# Talk:0.999...

(Redirected from Talk:0.999)
Frequently asked questions (FAQ) edit Q: Are you positive that 0.999... equals 1 exactly, not approximately? A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research should never be added to a Wikipedia article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so. Q: Can't "1 - 0.999..." be expressed as "0.000...1"? A: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10-k toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of ${\displaystyle {\frac {0.000\dots 1}{10}}}$ would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0. Q: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1? A: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit ${\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}$ is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1. Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1. A: No. By this logic, 0.9<0.999...; 0.99<0.999... and so forth. Therefore 0.999...<0.999..., which is absurd. Something that holds for various values need not hold for the limit of those values. For example, f (x)=x 3/x is positive (>0) for all values in its implied domain (x ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that ${\displaystyle 0.x_{1}x_{2}x_{3}...}$ must be less than ${\displaystyle 1.y_{1}y_{2}y_{3}...}$ for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation. Q: 0.999... is written differently from 1, so it can't be equal. A: 1 can be written many ways: 1/1, 2/2, cos 0, ln e, i 4, 2 - 1, 1e0, 12, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a bijection from decimal representations to real numbers. Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount? A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; differentials, which are not numbers at all; differential forms, which are not real numbers and have anticommutativity; 0+, which is not a number, but rather part of the expression ${\displaystyle \lim _{x\rightarrow 0^{+}}f(x)}$, the right limit of x (which can also be expressed without the "+" as ${\displaystyle \lim _{x\downarrow 0}f(x)}$); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application. Q: Are you sure 0.999... equals 1 in hyperreals? A: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given here, the problem is that almost same sequences give different hyperreal numbers, ${\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}$, and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...). Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly? A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a WP:POV desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system. Q: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right. A: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on real analysis, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in engineering, mathematics, statistics, computer science, or a natural science, it would probably help you in the future anyway. Q: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article? A: The criteria for inclusion in Wikipedia is for information to be attributable to a reliable published source, not an editor's opinion. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Wikipedia policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Wikipedia policy on original research.
0.999... is a featured article; it (or a previous version of it) has been identified as one of the best articles produced by the Wikipedia community. Even so, if you can update or improve it, please do so.
This article appeared on Wikipedia's Main Page as Today's featured article on October 25, 2006.
Article milestones
Date Process Result
May 5, 2006 Articles for deletion Kept
October 10, 2006 Featured article candidate Promoted
August 31, 2010 Featured article review Kept
Current status: Featured article

## 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 Wikipedia: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 Wikipedia: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: [1], [2]. 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. 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. 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 ${\displaystyle 0.999...=1}$ 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. 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:

"Any material lacking a reliable source directly supporting it may be removed and should not be restored without an inline citation to a reliable source. Whether and how quickly material should be initially removed for not having an inline citation to a reliable source depends on the material and the overall state of the article. In some cases, editors may object if you remove material without giving them time to provide references; consider adding a citation needed tag as an interim step. When tagging or removing material for lacking an inline citation, please state your concern that it may not be possible to find a published reliable source for the content, and therefore it may not be verifiable. If you think the material is verifiable, you are encouraged to provide an inline citation yourself before considering whether to remove or tag it."

Be advised that continuing to insert this unreferenced OR against policy will likely result in sanctions. 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. 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:

"However, many editors misunderstand the citation policy, seeing it as a tool to enforce, reinforce, or cast doubt upon a particular point of view in a content dispute, rather than as a means to verify Wikipedia's information. This can lead to several mild forms of disruptive editing which are better avoided. Ideally, common sense would always be applied but Wiki-history shows this is unrealistic."

WP:CALC:

"Routine calculations do not count as original research, provided there is consensus among editors that the result of the calculation is obvious, correct, and a meaningful reflection of the sources. Basic arithmetic, such as adding numbers, converting units, or calculating a person's age are some examples of routine calculations. See also Category:Conversion templates.."

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)

## 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 tokensrevised 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 ${\displaystyle 1/3=0.333...}$ (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 ${\displaystyle 0.999...}$ 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×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 ${\displaystyle 0.999...}$ 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 ${\displaystyle 0.999...}$ "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)

Hello fellow Wikipedians,

I have just modified one external link on 0.999.... Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

You may set the |checked=, on this template, to true or failed to let other editors know you reviewed the change. If you find any errors, please use the tools below to fix them or call an editor by setting |needhelp= to your help request.

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

If you are unable to use these tools, you may set |needhelp=<your help request> on this template to request help from an experienced user. Please include details about your problem, to help other editors.

Cheers.—InternetArchiveBot 20:38, 27 August 2017 (UTC)

## 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)

Hello fellow Wikipedians,

I have just modified one external link on 0.999.... Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

You may set the |checked=, on this template, to true or failed to let other editors know you reviewed the change. If you find any errors, please use the tools below to fix them or call an editor by setting |needhelp= to your help request.

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

If you are unable to use these tools, you may set |needhelp=<your help request> on this template to request help from an experienced user. Please include details about your problem, to help other editors.

Cheers.—InternetArchiveBot 07:43, 30 September 2017 (UTC)