Talk:0.999...

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view · edit Frequently asked questions 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.

Yet another anon

Moved to Arguments subpage

Ambiguity

0.999... is ambiguous. An infinite number of sequences could start with nines, yet only one of them has all nines. Furthermore, even if it were guaranteed to continue with nines, it is ambiguous whether it refers to the limit of the summation of a geometric series as the number of terms approaches infinity or some other definition. Lxvgu5petXUJZmqXsVUn2FV8aZyqwKnO (talk) 22:14, 18 August 2025 (UTC)[reply]

It is no more or less ambiguous than 0.33333... or 0.50000... which are clearly equal to 1/3 and 1/2.

It is just as unambiguously clear that 0.9999... is equal to one. Mr. Swordfish (talk) 22:57, 18 August 2025 (UTC)[reply]

I think you're missing the OP's point. In the lead sentence, it does say that this is notation for a repeating decimal, but it could maybe stand to be a bit more explicit that this is shorthand for a repeating sequence of 9s, rather than simply a truncation of some other decimal that starts with three 9s (repeating or not). 35.139.154.158 (talk) 23:08, 18 August 2025 (UTC)[reply]

The lead sentence says that it is a repeating decimal, which is what the "..." notation means. Perhaps we should explicitly state that it is an infinite sequence of nines, but it seems unnecessary to me. Mr. Swordfish (talk) 23:25, 18 August 2025 (UTC)[reply]

The caption to the figure in the lede does say so. Tito Omburo (talk) 12:10, 19 August 2025 (UTC)[reply]

I have pushed into a footnote the altenative notations and the claification of the meaning of the ellipsis. IMO, adding these details to the text is distracting; nevetheless, they may be useful to some readers. D.Lazard (talk) 12:56, 19 August 2025 (UTC)[reply]

What do folks think of this opening sentence:

In mathematics, 0.999... is a repeating decimal (i.e. there is an infinite string on 9s after the decimal point) that is an alternative way of writing the number 1.

Instead of making the reader click on a wikilink or a footnote to see the meaning of "repeating decimal", it is explained right in the text. As MOS:LINKSTYLE says:

Use a link when appropriate, but as far as possible do not force a reader to use that link to understand the sentence. The text needs to make sense to readers who cannot follow links.

If we assume that some readers don't already understand what "repeating decimal" means, then the parenthetical sentence above would conform to the MOS. Mr. Swordfish (talk) 14:10, 19 August 2025 (UTC)[reply]

Someone who doesn't know what a repeating decimal is is unlikely to know that a "string" means a list of symbols. You should use a common word like "list", "sequence", or "succession" instead of a computer science jargon word. I'd also recommend offsetting such an explanation by commas rather than parentheses, or using a separate sentence, and avoiding "i.e.". So something instead like:

In mathematics, 0.999... is a repeating decimal that is an alternative way of writing the number 1. The three dots represent an infinite list of "9" digits.

would be better than this proposal. I'll leave it to others to decide whether such an explanation is necessary. –jacobolus (t) 14:43, 19 August 2025 (UTC)[reply]

From experience with cranks on sci.math, I think "unending" is better than "infinite". Otherwise there is always the "after the infinite sequence of xs there must be a y" line; with "unending" you simply point out that this means there isn't an end. Imaginatorium (talk) 16:16, 19 August 2025 (UTC)[reply]

I think it is fine as is. Tito Omburo (talk) 16:43, 19 August 2025 (UTC)[reply]

I prefer the version proposed by jacobolus (t) over my proposal. "Unending" is also probably better than "infinite".

Also agree that it's fine as it is. Mr. Swordfish (talk) 16:54, 19 August 2025 (UTC)[reply]

A few years ago, the article started:

In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. [...]

–jacobolus (t) 18:20, 19 August 2025 (UTC)[reply]

I think that phrasing is problematic. For example, File:Ordinal ww.svg with nines at every site. Tito Omburo (talk) 18:25, 19 August 2025 (UTC)[reply]

I have implemented this suggestion with "unending" instead of "infinite". Stepwise Continuous Dysfunction (talk) 22:44, 19 August 2025 (UTC)[reply]

I'm not a fan of having a footnote in the middle of the first sentence. That feels pedantic and distracting. Stepwise Continuous Dysfunction (talk) 22:37, 19 August 2025 (UTC)[reply]

(imo) mathematics, just like every other field of science, can have facts/theories that are counter-intuitive. however as wikipedia is an aggregation, not the source of the knowledge, defending / asserting its correctness is probably not our *primary* job, in so far as to spend too much room addressing it *up front in the intro*.

i’d write somewhere “…see [[below|#misconceptions]]” and then put the bulk of clarification there, e.g. infinite means “unending”, etc.. - the problem is we already addressed this into detail later in article, and some readers didn’t seem to reach that part. 海盐沙冰 (talk) 10:28, 20 August 2025 (UTC)[reply]

In this case, most of the sources addressing the subject of the article are correcting the misconception or highlighting its role in education, so the focus is appropriate. Tito Omburo (talk) 12:40, 20 August 2025 (UTC)[reply]

Problems with 10-adic section

The first thing I note here is that the section heading is "p-adic numbers", and the body of the section also keeps using the term p-adic. But when you say p-adic, it's understood that p is prime. The 10-adics are a legitimate algebraic structure but they aren't usually included in the study of the p-adics.

More generally the section suffers from a tenuous connection to the subject of the article. It's interesting stuff but it doesn't seem to have much to do with 0.999.... --Trovatore (talk) 23:16, 21 August 2025 (UTC)[reply]

Fwiw, the 10-adics are canonically isomorphic as a topological ring to ${\displaystyle \mathbb {Z} _{2}\oplus \mathbb {Z} _{5}}$. Tito Omburo (talk) 10:42, 22 August 2025 (UTC)[reply]

Also, the image is not only cryptic, but it is misleading, as ignoring that 4-adic numbers are the same as 2-adic numbers and are obtained by grouping the binary digits by two. D.Lazard (talk) 15:20, 22 August 2025 (UTC)[reply]

Maybe something showing the compactness of the set, as it can be homeomorphically embedded in R^2 (or even R^1), but not much value in a picture attempting to show some particular value. Tito Omburo (talk) 20:11, 22 August 2025 (UTC)[reply]

I think the first source does discuss how 10-adics can arise in education, with an anecdote about how this arose in an elementary school class in which the teacher was stumped by the apparent fact, raised by a student, that ...999 = -1, in a discussion about 0.999... = 1. I think the last paragraph of the section is undue weight/fringe. Tito Omburo (talk) 20:11, 22 August 2025 (UTC)[reply]

Besides JSTOR 2309468 and doi:10.1080/07468342.1995.11973659 cited already, here are a couple other sources: doi:10.1007/s00591-022-00322-1; doi:10.1111/j.1949-8594.1992.tb15623.x; doi:10.1080/07468342.2008.11922313; CORE output ID 83041701 (doi:10.17877/DE290R-17740), p. 134. –jacobolus (t) 20:38, 22 August 2025 (UTC)[reply]

Yes, I think these support relevance to the article, but (based on the previews I've seen), also support the last paragraph as undue weight (although the reference – your JSTOR link – should be retained to support the overall link with the subject of the article). Tito Omburo (talk) 20:54, 22 August 2025 (UTC)[reply]

Agreed. It may also be worth mentioning the Method of complements or two's complement in this section. –jacobolus (t) 22:27, 22 August 2025 (UTC)[reply]

Issues with using algebraic manipulations

x=0.999...

10x=9.999...

10x=9+0.999...

10x=9+x

9x=9

x=1

Although the approach is subject to debate, it continues to be widely used.

From my perspective, although the criticisms may all be valid, the approach remains fundamentally flawed.

First of all, multiply by 100(10²).

x=0.999...

100x=99.999...

100x=99+0.999...

100x=99+x

99x=99

x=1

Then multiply by 1000(10³).

x=0.999...

1000x=999.999...

1000x=999+0.999...

1000x=999+x

999x=999

x=1

And keep going(10ⁿ , n:positive integer, n>3).

It seems intuitively correct when it's 10ⁿ.

But what about when it's 2? What about 3? What about 4?...

While it seems intuitively correct for certain values(10ⁿ,n:positive integer), no one has verified whether it holds for others(2,3,4,...,8,9,11,12,13,...,98,99,101,...).

As I see it, 0.999...=1 is valid only if the following criteria are met(when using Algebraic arguments).

x=0.999...

p*x=p*0.999..., p: integer or real number, p≠-1,0,1

p*x=(p-1).999...

p*x=(p-1)+0.999...

p*x=(p-1)+x

(p-1)*x=(p-1)

x=1 Leo92kgred (talk) 08:55, 20 October 2025 (UTC)[reply]

It only needs to work for one value (for example, 10). — xo Ergur (talk) 09:50, 11 December 2025 (UTC)[reply]

I wouldn't call the approach the approach fundamentally flawed (hence it remains in use), but it is somewhat problematic as it relies on hidden assumptions (computation rules for infinite digits/sums), which are usually not known (or are not formally introduced) to people to which the proof is presented (say mid schoolers or high schoolers) and the hidden stuff is already introduced you already know that the equation is true and the algebraic proof is just rewriting stuff you already know. However from a less rigorous perspective the proof might still convincing and sort of provides a stepping to properties you want to preserve when moving to rigorously dealing with infinite digits/sums. It is a bit like historic incomplete/fallacious proofs by historic mathematician that lead to correct result and which relied on hidden assumptions that still needed formalization and proof (and potentially some additional conditions/restrictions) from a modern perspective..--Kmhkmh (talk) 12:54, 11 December 2025 (UTC)[reply]

"0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999" listed at Redirects for discussion

The redirect 0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 December 14 § 0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 until a consensus is reached. NebY (talk) 17:40, 14 December 2025 (UTC)[reply]