# Talk:0.999...

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.
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Current status: Featured article
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.

## 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)

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## 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)

## 0.999... <> 1

Moved to talk:0.999.../Arguments#0.999..._<>_1

I don't think we need that again—see Talk:0.999.../Arguments/Archive 11#Time to face reality

## 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 Wikipedia:Short description, and more information at Wikipedia: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, · · · 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)
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, · · · 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, · · · 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. · · · 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)

## 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)