal, is likely to be difficult as well.  I've presented several ideas for how this section might be improved.  Yet another is to leave out not only "proofs" but also "algebraic."  Such a section ("common informal arguments"?) could be presented as common methods used to illustrate 0.999...=1 while emphasizing that their use was a matter of trying to illustrate to students who couldn't be assumed to have exposure to the subjects necessary for rigorous proofs: real analysis, limits, the Archimedean property, etc.  And, as such, the proofs were necessarily incomplete (indeed, the gaps requiring as much effort to fill as to prove 0.999...=1 itself).
 * One thing that I am curious about, but which you need not answer, is why you were supportive of this article staying FA 7 years ago, but now find several parts of it fatally flawed. People are free to change their minds, but I must admit I am curious why it appears as though you've changed yours so dramatically.  Again, this is not an argument against your opinions, just a curiosity on my part. Calbaer (talk) 04:02, 8 August 2017 (UTC)

This is another deflection. A 200+ year old source is not reliable for mathematics education either. Sławomir Biały (talk) 10:42, 8 August 2017 (UTC)
 * I've rewritten the section based on reliable sources on mathematics education, per your suggestion.  S ławomir  Biały  13:00, 8 August 2017 (UTC)
 * I like the spirit of your change, but I feel it needs to be hashed out in the talk section, since - as is - it seems deceptive and flawed, with several errors. (I've thus reverted it; the edit can be seen here.)  References are improperly done, too; they point to non-existent anchors within the page.
 * But it's the way it's presented that's even more worrisome. Byers does not say what you say he says.  "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999..." refers to the proposition 0.999...=1, not to the algebraic argument.  (From a quick reading, he states no technical flaw in this argument.  In fact, he claims that it actually convinces students.  The only flaw he finds is that the students now believe the equality but don't understand it.  That's very different than the characterization given in the recent edit, that the argument is unconvincing.)
 * Furthermore, the notion that, "most undergraduate mathematics majors, according to Byers," believe that 0.999... is not 1, is slightly different than what Byers says in his book. Instead, it's that "in my experience" this is the case.  That's the difference between anecdotal evidence (which is biased, e.g., by the voices of the loudest students in the class or by the type of students at his school) and scientific proposal.  Not to mention that it's unnecessary to state in this section.
 * There's also the problem of why you're presenting Byers' opinion on this and that, a sudden jump from math to what is basically social science (or rather social anecdotes). That makes for jarring reading.
 * It seems there's still a contradictory motive to present this information, but also to say, "This is wrong!" so fervently that such modifications leave the impression that the math is wrong or the result is wrong - neither of which is true. The problem is rather that there are portions left undefined or unproved, so that algebra alone doesn't form a formal proof.  We should be able to preserve these most common of arguments, and say why they are both true and insufficient. Calbaer (talk) 13:58, 8 August 2017 (UTC)
 * Nonsense. Byers asks, just after presenting the proof "What is the meaning of these equations?" meaning the identity $$0.999...=1$$ and the equations that preceded it (i.e., the proof he just offered).  It's clear that we're citing Byers viewpoint, on the issue of why students are unconvinced.  But if it's unclear, you can always ask for clarification.  In particular, we now have a version of material that is entirely uncited with a request for citations, and a version which is now completely cited.  I will now be removing the uncited content as having failed verification.  Please see WP:V.  Further reverts must now be justified under Wikipedia policies.   S ławomir  Biały  14:04, 8 August 2017 (UTC)
 * As I've noted, we've been discussing how to improve this section for weeks. You've decided to ignore all that and eliminate the section because you didn't get your way via the means you initially proposed to do so, effectively taking your ball and going home.  It's one thing to add citations.  But a section without citations is superior to one that gives citations then misinterprets them.  Most people aren't going to follow the citations - especially when the links to them are broken! - to see what's really said here. Calbaer (talk) 14:45, 8 August 2017 (UTC)
 * The cited material does not misrepresent the sources. That is just a bald lie, as was your recent edit summary.  This is clear disruption.  Please see WP:CHALLENGE:   Be advised that continuing to insert this unreferenced OR against policy will likely result in sanctions.   S ławomir  Biały  15:11, 8 August 2017 (UTC)
 * You call me a liar and attempt to override the results of your own RfC, and then threaten me with sanctions for undoing the latter? Come on.... Calbaer (talk) 15:41, 8 August 2017 (UTC)
 * The mandate of the verifiability policy is incontrovertible. The rest is noise.   Sławomir Biały  (talk) 15:44, 8 August 2017 (UTC)
 * That means you spend hundreds of edits filling this page with "noise." Why?  (By the way, there are a bunch of reliable sources.  The fact that the are no footnotes for them within the section is insufficient reason to suddenly wipe the whole section after you fail to get your way on it.  Wikipedia is about converging to consensus, not citing one policy after another in order to justify why you should have the final word on all matters.) Calbaer (talk) 15:49, 8 August 2017 (UTC)
 * I have no interest in further discussion with you if you are going to blatantly lie. WP:V is a bright line rule.  Full stop.  If you want to have a reasonable discussion, begin it by showing some good faith and not lying.   S ławomir  Biały  16:31, 8 August 2017 (UTC)
 * It's too important to the showing the plausibility of the topic to most readers to leave out. It has been in forever, has concensus, and should stay in.  If anything, about 2/3rds of the words can come out and retain the meaning.  The math examples don't need to be removed since most can easily fall under routine calculations.  If you were going to explain this topic to someone with hardly any math knowledge, starting with one of those would be the place to start.  Please see WP:You don't need to cite that the sky is blue:
 * WP:CALC:   TVGarfield (talk) 02:50, 9 August 2017 (UTC)
 * WP:CALC and "the sky is blue" do not apply here. I have reinstated the sourced revision of the section.  Sources were requested by several editors.  I hope this is a suitable compromise, subject to improvement by normal editing (while still maintaining close fidelity to the sources).  Otherwise, an alternative is to remove the entirely unsourced section as having failed WP:V.   Sławomir Biały  (talk) 10:28, 9 August 2017 (UTC)
 * Note though, that it is still unsourced. Harv error: link from #CITEREFByers2007 doesn't point to any citation. Harv error: link from #CITEREFRichman1999 doesn't point to any citation. Harv error: link from #CITEREFPeressiniPeressini2007 doesn't point to any citation. Harv error: link from #CITEREFBaldwinNorton2012 doesn't point to any citation. Harv error: link from #CITEREFKatzKatz2010a doesn't point to any citation. You need top add three references to the References section.   Hawkeye7   (talk)  12:24, 9 August 2017 (UTC)
 * The references are there, but the citation anchors are broken. I have fixed the citation anchors.   Sławomir Biały  (talk) 13:07, 9 August 2017 (UTC)

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dots vs. overline
I mostly agree with the revert about this, but I actually think overlines would be better for those three long-period decimals: 1/7, etc., by making it immediately clear visually which digits repeat (while for 0.999..., it's not really necessary). Would anyone object to changing just those? --Deacon Vorbis (talk) 15:58, 6 September 2017 (UTC)
 * Indeed, good idea. I went ahead, but not the strike-through part that the anon had done. - DVdm (talk) 16:06, 6 September 2017 (UTC)

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Origins of recurring decimals
In follow-up to the comments above about considering the origin of the ellipsis notation and the concept of decimals with infinitely many digits. Decimal notation is usually credited to Bartholomaeus Pitiscus (1561–1613). John Wallis (1616–1703), wrote about recurring decimals and sexagesimal fractions in his Mathesis Universalis (1657) and Algebra (1685). He worked with them, and introduced the term continued fraction in his Opera Mathematica (1695), and did a lot of work with infinite series. Johann Heinrich Lambert (1728 – 1777) worked out when a number had a finite decimal representation, and realised that irrational numbers have infinite decimal representations. The ellipsis was already in use back then. Hawkeye7  (talk)  01:22, 25 August 2017 (UTC)


 * I think that the above valuable facts do belong in a good article about number representation via decimals, and that this here article, necessarily dealing with the inherent delicate finesses of infinity, should engage itself just with the difficulties of "repeating" decimal representation of real numbers, and the difficulties this causes when confronted with an informal, only intuitive understanding of real numbers. Hints to constructions of other number systems (hyperreals, etc.), in which the equality under consideration might not hold, should make clear that this article is based on the real number system, in which the number represented by "0.999..." unavoidably equals the value of "1.000..." and all of its truncations, and that this system is not only the one overwhelmingly in use, but also the overwhelmingly convenient one to use.
 * Considering the wide proliferation of the "algebraic" pseudo-proofs, I think it is necessary to not only mention them in an article suiting my needs, but also to carefully point to the loopholes left in their sequence, which make up for most of the difficulties among the non-initiated, not accessible via informal historic use of "infinitely repeating".
 * In fact, I am advocating for trimming this article, and dramatically improving the coverage of decimal representation of reals methods used to denote reals by strings containing decimal number tokens revised for non-technical use of representation 09:38, 26 August 2017 (UTC) , but outside of this article, which should focus on the title induced topic. Purgy (talk) 07:43, 25 August 2017 (UTC)
 * In fact, decimal representation of reals is an oxymoron, as a representation is necessarily finite, and there are too many real numbers for having a finite representation for all of them. Thus the correct term is decimal expansion of real numbers. This misnomer is probably the origin of the misconception appearing in several articles, consisting of considering finite decimal representation as a (minor) special case of infinite representation. In fact, the important concept is the concept of decimal numeral, which is finite and used everywhere, while the concept of infinite decimal expansion is used only in mathematics. Even in mathematics, infinite decimal expansion is used only in some constructions of real numbers, in the proof of non-enumerability of real numbers, and, apparently in mathematical education in some countries. I have partially rewritten Decimal for making clear the distinction between finite decimals and infinite decimal expansions. Sections on infinite representations decimal computation and history deserve also to be rewritten, as well as the articleDecimal representation. D.Lazard (talk) 09:33, 25 August 2017 (UTC)
 * I apologize for any inconvenience or embarrassment, caused by me using the word representation in a non-technical meaning. I did not want to present any oxymorons. Purgy (talk) 09:38, 26 August 2017 (UTC)
 * I think a persistent problem has been that there is a tendency to regard "recurring decimals" as the same thing as "rational numbers". A recurring decimal is an infinite expression.  It is not necessarily interpreted as a number.  When Wallis writes $$1/3=0.333...$$ (if he ever writes that), then I'm fairly certain he means that the result of dividing one by three gives three tenths, with a remainder leaving three hundredths, and so on.  Thus "0.333..." is a process rather than an object, and the equals sign is not reporting the identity of two objects, but rather the outcome of a particular numerical computation.  This is very different from what is meant by the modern notion of repeating decimal, so should not be discussed here without sources clarifying the ontology.   Sławomir Biały  (talk) 11:39, 25 August 2017 (UTC)


 * A recurring decimal is indeed a representation of a rational number. It is not an expression, but a representation of a numerical value, i.e., a number. There is no "process" associated with such a representation, because it does not represent any operational steps. You can, however, argue that the concept of a mathematical limit is implied by the notation (but a limit is also not a process). You're going to have to provide a reliable source for your assertion that Wallis thought otherwise. — Loadmaster (talk) 20:35, 18 October 2017 (UTC)


 * The notation $$0.999...$$ absolutely is an expression, not a number: it is a zero followed by a period, followed by three nines and an ellipsis. To say that it "represents a rational number" fails to specify the means by which such a notation may represent a number.  We might say that x=0.999... is the unique rational number with the property that 10&times;x=9+x.  (Presumably this is the sense in which you mean that the notation "represents" the number?  Or do you have some other idea in mind?)  Or it might be a limit. We have sources that often recurring decimals are interpreted operationally (e.g., that 1/3=0.333... is reporting the result of a computation that can be carried on indefinitely).  Would it be surprising if this was an interpretation to be found in Wallis? In any case, we equally well would need a source saying Wallis meant something else by this notation. Merely noting that Wallis used the notation solves nothing.  (And Wallis even believed in infinitesimals, further complicating matters.)  And regarding a "reliable source that Wallis thought otherwise", our current understanding if decimals is based on the work of Cauchy and Dedekind.  It is simply absurd to suggest that Wallis' thoughts on the matter of decimals is anything like our modern understanding.  It's that Whiggishness that I am responding to here.   Sławomir Biały  (talk) 21:48, 18 October 2017 (UTC)


 * You are muddying the waters unnecessarily here. How does the notation 9 represent a number? Even if Wallis did have a different interpretation of what the ellipsis in 0.999... means, how does that affect our current use of it; specifically, how does it change how we use it in this article? — Loadmaster (talk) 21:27, 19 October 2017 (UTC)


 * Well, someone suggested Wallis' use of the ellipsis is somehow relevant to "our current use of it". I don't see the question "How does the notation $$0.999...$$ represent a number?" is muddying the waters in any way.  When you say "Repeating decimals represent rational numbers", you haven't said what "represent" means.  In exactly what way does the notation $$0.999...$$ "represent" a rational number?  Is it the solution of some equation?  Is it a limit, relying on completeness properties?  Be specific!  (And one need only looks in Wallis to see that he thinks of infinite series as a process that can be continued indefinitely, rather than a number.  Significantly, for this reason Thomas Hobbes objected to Wallis' use of induction to establish "identities" involving infinite expressions.)   Sławomir Biały  (talk) 22:07, 19 October 2017 (UTC)
 * May I point to the remarks on "representation" by D.Lazard above? Imho, the inappropriate belief that ellipses weren't deepest mud, but already the clarification they're in dear need of, is the reason for much of the ongoing debate on this topic. Purgy (talk) 06:07, 20 October 2017 (UTC)

Archive index broken?
I noticed by chance that this talk page has meanwhile 19 archives, but the index and the table of contents for archives mention only 14. Furthermore, the index page was last updated in 2013. Possibly, the "Arguments"-subpage caused this? Then there is some residue from an edit in semi-protected state. Could someone knowledgeable have a look, please? Purgy (talk) 15:49, 20 December 2017 (UTC)
 * There are two different links to archives, one in the headers, and one in a box. The link in the headers seems correct, although the linked index seems incomplete, at least for archive 19. I'll reorder the headers for having the archive link near the TOC, and remove the bugged boxes. Feel free to revert if I makes some errors, D.Lazard (talk) 16:37, 20 December 2017 (UTC)

9/9 listed at Redirects for discussion
An editor has asked for a discussion to address the redirect 9/9. Please participate in the redirect discussion if you have not already done so. -- Tavix ( talk ) 14:59, 22 March 2018 (UTC)

...smallest number no less than all decimal numbers...
What does this phrase mean? Any numeral written in base 10 is a decimal numeral. 0.999... is less than all decimal numbers (or numbers in any other base) greater than 1. Should the phrase be no less than all decimal fractions?. Koro Neil (talk) 06:06, 9 August 2018 (UTC)
 * No less than means the same as not less than. The article is about 0.999... being not less than 1. It is also not greater than 1 so it is 1 with the usual axioms for the real numbers. Dmcq (talk) 10:09, 9 August 2018 (UTC)
 * Nevertheless, the formulation was confusing: on some browsers (including mine), there is a line break between "all decimal numbers" and the list to which this phrase must be restricted. I have edited this sentence for fixing this confusing formulation. D.Lazard (talk) 17:10, 9 August 2018 (UTC)

Short descriptions
Hi all, My recent addition of a short description in the article was reverted with a request for explanation and reason why it is neccessary. As this is a recent requirement, the request is entirely reasonable. However, the detailed explanation would take a wall of text which would not be appropriate here, as this is a requirement for all Wikipedia articles. A reasonably detailed explanation can be found at Short description, and more information at WikiProject Short descriptions. If anyone has further questions, please ask them at Wikipedia talk:Short description so that the answers can be available for everyone at a central place. Cheers, &middot; &middot; &middot; Peter (Southwood) (talk): 06:40, 21 February 2018 (UTC)


 * Not having been aware of its necessity, I reverted this (twice already, won't do again). Certainly, I am not in charge to sculpture how to deal with this necessity, but for the time being, and only if this is a really, really unavoidable necessity, I suggest "mathematical treatment of the ellipsis in 0.999...". Despite fundamental opposition to many spirits (sic! ghosts?) of WMF, I am not after any kind of fight. Good luck for the scuba project. Purgy (talk) 07:56, 21 February 2018 (UTC)
 * If I'm understanding this correctly, the current description in WD is "real number that can be shown to be the number one". I think that's probably better than Peter's version, "The number represented by infinitely repeating 9 after the decimal point preceded by zero".  Peter's version is certainly literally accurate, but the current WD version is more to the point. --Trovatore (talk) 08:13, 21 February 2018 (UTC)
 * Having looked at some of the documentation pages, and in particular the image at right, I think conciseness is more important than completeness of description. So I'm going to suggest "alternative decimal expansion of one".  --Trovatore (talk) 08:47, 21 February 2018 (UTC)Top_read_sep_17_2017.png
 * I totally agree with this one. Slightly provocative to some of our friends maybe, but to the point and correct . - DVdm (talk) 10:31, 21 February 2018 (UTC)
 * I have no special attachment to any version of the short description and am quite happy to go with the editors who know the subject better, as long as there is one on Wikipedia that is appropriate to its purpose. I leave it in your capable hands. Cheers, &middot; &middot; &middot; Peter (Southwood) (talk): 12:55, 21 February 2018 (UTC)
 * There are good arguments for having a short description, and many valid objections to the way it has been implemented without proper consultation and against consensus by the Reading team at WMF but this is not the place to discuss those. Cheers, &middot; &middot; &middot; Peter (Southwood) (talk): 13:02, 21 February 2018 (UTC)
 * If the talk page consensus is that there is no need for a short description or that it is undesirable to have one, please put a non-breaking space in the template in the place where a description would go. &middot; &middot; &middot; Peter (Southwood) (talk): 13:05, 21 February 2018 (UTC)
 * While not everyone may be convinced of the benefits of the short descriptions, I'm having trouble seeing much downside. There'll be a little watchlist churn, and of course the content of the description is another thing to argue about, but other than that, if you don't see the descriptions then they shouldn't bother you, and if you do see them then it seems like they're good to have.  I'm not thrilled with the Foundation on some other issues, but that's not a general argument for opposing all their initiatives. --Trovatore (talk) 21:12, 21 February 2018 (UTC)

I agree with the short description suggested by above and adopted since. I just updated it to "Alternative decimal expansion of the number 1 ", which is a tad bit clearer grammatically then "expansion of one". — JFG talk 14:52, 9 August 2018 (UTC)
 * Definitely better. - DVdm (talk) 15:25, 9 August 2018 (UTC)
 * It is clearer but it does break the 40-character limit. I know that's a "soft limit", so maybe it's justified here, but it concerns me that a lot of "short descriptions" I've been seeing aren't really all that short.  I think it might be good to try to hold the line if we can. --Trovatore (talk) 16:59, 9 August 2018 (UTC)
 * The limit does not seem to be an issue, and at 45 characters we are far from abusing the guideline. — JFG talk 17:54, 9 August 2018 (UTC)
 * I'm concerned that a lot of people adding short descriptions don't really understand what they're about. The goal should be to give enough information that a user on mobile knows whether he/she wants to click on a link, and no more than that.  For this article in isolation, I agree, the one you added seems OK, but the more we can keep to the limit, the better example we set for others. --Trovatore (talk) 19:19, 9 August 2018 (UTC)

Geometric proof
I have discovered that it is possible to construct an obtuse scalene triangle which has angles that measure 101.010101... and 50.505050... and 28.484848... Added together, these measures are equal to 179.999999... degrees.

Does this triangle constitute a "geometric proof" that 179.999999... degrees is equal to 180 degrees (the given number of degrees in a triangle) and that 0.999... is equal to 1? Scott Gregory Beach (talk) 21:05, 29 August 2018 (UTC)


 * Not likely. We have this policy about wp:original research and wp:reliable sources... - DVdm (talk) 21:08, 29 August 2018 (UTC)
 * This wouldn't be appropriate to add to the article, but please do post it (maybe on my talk page if it's not appropriate here) as it sounds fascinating. a11ce (talk)

Fundamental Misconceptions embodied in article
Moved to Talk:0.999.../Arguments. D.Lazard (talk) 08:14, 10 September 2018 (UTC)

Redirect
To save all these pointless conversations shouldn't the article just redirect to 1? ;) JZCL 02:56, 27 September 2018 (UTC)


 * Please, note the content of this article that is beyond the information given at 1, which is imho not pointless and also should not be accumulated there. I oppose to the suggestion of a redirect. For the discussion about deleting the "arguments"-page, please see there. Purgy (talk) 06:37, 27 September 2018 (UTC)
 * Oppose -- the theorem that the infinite series denoted by this notation converges to 1 is basically the subject of the article, distinct from the number 1 itself.--Jasper Deng (talk) 06:40, 27 September 2018 (UTC)

Oh goodness, it wasn't a serious suggestion! I was only trying to give some light relief for the editors here who so tirelessly have to explain the same arguments again and again... I am not meaning to waste anybody's time. JZCL 10:45, 27 September 2018 (UTC)
 * I had seen it, but your smiley ;) must have been looked over. Next time try . - DVdm (talk) 17:20, 27 September 2018 (UTC)

Feynman Point
Should a comment on how 0.999... is not like the Feynman Point be placed in article? John W. Nicholson (talk) 10:38:40‎, 21 March 2019 (UTC)
 * A much better example for this article is
 * $$e^{\pi\sqrt{163}}\approx 262537412640768743.999999999999250\dots.$$
 * That is, 12 nines after the decimal point, and one needs more than 30 exact digits for knowing that it is not an integer. For proving that it is not an integer without numerical computation, one needs Gelfond–Schneider theorem, which states that this is a transcendental number (for applying this theorem, one can remark that $$e^{\pi\sqrt{163}} = \left(e^{i\pi}\right)^{-i\sqrt{163}}=(-1)^{-i\sqrt{163}}$$). D.Lazard (talk) 19:03, 21 March 2019 (UTC)

OK, maybe both? Simply state you thought but add "Even pi alone has what is called a Feynman Point which shows the error of assumption without checking the values continuance." John W. Nicholson (talk) 23:13, 21 March 2019 (UTC) Maybe a better way is to do it this way:
 * $$e^{\pi\sqrt{163}} - 262537412640768743 = .999999999999\dots$$ does not equal $$1$$ because with the next three digits we have $$.999999999999250\dots.$$

John W. Nicholson (talk) 23:30, 21 March 2019 (UTC)


 * I oppose to including any example, how simple or promi-named it may be, illustrating the possibility of a finite number of nines in its decimal representation. These examples seem to bypass the fundamental difference between any finite string and the construction required for infinite representations. I think they lead astray in not focusing on the effects of somehow formalizing infinity.
 * How many consecutive nines are in $$1-\pi\cdot 10^{-100},$$ in general, and immediately after the decimal separator? Purgy (talk) 09:17, 22 March 2019 (UTC)
 * In a first thought, I agreed with Purgy. However, in a second thought, it appears that ellipse notation (...) can be ambiguous, and this may explain the pedagogical difficulties illustrated by this article, and the size of the "Arguments" subpage. In fact, ellipses are not only used for indicating a simple pattern that is implicitly continued infinitely, such as for 0.999...; it is also used for truncation of a finite or infinite sequence of digits, such as $$\pi=3.14159\dots$$ instead of $$\pi\approx 3.14159$$ (in the article on $\pi$, this use of an ellipse is not used in decimal base, but it is used for the expression of π in other bases). IMO, this possible (and even common) confusion between two meanings of ellipses deserves a section that should be placed just after the lead. This section should makes clear that, in the case of a possible confusion, ellipse must not be used for truncation, and this could be illustrated by writing: For example, $e^{\pi\sqrt{163}} - 262537412640768743 \approx .999999999999,$ but $e^{\pi\sqrt{163}} - 262537412640768743 \neq .999999999999\dots,$ since, with three more digits, one gets $e^{\pi\sqrt{163}} - 262537412640768743 \approx .999999999999250.$|undefined D.Lazard (talk) 10:45, 22 March 2019 (UTC)


 * I was reprimanded elsewhere already (approx. value of 𝜏?) for calling an ellipsis a method of notation, unworthy for any mathematician with some amour propre (or similar). Maybe this section in the article on ellipsis suffices, when linked to, but I also oppose to cover the variants of this notational flaw within this here article, since here the dots are better replaced by an overbar or by parenthesizing the last nine, both better excluding the inapplicable variants in the possible meanings of an ellipsis; and again, discussion of the variants distracts from the problem at the heart of this article. I do not deny that the ellipsis makes up a problem on its own, but I doubt that its ambiguity contributes to the reported problems with 0.999... = 1 . BTW, the inequality above may also be considered an equality when taking ellipsis as "some (possibly infinite) continuation". Purgy (talk) 13:39, 22 March 2019 (UTC)

0.999... <> 1
Moved to talk:0.999.../Arguments
 * I don't think we need that again—see Talk:0.999.../Arguments/Archive 11

I believe there is a missing " in the p-adic number section. I do, however, not know where it is meant to be placed. — Preceding unsigned comment added by 2001:7C0:31FF:3:9442:1FCB:771B:EF91 (talk) 10:55, 28 January 2020 (UTC)

0.999... in history.
Who was the first person to formulate 0.999... and ponder it's meaning? What is Isaac Newton? He invented calculus, but did not actually resolve the issue of infinitesimals or formulate the Real set. Searching google on this does not help, since it only goes to modern interpretations and irrelevant links. Understanding a concept's history can be a very powerful learning tool - knowing what was tried and DIDN'T work can bring a lot of confidence. This is the closest thing I was able to find, but it does not answer the question: https://www.youtube.com/watch?v=WYijIV5JrKg&ab_channel=Numberphile --Algr (talk) 22:28, 9 September 2020 (UTC)


 * Decimals had been around since ancient times, particularly among the Arabs. They were well known to mathematicians in Europe in the 16th century like Fibonacci, but European mathematicians preferred using fractions. The benefits of decimals for arithmetic was propounded by Simon Stevin in the late 16th century. He used an awkward notation though; the one we use today was invented by Bartholomaeus Pitiscus for his trigonometrical tables, and was then adopted by John Napier for his logarithmic tables. Newton used decimals for his calculations of planetary motions. The rigorous construction of the rational numbers was undertaken by Dedekind and Cauchy in the 19th century. Dedekind, Eduard Heine and Georg Cantor then extended this work to the Reals. The Reals are not actually needed in the article though, since 1 is a rational number after all. Hawkeye7   (discuss)  23:51, 9 September 2020 (UTC)
 * That doesn't really answer the question at all though. They had the symbols in ancient times, but the entire article is 19th century math.  What happened before then?  James Grime says "they just ignored them, and then limits made them go away, and then they came back." --Algr (talk) 12:34, 10 September 2020 (UTC)
 * The concept of infinite decimal requires the concept of actual infinity. It was not acceped by mathematicians before the work of Georg Cantor at the end of the 19th century. Before that the only thing that mathematicians said was that 1 is the limit of the sequence $0.9, 0.99, 0.999, ...$. It is a pity that the concept of infinite decimal that involve the difficult and counterintuitive concept of actual infinity is taught in US schools, when much more useful concepts such as the accuracy of approximations receive much less attention. D.Lazard (talk) 12:58, 10 September 2020 (UTC)
 * Okay. So how did they cope with the decimal equivalent of 1/3? --Algr (talk) 20:00, 10 September 2020 (UTC)

0.999... in history (again)
I support the notion that this article misses opportunities to put the problem into a historical context. (I have not just now re-read the whole article; I may have overlooked if it actually does.) Obviously, the existence of the article is a bit of a curiosity; we also have the article 1. What sets the subject of the present article apart from that one is not really about 0.999... (i.e., 1), but about certain finer points in the nature of decimal notation. Most of these these points are equally relevant for 2.74999..., 0.142857142857..., or even 3.14159265..., so why this article, when we have Repeating decimal, Decimal representation, Decimal, and others? But it is here, I (for one) like it. And, actually, Decimal representation link to the present article for these finer points. In the articles I mention, history is only (and briefly) covered in Decimal, and that only seems to be about finite decimal.

I am not competent to write what seems to be missing, but it is a valid point, that shouldn't be closed down simply because no specific proposals have been put forward.--Nø (talk) 08:41, 11 September 2020 (UTC)
 * PS. No, this is not a general discussion forum, and the author of the previous post with the same heading perhaps failed to clarify that (s)he was missing answers to these questions in the article. But I assume that was the intention.--Nø (talk) 08:44, 11 September 2020 (UTC)
 * The history of decimal notation does not belong here; it is properly covered in decimal. Hawkeye7   (discuss)  22:21, 11 September 2020 (UTC)
 * But it is not there either. The decision to assign a certain meaning to a certain set of symbols is always of historical significance.  The symbols have been around since antiquity, but the article addresses only 19th century mathematics, and ignores everything that happened before that.  --Algr (talk) 00:59, 12 September 2020 (UTC)
 * Decimals have been around since antiquity. The notation we use dates from 1595. This article does not ignore mathematics before the 19th century; that is covered in the Elementary Proofs section. The 19th century was the era in which mathematicians attempted to place mathematics on a more logical and rigorous basis. The article is carefully written so the further you read down, the more advanced the treatment becomes. Hawkeye7   (discuss)  02:38, 12 September 2020 (UTC)
 * Hawkeye, The article as it stands is like teaching relativity without mentioning Einstein. I wish you understood history well enough to appreciate the magnitude of what those proofs represent. In ancient times, the infinite was closely connected to the divine.  Treating infinity as something a mortal could accomplish with symbols on a page could get you executed for blasphemy.  Who was the first person to say that an infinite number of digits after a decimal point was something useful to imagine?  How was that received by his peers?  THAT is what history is. --Algr (talk) 05:28, 12 September 2020 (UTC)
 * WP:SOFIXIT, or stop complaining about it. –Deacon Vorbis (carbon &bull; videos) 12:31, 12 September 2020 (UTC)
 * Why that comment? Is it not legitimate to point out something that would benefit an article, without being able to fix it yourself? You can ignore the request if you like, but why denigrate it?--Nø (talk) 13:09, 12 September 2020 (UTC)
 * The thing that is special about decimal representations that are eventually all nines is that they are the only ones where two different standard decimal representations can denote the same number. That doesn't happen with your $$0.\overline{142857}$$; it's the only standard decimal representation of 1/7 the reason I say "standard" is to exclude things like decimal representations that allow negative digits or digits greater than 9 &mdash; these are legitimate things, but few in the audience of this article are likely to have come across them.
 * This seems to confuse people; most people, I hypothesize, have the idea that for each real number there's exactly one decimal representation (indeed, they may not distinguish the representation from the number in the first place). That confusion is the reason for this article to exist. --Trovatore (talk) 00:38, 12 September 2020 (UTC)
 * True; one of the things that are peculiar about 0.999... only applies to numbers ending in 999... in decimal form (or numbers ending in 111... in binary, etc.). However, this arises because infinite decimals exist and are assigned a meaning as a limit, so that is also what this article is about, 0.999... epitomizing the interpretation of infinite decimals. -- Anyway, I did not wish to start a discussion about that, but to point out that we do lack coverage of the history of infinite decimals - in this article, or in another (unless I just haven't found it - in which case we lack a wikilink in this article).--Nø (talk) 13:09, 12 September 2020 (UTC)
 * The history of finite decimals is well summarized in Simon Stevin. As I said above, infinite decimals did not exist before 20th century and the introduction of infinite sets by Georg Cantor (here, infinite sequences of decimal figures). If you have reliable sources on the history of infinite decimals during 20th, be bold and WP:SOFIXIT. If such sources do not exist there is nothing to discuss here. D.Lazard (talk) 13:39, 12 September 2020 (UTC)
 * This is off-topic of course but &mdash; I think it's unlikely that infinite decimal expansions were unknown before Cantor. The actual infinite is implicit in such expansions, but it's implicit in the notion of real number in the first place &mdash; people just hadn't entirely come to terms with that before the mid-19th century (this development somewhat predates Cantor; we're talking at least about Weierstrass and Dedekind).  But I bet you would find that people were talking about the decimal representation of 1/3 being $$0.\overline{3}$$ long before that, though they likely didn't linger over explaining exactly what they meant by it. --Trovatore (talk) 17:23, 12 September 2020 (UTC)
 * Nothing off-topic about that - on the contrary, it is exactly what I think is the topic. It would be great if someone, knowing this stuff and which sources to cite, could actually write it. I am not sure which of the articles about decimal numbers it would belong in, though (properly linked from other articles).--Nø (talk) 18:03, 12 September 2020 (UTC)
 * I can possibly buy that some content of that kind might be useful in this article, but I'm not the one to write it. I was mainly responding to what I thought was an unsound inference, from infinite decimal expansions implicitly assume completed infinities and Cantor was the first mathematician to treat completed infinities as first-class objects in a way that really seemed like doing mathematics, and that really led somewhere, both of which are true, to, which I find highly doubtful, though I don't have a source to directly refute it. --Trovatore (talk) 20:40, 12 September 2020 (UTC)
 * From the 1608 English translation of Stevin's book: It sometimes happens that the quotient cannot be expressed by whole numbers, as in the case of 4 divided by 3. Here, it appears that the quotient will be infinitely many threes with always one-third in addition.  Hawkeye7   (discuss)  22:34, 12 September 2020 (UTC)
 * Ah ha! Time for some reading.  Thank you!  --Algr (talk) 00:00, 13 September 2020 (UTC)

Remove formal proof from "Elementary Proof" section. This isn't where it belongs.
I made an edit. It was reverted. I don't think that the formal proof given belongs in the Elementary Proof section, I don't have a PhD in the subject, but it certainly doesn't seem elementary, in any sense of the word. I'm not saying it has no place on the page, simply that it doesnt' belong there. Cliff (talk) 14:11, 16 June 2021 (UTC)
 * This formal proof is simply the formalization of the content of the two preceding subsections. So, if it would not belong to this section the two preceding subsections should be removed also. If you know of a proof that is mathematically correct, and more elementary, please produce a reliable source for it. I doubt that such a proof exist, as there were many discussions, involving many editors, on this subject on this talk page, and nobody has proposed a more elementary correct proof. D.Lazard (talk) 15:24, 16 June 2021 (UTC)
 * It is an elementary proof in the sense that it uses only basic techniques. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated. The "Algebraic arguments" and "Infinite series and sequences" of "Analytic proofs" use elementary maths of the kind taught in primary school, but are not formal proofs. Dedekind Cut and Cauchy Sequence require undergraduate university maths. No part of the article requires postgraduate maths. Hawkeye7   (discuss)  20:49, 16 June 2021 (UTC)
 * Perhaps we're hitting a language barrier then. In the US, "elementary" is not commonly used this way, and strongly connotes simplicity and ease of understanding. I understand your meaning, and your use, and do not disagree with it. Maybe the section title needs to be changed? Cliff (talk) 14:04, 17 June 2021 (UTC)
 * I've removed it completely, as it doesn't really add anything to the argument beyond the informal discussion a few paragraphs before; it just performs some trivial symbol manipulation before pulling the Archimedean property out of a hat to give the final result. The Archimedean property is the real meat here, and without incorporating a proof of that, this isn't really a proof at all. -- The Anome (talk) 08:45, 20 August 2021 (UTC)


 * I've restored it. It has been there a long time and there is no consensus to remove. Hawkeye7   (discuss)  09:24, 20 August 2021 (UTC)


 * Just because material is long-established is not a good reason to keep it. The "formal proof" is neither formal, nor is it a proof, since it hinges entirely on the Archimedean property of the reals, which is neither taken as an axiom, nor proved within it as a lemma. It in effect says only "1 - 0.999... = &epsilon; < 1/10$n$ for all n in Z$+$; the Archimedean property holds, therefore &epsilon; = 0, therefore 1 = 0.999..." only in a more wordy way that illuminates nothing. And you can't prove the Archimedean property of the reals without formalizing the reals in some way, which leads us to the proper formal proofs further down the article, which address all of this properly. -- The Anome (talk) 14:49, 20 August 2021 (UTC)
 * As the standard definition of $0.999...$ is to be the least upper bound of its truncations. This is a theorem that such a least upper bound exists and equals $1$; the equation $0.999... = 1$ is a short statement of this theorem. As for every theorem, a proof is needed, and, as it is a theorem on real numbers, every proof must rely on basic properties of real numbers, here the Archimedean property. Generally, proofs are not given in Wikipedia, but there are needed for counterintuitive results. Specifically in this case, as the article is intended to beginners in mathematics, a proof is required. In mathematics, the only way to contest a proved counterintuitive result is to show that the proof is wrong. This is fundamental, but apparently not always taught in elementary courses.
 * So, this section is required early in the article, as the only correct answer to people who contest $0.999... = 1$ is to say: "Here is a proof. If you disagree, you must show that this proof is wrong".
 * However, because of the modern meaning of "formal proof" (proof that can be checked on a computer), I'll change "formal" into "rigorous". D.Lazard (talk) 11:31, 21 August 2021 (UTC)


 * You say " In mathematics, the only way to contest a proved counterintuitive result is to show that the proof is wrong." This is nonsense; as Christopher Hitchens said, that which can be asserted without evidence, can be dismissed without evidence. And what is asserted here without evidence is the Archimedean property. As students are taught, even in the most elementary logic classes, there is a difference between a sound argument and a valid argument.


 * For example, "if something-or-other, then 0.999... = 1" is an entirely valid argument. But it's not a proof of anything, because it pulls "something-or-other" our of the ether by mere bold assertion.


 * In the case of the "formal elementary proof" in the article, the Archimedean property is the missing link and the whole argument collapses at that point; one of the premises of the argument is simply missing. The Archimedean property is by no means obvious, and simply does not hold in some cases; for example, in the surreal numbers (although, of course, 0.999... = 1 is still true in the surreals, it's just that you can't pull the Archimedean property out of a hat to prove it).


 * And the reason it has to be pulled out of a hat is that you have no real definition of "real number" at that point in the article. You are, of course, quite right in your discussion of the standard construction of the reals, and the Archimedean property is indeed a real thing, and the proof is then correct. But the moment you start entertaining concepts like least upper bounds, you are no longer in the world of elementary mathematics, and you are in the realm of more sophisticated arguments, which the article provides just a few paragraphs below. -- The Anome (talk) 14:21, 21 August 2021 (UTC)
 * Seems elementary, since it already defined at the top of the article that this is real numbers, so a property of real numbers is already elemental. -- Alanscottwalker (talk) 14:36, 21 August 2021 (UTC)
 * Yes, the Archimedean property is indeed a property of the (standard) real numbers. But so is 0.999... = 1. The "proof" relies entirely on the Archimedean property, and the lack of proving that makes a nonsense of calling this a rigorous proof.
 * I've now re-cast the exact same argument in terms of derivation from the Archimedean property; and re-titled the subsection to match. -- The Anome (talk) 14:53, 21 August 2021 (UTC)
 * There are two problems in your way of reasoning. Firstly, 0.999... = 1 is a property of real numbers. So, nothing can be said on this subject without supposing that the real numbers are known by readers. Archimedean property is a property that is assumed in every treatment of the real numbers, either as an axiom, or as a direct consequence of the axioms, or as a property that results from experimental evidence; the latter was the case before 20th century, and is still the case at elementary level, since an axiomatic treatment is far too technical for most users of real numbers.
 * The second problem here is that 0.999... is an infinite decimal, and some teachers do not understand that "infinite decimal" is a non-elementary concept. It requires the concept of actual infinity, which was so unnatural that it was not accepted before the 20th century. To be properly defined an infinite decimal requires either the concept of least upper bound or the less elementary concept of limit. So nothing proper can be said on infinite decimals without the concept of least upper bound.
 * Finally, some of your sentences suggest that you do not accept that the "real numbers" is the standard, and nonstandard real numbers are not real numbers. This may be your opinion, but it is WP:OR and cannot be used in Wikipedia. D.Lazard (talk) 16:50, 21 August 2021 (UTC)

Another representation of the algebraic proof
Would it be helpful to write a second proof in the section "Algebraic arguments"? Here is an alternative representation using infinite sums for readers who do not understand or accept repeating decimals: $$x = 0.(9)_n = \frac{9}{10} + \frac{9}{10^2} + \frac{9}{10^3} + ... = \sum_{i=1}^{\infty} \frac{9}{10^i}$$ $$10x = \sum_{i=1}^{\infty} \frac{9}{10^{i-1}} = \sum_{i=0}^{\infty} \frac{9}{10^i}$$ $$10x - x = 9x = \sum_{i=0}^{\infty} \frac{9}{10^i} - \sum_{i=1}^{\infty} \frac{9}{10^i} = \frac{9}{10^0} = 9$$ $$x = 1$$ OneToZero (talk) 12:09, 3 November 2021 (UTC)

Supremum or infimum?
Hi - shouldn't the link in the second sentence of the article link to the supremum (smallest number no less than all elements of a set), not the infimum (largest number no greater than all elements of a set)? (Not very good at Wikipedia, don't understand the etiquette - please excuse my not signing this.) — Preceding unsigned comment added by 77.101.228.252 (talk) 16:52, 3 December 2021 (UTC)
 * The second sentence links to Infimum and supremum, where the second paragraph defines "supremum". I do not see any problem. D.Lazard (talk) 17:40, 3 December 2021 (UTC)
 * At a second thought, there is problem, described in WP:SUBMARINE, and I have fixed it. D.Lazard (talk) 19:12, 3 December 2021 (UTC)

The article seems incorrect, at the beginning
Firstly, saying ".9 repeating equals 1" is not necessarily true. It is true for real numbers, but not necessarily for hyperreal numbers. I think the statement should be clarified. If .9 repeating holds for real numbers but not elsewhere, why could I not say 3 = 1, because 3 = 1 mod 2 and 1 = 1 mod 2? I believe the article should add "In standard analysis" before "This number is equal to 1.", due to the fact that it is not necessarily true.

--The big parsely (talk) 15:59, 18 January 2022 (UTC)The Big Parsely


 * Could you say which sentence exactly you are unhappy with? Gesturing at hyperreals isn't enough, because repeating decimals give no way of coherently picking out hyperreals with nonstandard parts. The article already has 0.999..., which I think you are aware of. &mdash; Charles Stewart (talk) 18:49, 18 January 2022 (UTC)
 * Firstly, saying ".9 repeating equals 1" is not necessarily true. It is true for real numbers, but not necessarily for hyperreal numbers.
 * Hyperreal numbers are not only nonstandard and not necessary to account for in the lead section. If we took this approach here, to be logically consistent we would have to provide alternative definitions of the limit and derivative (which are not elegantly expressed using limits with a hyperreal framework) for every possible construction of math in every article where they are defined and in every section (even lead sections). Just to list 2 basic concepts that would be affected, not to mention series and all related articles on them. It's enough for these constructions to be explained in a subsection, in my view. Also, by the transfer principle statements of this kind (first order) such as .9~ = 1 that are true in the real numbers must also be true in the hyperreals.
 * why could I not say 3 = 1, because 3 = 1 mod 2 and 1 = 1 mod 2?
 * That's an incorrect statement, in modular arithmetic we would say 3 is congruent to 1 (modulo 2), not that they are equal. Freeze4576 (talk) 17:13, 2 April 2022 (UTC)
 * ".9 repeating equals 1" is not necessarily true. It is also not necessarily false. It is simply nonsensical if the definition of ".9 repeating" is not given or if the context where this definition applies is not given. The article describes clearly the meaning of the notation .999... (this is nothing else as a notation) and proves that the number represented by this notation is the number 1. It is never said that the notation is used in other contexts. So there is definitively nothing incorrect there. D.Lazard (talk) 20:55, 2 April 2022 (UTC)
 * It is also not true that the population of the Earth is 7.9 billion, if we e.g. don't use short scale, or if we use undecimal. It's sensible, however, to assume the standard meaning af words and notations, at least for a start (i.e. in the lead) - if relevant, we can get into the more exotic things further down in the article.--Nø (talk) 11:14, 3 April 2022 (UTC)

"Liking" the subject
I don't "like" the subject of this article. I'm used to the idea that "a number can be represented in one and only one way by a decimal", so it's uncomfortable, but I accept it because I know it's true.

Several authors are cited in the "Skepticism in education" section. Does any of them also talk about people who "dislike" the concept but still accept it, for example Tall's case studies? If so, could anything be added to the "Cultural phenomenon" section? A scholarly sentence/paragraph about popular opinion would be at least as good in this section as the current remarks about UseNet and World of Warcraft. 49.198.51.54 (talk) 02:19, 4 April 2022 (UTC)

Adding a section on representation
I think the introductory statement on the number represented by 0.999... will be mostly ignored or not adequately understood by the typical reader. I recommend adding the following to the Wikipedia page, particularly somewhere above where the proofs start.


 * There's a recurring complaint by editors who don't like to see highly technical content represented in Wikipedia. They say that the only people who can understand it don't need it anyway.
 * Most of the time they're just wrong about that. There is lots of technical content in Wikipedia that is not readable without some fairly meaty background, but that is very useful to those who have the background.
 * For the purposes of this article, though, the complaint has some merit. If you understand the real numbers rigorously, the identity of 0.999... and 1.000... is a trivial fact of minor interest (the topological implications can be a little more interesting, but this would not be the right title for that article).
 * The audience that will benefit from this article is composed primarily of people who don't understand limits. This is probably not the right article to teach them about limits.
 * That's why I think the best chance of delivering value from this article is to emphasize the Archimedean principle. --Trovatore (talk) 18:19, 11 September 2022 (UTC)

Representation of a real number
To understand the equality, and its proofs, it is necessary to know exactly what is meant by the standard real number 0.999... Failure to understand what 1 = 0.999... means, often results from a failure to understand what the number 0.999... means.

This representation of a real number is unlike more basic representations of 2, or 2.1, or 1/3, or $$\sqrt 2$$, each of which can be understood in more immediate and intuitive terms. By contrast, 0.999... must be understood in terms of a limit of a sequence.

Note that, the statement 1 = 0.999... claims that the number represented by 1 is the same as the number represented by the limit 0.999... The fact that a single number can have many representations is not unique to limits. In fact the number 1 can be represented as $$2-1$$ or $$2 \div 2$$, or by the English phrase "the least integer greater than zero". The fact that a real number can be represented by the decimal sequence 1.000... and the decimal sequence 0.999..., or that it may be represented in any number of other ways, is not problematic or unique to numbers.

To understand which number is represented by 0.999... consider the sequence of numbers, 0.9, 0.99, 0.999, and so on. Put intuitively, "the limit of this sequence" is the number which the sequence becomes close to, eventually. So by saying that 1 = 0.999..., this means the same as saying that 1 is the limit of the sequence 0.9, 0.99, 0.999... And then by saying that 1 is the limit of this sequence, this means that 1 is the number which the sequence becomes close to, eventually.

More rigorously stated, the number 0.999... is defined as the following limit.


 * $$\lim_{n\to\infty}\sum_{n=1}^\infty \frac{9}{10^n} $$

Therefore the statement 1 = 0.999... is the same as the statement 1 = $$\lim_{n\to\infty}\sum_{n=1}^\infty \frac{9}{10^n} $$.

Addemf (talk) 17:23, 11 September 2022 (UTC)


 * In fact, as I scan the arguments sub-page, almost every single argument against 1 = 0.999... seems to stem, not from any part of any of the proofs. They all come from simply misunderstanding what is expressed by 0.999... Addemf (talk) 17:43, 11 September 2022 (UTC)


 * Yes, but part of that is a misunderstanding of limits. The article deliberately avoids language like "becomes close to, eventually" as it is liable to be misinterpreted by the ignorant to mean that 0.999... is close to but not exactly 1. Hawkeye7   (discuss)  18:46, 11 September 2022 (UTC)
 * I'm open to finding a better way of expressing the meaning of 0.999... I'm not committed to how I did it here. But I do think a section that emphasizes
 * (1) most people assume various meanings of 0.999... which are not the ones used in the claim 1 = 0.999... and
 * (2) the correct definition is [insert here]
 * would do a LOT more for resolving confusion, than analysis proofs for people who almost certainly are unequipped to read an analysis proof. Addemf (talk) 20:49, 11 September 2022 (UTC)


 * It's all in section 0.999.... - DVdm (talk) 21:24, 11 September 2022 (UTC)
 * Alright, if you think that's enough, so be it. Just seems like precisely zero people are going to get what's going on there, except people who already know analysis. Addemf (talk) 23:17, 11 September 2022 (UTC)
 * I agree. The last paragraph of that section is i.m.o. the most — and perhaps the only — important part of the subject and it's hidden way too deep in the remainder of the article. I think it deserves a more prominent place, but consensus seems to have sort of hidden it a bit . - DVdm (talk) 08:17, 12 September 2022 (UTC)

Why are algebraic proofs only listed as "arguments"?
1/3 = 0.333..., so multiplying both sides by 3 we get 1 = 0.999...

This is the simplest way to show this identity, and I think it is very important for this article. Why is it only listed as an "argument" and not as a "proof"? Are there problems with this proof?

If there are indeed problems with this proof, I think we should include this proof and explain its problems in this article, because I think it's important for this topic. Cooper2222 (talk) 04:42, 31 October 2022 (UTC)


 * It lacks the rigor of a true formal mathematical proof. We include it for pedagogical reasons. As the article states, it is easily understood, and satisfies most readers. Hawkeye7   (discuss)  05:31, 31 October 2022 (UTC)


 * Another answer that may (or may not) better address User:Cooper2222s doubts: The argument based on 1/3 is not wrong as such, but how do you know that 0.333... * 3 = 0.999...? It's true, but is it completely obvious, with no room for doubt? How? One can prove that usual algorithms for arithmetic operations with decimal numbers are correct for terminating decimals, but do we know they are valid for non-teminating ones too? The proper proof of 0.999... = 1, as given in the article, goes to first principles, therefore we know that 0.999... = 1, but it is satisfying and reassuring that less rigorous arguments like the one based on 1/3 get it right too. -- Note that you could easily construct a "proof" that 0.999... < 1, based on principles that are valid for terminating decimals, but the proof from first principles show this to be wrong.--Nø (talk) 08:27, 31 October 2022 (UTC)
 * The claim 0.333... * 3 = 0.999... is going to be true under pretty much any interpretation. The harder step is actually 0.333... = 1/3.  People accept that because they're used to it and it's the answer that comes out of short division, but what if 1/3 just doesn't have a decimal representation, and short division just doesn't give an exact answer here?
 * That would be the situation, for example, working in Fred Richman's "decimal numbers" (which don't allow subtraction or negative numbers).
 * But as I said above, we should try to explain the situation without requiring explaining limits, which are a big step for readers who need this article. The Archimedean principle is easier to explain, I think. --Trovatore (talk) 19:02, 1 November 2022 (UTC)
 * User:Trovatore, you argue from the point of view of someone knowing a lot about mathematics and different number constructions. I try to argue from the point of view of the reader we are trying to help here. It is - such a reader would say - common knowledge that 1/3 = 0.333... (and it is true, too, even if the arguments are complicated and only valid for some constructions of the numbers). Allthough most number constructions that include 0.333... also agree that 0.333... * 3 = 0.999..., it is not trivial.--Nø (talk) 08:24, 10 November 2022 (UTC)
 * Well, it's more trivial than the other equality involved, namely 1/3 = 0.333.... --Trovatore (talk) 07:02, 23 November 2022 (UTC)

We should include them and explain why they are not rigorous
If these arguments are not rigorous, I think we should include them and explain clearly why they are not rigorous: what assumptions are implicitly made or what steps are omitted. A lot of people may have the same doubts and want to know. So it's important to include these in this article. Cooper2222 (talk) 02:27, 1 November 2022 (UTC)


 * We already provide several rigorous proofs. Hawkeye7   (discuss)  02:30, 1 November 2022 (UTC)
 * But those don't explain why the algebraic proofs are not rigorous, do they? Cooper2222 (talk) 02:33, 1 November 2022 (UTC)
 * I agree with User:Cooper2222. Another thing is how to accomplish this in a clear way, and once it is there, how to keep math nerd away from edting it to satisfy themselves rather than the lay reader.--Nø (talk) 08:28, 10 November 2022 (UTC)
 * Rigorous proofs are provided. The simple arguments are provided because they are more readily understood by the some readers. I would object to their removal. The pedagogical point is that they more easily accept that 1/3 = 0333... that 1 = 0.999... That is because (1) they more easily accept 1/3 as a mathematical construct than 1, which is more familiar and has religious implications and/or (2) they see 1/3 and 0.333... as processes (verbs) but 1 as a number (noun). As noted above, a proof would be required that 1/3 = 0.333... To do that we would use one of the techniques we use below to show 1 = 0.999... Hawkeye7   (discuss)  09:19, 10 November 2022 (UTC)
 * User:Hawkeye7, I'm not aware of anyone suggesting removing them; the question is how to present them. And once they are there, how to make the reader understand the distinction between proofs and arguments.--Nø (talk) 13:20, 10 November 2022 (UTC)
 * I have added an introduction to section for explaining why they are not proofs. Nevertheless, I cannot understand why infinite decimals are so often taught before the distinction between a (rigorous) proof and an (informal) convincing argument: infinite decimals are a concept that is rarely used in applications as well as in pure mathematics, while the concept of a proof is the core of mathematics. D.Lazard (talk) 12:39, 11 November 2022 (UTC)
 * As for why infinite decimals are taught early on, the obvious reason is that once long division has been taught, the case of 1/3 is impossible to avoid. After all, maths is not taught primarily to future matematicians, who might benefit from a rigorous approach, but to the common population, who need a basic idea about what holds true and how to manipulate numbers.Nø (talk) 12:11, 22 November 2022 (UTC)
 * I agree that elementary maths course are not taught for future mathematicians. This is the reason for which infinite decimals should not be taught too early, as only mathematician use them. So, for students that are not future mathematicians, it is must more useful to put emphasis on approximations than on infinite decimals. After all, finite decimals were not considered before the second half of the 19th century, and before that, finite decimal were sufficient for everybody. D.Lazard (talk) 12:35, 22 November 2022 (UTC)
 * If you have a bright student in class, there's no way you can escape from saying something about long division for 1/3 (and other repeating decimals). Teachers at that levet are often not very knowledgable about more advanced maths, so what they choose to say about such things may be a problem. (And even if you had an expert matematician to teach, he/she might still be tempted to say something that is not fitting.) But yes, saying that any finite decimal in these cases is an approximation, and leaving it at that, would be appropriate at an elementary level. Though ... my interest in maths was not least piqued by teachers hinting at things beyond what they were supposed to be teaching at the given level (as well as by books by e.g. Martin Gardner, also not too concerned about rigour). Nø (talk) 12:59, 22 November 2022 (UTC)
 * It just isn't true that "only mathematicians" use infinite decimal expansions. I was introduced to them in the second grade, when we read A Wrinkle in Time, and I don't think I was a mathematician at that point.  It might be true that only mathematicians use them in ways such that the infinitude of digits is actually essential to the use, but that's a different claim, and betrays what I consider an overly austere mindset. --Trovatore (talk) 18:39, 22 November 2022 (UTC)
 * Same here. I also remember my mind being blown by my fourth grade text, which said: "a circle is a set of points". Whoa! My classmates could not see what my problem was, because they thought of a point as being a blob of ink on a page, and having a size. But what I don't recall from high school is being taught about proofs. Instead teacher would write proofs up on the board. (Byers says this too - see p. 363) Hawkeye7   (discuss)  20:11, 22 November 2022 (UTC)

It might be worth to note that using a not quite rigorous argument or using infinite decimals loosely where it fits is kinda mimicking the historical development. Practical, intuitive use and "proof" usually preceded a rigorous analysis (and famous mathematician arrived at famous result with "dubious" arguments form a rigorous perspective). Learning math in the earlier stages often mimicks historical development to degree and that's why such things occur in math education before a rigorous base/definition is given/available.

As far as our article is concerned the algebraic proofs definitely should be mentioned as they are widespread in (educational) literature, but their issues of course should be mentioned as well. In that sense the current version seems reasonable to me.--Kmhkmh (talk) 16:20, 22 November 2022 (UTC)

My understanding of why they are not rigorous
Proof 1: 1/3 = 0.333... ⇒ 3 * 1/3 = 3 * 0.333... ⇒ 1 = 0.999...

Proof 2: Let x = 0.999... ⇒ 10x = 10 * 0.999... = 9.999... = 9 + 0.999... = 9 + x ⇒ 9x = 9 ⇒ x = 1

Problems:

1) In Proof 1, "1/3 = 0.333...", if proved rigorously, requires the same proof as we rigorously prove "1 = 0.999...".

2) In Proof 2, when letting x = 0.999..., the existence and uniqueness of 0.999... is not established.

3) In both Proof 1 and Proof 2, we assumed infinite decimals can do the same usual arithmetic operations, and we can follow the same rules to manipulate the digits of infinite decimals, and to manipulate the expressions and equations involving them.

Cooper2222 (talk) 23:54, 7 May 2023 (UTC)


 * In (2) we do not assume that 0.999... is unique, which is good because it is not. Hawkeye7   (discuss)  00:03, 8 May 2023 (UTC)
 * If 0.999... is not unique, it's problematic that we can change 0.999... back to x in the later steps. The new x might not be the same as the initial x.
 * For example, let x = ±1 ⇒ -x = ±1 ⇒ -x = x ⇒ x = 0.
 * Cooper2222 (talk) 00:56, 8 May 2023 (UTC)
 * "... because it is not": Why not?
 * Cooper2222 (talk) 04:44, 8 May 2023 (UTC)
 * I think this comment gets to the heart of the matter. 0.9999... is "unique" in that it represents exactly one particular real number.  However, the decimal representation of many real numbers is not unique; in particular the number represented by 0.9999... does not have a unique representation since it has another decimal representation i.e. 1.00000...
 * The article states the fact that many real numbers do not have a unique decimal representation, but perhaps that fact should be featured more prominently since (at least to me) it is the key to understanding the equality of 0.9999... and 1.0.
 * Agree with the sentiment expressed above that this article should be aimed more at lay readers than trained mathematicians. That's not an excuse to be sloppy, but the primary audience for this elementary topic is not readers with math degrees. Mr. Swordfish (talk) 12:45, 8 May 2023 (UTC)

Editorial dispute over ultrafinitism
Since there appears to be some dispute over the section on ultrafinitism, I will say that I favor removing that section altogether, as the currently cited source from Sazonov does not mention 0.999... at all. It is trying to formalize a notion of numbers "not too big" to conceive of, so has no bearing on the current article whatsoever. It would take some significant original research to apply this to 0.999..., and at any rate ultrafinitism is a sufficiently fringe theory that there is no need to include it in the article. AristippusSer (talk) 07:56, 11 August 2023 (UTC)r.


 * I agree to remove this section. D.Lazard (talk) 09:15, 11 August 2023 (UTC)


 * Concur. Hawkeye7   (discuss)  11:23, 11 August 2023 (UTC)
 * Agree to remove the section. I don't think it improves the article and it may confuse readers by giving untoward promotion of a decidedly fringe viewpoint. Mr. Swordfish (talk) 14:05, 11 August 2023 (UTC)
 * I do not agree to remove the section. It seems fine and relevant when cited well, and I see no reason to throw the baby out with the bath water if you will - even if you happen to disagree with the view of ultrafinitism (or even feel it is not worthy of being a valid "view"). I believe it may be best for some users with strong feelings either way on the topic (as it seems to bring out, humorously enough) to distance themselves from the article.
 * A MINOTAUR (talk) 16:49, 11 August 2023 (UTC) A MINOTAUR (talk) 16:49, 11 August 2023 (UTC)
 * If nothing else I think we should allow for the talk page to ruminate before making any decisions. The last edit citing a "consensus" formed over roughly 12 hours by four people (3 of who were directly involved in original editing) seems a bit silly. A MINOTAUR (talk) 16:54, 11 August 2023 (UTC)
 * WP:SNOWBALL. D.Lazard (talk) 17:09, 11 August 2023 (UTC)
 * I actually do think that this POV is likely worthy of mention in some form. Ideally it should be done in a way that makes it clear that the ultrafinitist doesn't consider 0.999... to be something different from 1, but rather simply not a legitimate "thing" at all.
 * That said, I do not know where to find high-quality sources. At some point this article quoted a nearly incomprehensible rant from Doron Zeilberger, who by all accounts is a serious mathematician, but who was clearly not writing in a usable way in that particular piece of venting. --Trovatore (talk) 17:15, 11 August 2023 (UTC)
 * The problem is not just with ultrafinitism being a minority viewpoint; it is that the Sazonov paper cited in the article does not support the section in any way, and thus the entire section as it was written was WP:OR.


 * In any case, I still fail to see how ultrafinitism is relevant here, as opposed to, well, just regular finitism. For instance, finitists would be OK with 0.999... with $$2^{1000}$$ 9's after the decimal point, whereas ultrafinitists might possibly take issue with this, but it would depend on their particular formalization.


 * Again, I would be open to including a section on ultrafinitism provided it cites a reliable source by someone who is mathematically qualified and specifically addressing the application of ultrafinitism on 0.999..., but the source cited before was not that. And again, WP:OR means that this website is decidedly not the place to publish any original research or interpretation. AristippusSer (talk) 17:12, 11 August 2023 (UTC)
 * Narrowly addressing the ultra-vs-regular finitism question: I think the usual distinction is that the "regular" finitist accepts so-called "potential infinity", a concept going back at least to Aristotle, whereas the "ultra" finitist may not accept even arbitrarily large natural numbers (test question: Is it obvious that the function $$n\mapsto2^n$$ is total on the natural numbers?).  It's possible to frame the equality 0.999...=1 as a statement about potential infinity, so "regular" finitists should accept it. --Trovatore (talk) 17:20, 11 August 2023 (UTC)
 * How I see it is that finitists can accept "0.999..." as a notation and agree that it represents the same underlying number as 1, but they would disagree that it represents an actual decimal expansion the way one might analogize from the basis of finite decimal expansions like 0.9. A priori it seems to me that ultrafinitists would not have any philosophical difficulty in accepting the same thing, just as they would not have problems with $$\frac13$$ and $$\frac23$$ either, only the interpretation of their decimal expansions as analogous to finite decimals.
 * In any case, this would all make a very interesting topic to discuss on a math forum, but let me stop myself from straying too far from the editorial discussion at hand. My point is, we need a reliable source talking about this if we don't want to violate the policy against OR, and the fact that the editor of the original section did not really attempt to address the finitism/ultrafinitism distinction certainly would not fill one with confidence in their original research, to say the least! AristippusSer (talk) 17:40, 11 August 2023 (UTC)
 * I appreciate the concerns about drifting off topic, but you have to let me respond. The point is that an ultrafinitist may consider 0.999... going to infinity to be simply nonsense, since they don't accept the idea of "going to infinity" even potentially.  Keeping it on-topic, that's the possible ultrafinitist objection that could possibly be covered in the article.  For a person with such a viewpoint, presumably $$\frac{1}{3}$$ simply does not have a decimal representation. --Trovatore (talk) 18:27, 11 August 2023 (UTC)
 * I do appreciate this explanation a bit better, as I truly do largely just care about references being used accurately (and not somewhat haphazardly in order to justify various viewpoints). I suspect there are a few cases of the latter still present in the article, as this particular case of mathematical "pedantic-semantics" really seems to fire people up - but I suppose those are issues for another day.
 * Regardless, thank you for the more in depth reply. It shines a good light on your character. A MINOTAUR (talk) 17:25, 11 August 2023 (UTC)