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 0.000 … 1 10 {\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 0.999 … = lim n → ∞ 0. 99 … 9 ⏟ n {\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 0. x 1 x 2 x 3 . . . {\displaystyle 0.x_{1}x_{2}x_{3}...} must be less than 1. y 1 y 2 y 3 . . . {\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 lim x → 0 + f ( x ) {\displaystyle \lim _{x\rightarrow 0^{+}}f(x)} , the right limit of x (which can also be expressed without the "+" as lim x ↓ 0 f ( x ) {\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, 0. ( 9 ) < 0.9 ( 9 ) < 0.99 ( 9 ) < 0. ( 99 ) < 0.9 ( 99 ) < 0. ( 999 ) < 1 {\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.

Moved to Arguments subpage

This is a revival of the discussion Talk:0.999.../Archive 20#Greater than or equal to. Ping to previously interested parties: @Tito Omburo:, @D.Lazard:, @Mr swordfish:, @Trovatore:

The lead says (at the moment):

Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than every number in the increasing sequence 0.9, 0.99, 0.999, and so on.

While it is true in a sense, it may be read as implying (or relying on) the following "standard rule for representing real numbers in decimal notation":

The value of the decimal 0.abcdefgh... is the smallest number greater than every number in the sequence 0.a, 0.ab, 0.abc, and so on.

which is incorrect. It is (and should be, in my opinion) fixed by writing the following instead:

Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on.

I see no serious reason not to do this. "Greater than or equal to" may seem slightly more complicated than "greater than" (though mathematicians may beg to differ on that), but it actually strengthens the argument, and would be less confusing for a reader who happen to speculate "How would that work for a decimal like e.g. 0.25?" Such a reader might conclude the reasoning in the current lead is unconvincing.

One might say the lead does not need to reason at all; it just need to sitpulate facts (borne out by sources and by the article in its entirety). If so, we should leave the reasoning out, saying, e.g.:

According to the standard rules for representing real numbers in decimal notation, its value is exactly 1.

In my suggested version I left out the word "increasing". I'm not sure it had a function, but I may miss something here.

I've made this change more than once, but been reverted, or it has been edited out - therefore this post.Nø (talk) 11:59, 19 January 2026 (UTC)[reply]

Which part is "incorrect"? 0.999... is provably strictly greater than every number in the sequence 0.9, 0.99, 0.999, .... I don't believe in the existence of your hypothetical reader who has no trouble with the phrase "greater than or equal to" but finds the phrase "greater than" to be so confusing that they can't continue reading beyond it. –jacobolus (t) 16:33, 19 January 2026 (UTC)[reply]

"Greater than or equal to" is only necessary in the special case of a terminating decimal when expressed as an infinite string of zeroes. I don't see the need to add unnecessary verbiage to address that special case, and find it distracting.

I do think that it's worth keeping the "greater than part" as it explains the "standard rule" being cited. Mr. Swordfish (talk) 17:29, 19 January 2026 (UTC)[reply]

• My view continues to be (I think I've said this before but I don't see it in the linked discussion) that we should say "greater than or equal to" because it corresponds to the correct general rule, but we should not say why. Going into why would definitely be distracting. Some readers may wonder why we say "or equal to", and those readers should figure it out for themselves. That exercise may be the greatest value they get from this article. --Trovatore (talk) 20:37, 19 January 2026 (UTC)[reply]

This is not the second part of the sentence that can be confusing, but the first one, which refers to vaguely specified rules. I suggest to change the sentence into: Following the definition of non-terminating decimals, its value is the smallest number greater than every number in the increasing sequence 0.9, 0.99, 0.999, and so on. This change would prevent readers to try to applu the sentence to terminating decimals. So "greater than" would be always correct. "Greater than or equal to" would be also correct, but, in the lead, simpler is better. D.Lazard (talk) 22:07, 19 January 2026 (UTC)[reply]

No, I don't agree with that. The interpretation of decimal representations doesn't need a special case for "terminating" decimals. Really we should de-emphasize the whole notion of "terminating decimals"; they're really just infinite decimals that happen to have a final segment consisting of zeroes. --Trovatore (talk) 22:20, 19 January 2026 (UTC)[reply]

I think we are closer to consensus on "greater than or equal to" than on "greater than" (which is fewer words but mathematically the less simple concept, and in disagreement with the general meaning of decimal strings even if it holds in this case). I will now change it one more time. Nø (talk) 11:52, 28 January 2026 (UTC)[reply]

PS. I also, again, removed "increasing" in "increasing sequence" - I don't see any need for it. All in all, I've increased the wordcount by 2, while reducing the character count by 1 and the letter count by 2, and sharpening the logic. Nø (talk) 11:56, 28 January 2026 (UTC)[reply]

This article was the subject of a Wiki Education Foundation-supported course assignment, between 13 January 2026 and 17 April 2026. Further details are available on the course page. Student editor(s): Hasan Paul (article contribs).

— Assignment last updated by Hasan Paul (talk) 17:31, 7 February 2026 (UTC)[reply]

... compare infinite series to real numbers which is an oversight and cannot result in equivalence. Others require us to suspend our knowledge of advances in mathematics in the past 300 years post Euler. In his time this was accurate, however algebra cannot handle infinity properly which was addressed around 100 years after Euler’s passing. Your page doesn’t address either of these assumptions when making claims of proof which continues to give rise to the debate. Either claim your limited universe or replace equivalence with convergence.

Assumptions are not declared ~2026-13994-61 (talk) 12:54, 4 March 2026 (UTC)[reply]

@MartinPoulter and SamuelNelsonGISP: Article currently reads:

The three dots represent an infinite list of "9" digits.

I agree that this wording is better than the suggested

The ellipsis represent[s] an infinite continuation of the numeral "9".

because you can't "infinitely continue" a numeral. I prefer the word "ellipsis", but I can see that that would be unfamiliar to some readers. On the other hand, "infinte list" seems inappropriate. Maybe "infinite sequence"? --Macrakis (talk) 20:53, 30 March 2026 (UTC)[reply]

I updated it to Ellipsis instead of three dots because three dots is ambiguous and non-standard terminology; this is not the simple English wiki, just the English wiki, so confusing someone on a niche math article shouldn't be a high concern. Also, even the ellipsis article has a section about this mathematical notation: Ellipsis#In mathematical notation. Also see Repeating decimal#Background for another article that explicitly uses the term ellipsis in another similar explanation.

I am far less sure about how to word the "infinite continuation of the numeral "9". Digit makes more sense than numeral, thanks for the correction. I'm not sure I like sequence or list though.

Perhaps it can be rewritten:

0.999... is a repeating decimal that is equal to the number 1. The ellipsis represents an infinite succession of the digit "9".

I still like the word continuation more than list or sequence but this paywalled scholarly article seems to describe a repeating decimal of this nature as an infinite succession. SamuelNelsonGISP (talk) 21:18, 30 March 2026 (UTC)[reply]

I replaced represent in the first sentence of my suggestion with equal (because it is equal) and represents makes it seem as if its just some shorthand or alternative and not a completely valid and distinct way to represent the same value. I guess it is like saying 0.5 represents 1/2 instead of saying 0.5 is 1/2, is meaning equal to. SamuelNelsonGISP (talk) 21:20, 30 March 2026 (UTC)[reply]

I had to click on the three dots link to understand what was meant by "ambiguous". While "three dots" can mean many things, I don't think there is any ambiguity here. It obviously refers to the three dots in the expression being referenced.

Sequence is the standard mathematical term. We should use that unless there's a good reason not to. Mr. Swordfish (talk) 21:26, 30 March 2026 (UTC)[reply]

Fair enough, me saying its ambiguous is imprecise. But three dots is clearly not the most accurate description of the notation. Sequence is a standard term but it generally means a comma separated list as you can see when the sequence of digits for pi is depicted in that article, not as the normal decimal expansion of pi, but as a parenthetical list of digits. I think if we are talking about the digit "9" repeating, we should not use the term sequence. SamuelNelsonGISP (talk) 21:32, 30 March 2026 (UTC)[reply]

Sequence is a standard term but it generally means a comma separated list...

The delimiter used to express a sequence is irrelevant to the basic notion of "sequence"; the digits of pi, for example, are a sequence regardless of whether expressed as a comma delimited list, the usual way of writing its decimal expression, or any other way of representing it.

As explained at sequence:

In computing and computer science, finite sequences are usually called strings, words or lists, with the specific technical term chosen depending on the type of object the sequence enumerates and the different ways to represent the sequence in computer memory. Infinite sequences are called streams.

Sequence is the mathematical term. List and string are computer terms. Stream is an infinite sequence in the computer world, but I wouldn't advocate using that here. It's a mathematical article so lets use standard mathematical terms. Mr. Swordfish (talk) 21:59, 30 March 2026 (UTC)[reply]

Well, we're at an impasse over the term sequence. I'm curious about outside opinion. Maybe a more extension analysis of academic or pedagogical literature is in order to help determine which terminology is apt to be used in this case. SamuelNelsonGISP (talk) 22:31, 30 March 2026 (UTC)[reply]

Well, we're at an impasse over the term sequence.

This thread is barely two hours old. Way too early to declare an impasse. Mr. Swordfish (talk) 01:40, 31 March 2026 (UTC)[reply]

lol are you critiquing my use of impasse. I guess all I meant is both of us made our points and we're not budging, hence I call to hear out others. SamuelNelsonGISP (talk) 02:38, 31 March 2026 (UTC)[reply]

Perhaps we could say "standoff" instead? Hawkeye7 (discuss) 04:29, 31 March 2026 (UTC)[reply]

"List" is used in computing because it is an ordinary plain English word, which in relevant non-technical contexts is much more common than "sequence". (In non-technical contexts "sequence" is more often used in the sense of a "sequence of events" or sequence of temporal steps or stages of some process.) –jacobolus (t) 22:31, 30 March 2026 (UTC)[reply]

While my preference is for "sequence", I have no problem with using an ordinary plain English word such as "list" for a general interest article like this.

I don't think "infinite continuation" is optimal here. Mr. Swordfish (talk) 01:35, 31 March 2026 (UTC)[reply]

In case it's not clear though, I also think sequence would be more or less fine. I don't think readers will have excessive difficulty with either word. –jacobolus (t) 02:36, 31 March 2026 (UTC)[reply]

"infinite sequence" is the usual mathematical term, so I gravitate towards that over "list". There's an argument to be made to use the less rigorous term to use more familiar words, but I don't think "sequence" it that an obscure a term.

As for "elipsis" vs "three dots", I can't say that I have a strong opinion either way. Neither is a mathematical term, but "three dots" may be understandable by more readers. Mr. Swordfish (talk) 21:19, 30 March 2026 (UTC)[reply]

I agree that "three dots" is more likely to be understood by the intended audience of this article, who are... the mathematically unsophisticated. And if there is one thing the mathematically unsophisticated do not understand, it is the word "infinity" (or "infinite"). So "infinite sequence" communicates absolutely nothing to them. I suggest it is much better to write "unending sequence", because this is technically certainly not wrong, and makes it easy to reply to objections about what happens at the end of the sequence by pointing out that it does not have an end. Imaginatorium (talk) 05:39, 31 March 2026 (UTC)[reply]

Three dots and unending sequence works for me. Mr. Swordfish (talk) 12:08, 31 March 2026 (UTC)[reply]

You should both look briefly through the history of the page. 6 months ago, it said "The three dots represent an unending list of '9' digits." People keep regularly twiddling the wording one way or another. Before settling on a change it might be worth doing a survey of the past variations of the first few sentences and pinging all of the editors who have made changes to them. –jacobolus (t) 16:15, 31 March 2026 (UTC)[reply]

Well, if people keep twiddling with the wording, we're never going to settle on the language.

Might it be worth mooting the various recent versions and doing an RFC to see if we can reach consensus on one of them? Mr. Swordfish (talk) 17:04, 31 March 2026 (UTC)[reply]

Both "infinite sequence" and "infinite list" seem fine to me. The comp-sci distinction called out above seems not quite right to me. As I understand it a comp-sci "stream" is not so much an infinite sequence as a process from which you can keep getting data without limit. The data you get may not be deterministic (for example, it might come from stdin or from /dev/random) and you never have an infinite amount of data at any particular time. --Trovatore (talk) 01:00, 1 April 2026 (UTC)[reply]

Agree with your critique of the usage of the term "stream" in the sequence article. Of course, this is not the venue for addressing that article's shortcomings.

Also agree that both "infinite sequence" and "infinite list" seem fine. Mr. Swordfish (talk) 02:37, 2 April 2026 (UTC)[reply]

After reading Imaginatorium's contribution above, I agree that "unending sequence" might be better for the intended readership of this article. --Trovatore (talk) 01:14, 1 April 2026 (UTC)[reply]