Talk:0.999...

Definition of 0.99999...

Moved to Talk:0.999.../Arguments § Definition of 0.99999...

 – Hawkeye7 (discuss) 19:37, 25 April 2026 (UTC)

Wiki Education assignment: Proofs and Problem Solving

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 Aynelson15 (talk) 06:33, 27 April 2026 (UTC)

Bloat

Does anyone agree with me that this article is bloated?

It's hard to find anything specific to call out, so it's hard to say how I would want to fix it. There's no one thing where I can say, just get rid of that. Generally, everything looks true, probably reasonably well-sourced though I haven't personally checked the sources. There's nothing spectacularly out of place the stuff about the Cauchy sequences and Dedekind cuts are pretty out-of-band for the "intended audience", but I don't see how we can just not talk about these things. But if you just read the article, it's hard to find a mental model of a reader who finds the equality of 0.999... and 1.000... to be something worth reading about, who then reads the article, and is enlightened.

Mostly it's just too long.

I think it would be great if someone would take it as a project to propose a much-tightened rewrite, hitting the basic points but in a way that doesn't put people off by the sheer quantity of text. I suppose I should probably do it but I'm just not excited about it. Anyone? --Trovatore (talk) 19:23, 28 April 2026 (UTC)

The article has been like this for twenty years and has scarcely increased in size in that time. It follows the standard layout of mathematical articles in moving from the simple to the more complex for the benefit of increasingly sophisticated readers. With a prose size of 5,705 words, WP:SIZE says length alone does not justify division or trimming. I would strongly oppose any attempt at rewriting. Hawkeye7 (discuss) 19:46, 28 April 2026 (UTC)

It's certainly true that it's not too long for an article generally. I do think it's too long for an article on this topic. There's just not that much to say about it, but we say it anyway. I don't think that's useful for our readers. --Trovatore (talk) 19:54, 28 April 2026 (UTC)

My take is that the article could probably be trimmed, but since most readers won't read past the lead I'm not motivated to sharpen my editing pencil.

As for the lead, I don't think the final paragraph adds much, and the would be improved by removing that unnecessary and potentially confusing material. Mr. Swordfish (talk) 18:10, 23 August 2026 (UTC)

At 5,700 words, it comes nowhere near WP:SIZE. Agree with Swordfish about the value of trimming, and would support removal of the last sentence from the lead. Hawkeye7 (discuss) 19:46, 23 August 2026 (UTC)

Reference to WP:SIZE is a complete non-sequitur here; has nothing whatsoever to do with my point. --Trovatore (talk) 20:17, 23 August 2026 (UTC)

I was commenting on the last paragraph, not just the last sentence. In particular, I don't think we need to go into alternative number systems in this article, especially in the lead. Both Infinitesimal and P-adic number have their own articles, and the contentions/misconceptions are about the reals, not more esoteric mathematical objects.

I've added links to the main articles listed above, but we should give consideration to either remove or trim redundant material from the other number systems section. Mr. Swordfish (talk) 21:24, 23 August 2026 (UTC)

I think we do need to mention alternative number systems. In a very few words, this article is saying "0.999... is exactly equal to one, always, get over it!" -- and simplified like that, it is simply not true. We need to mention briefly that there are exotic contexts where it is not true, and in my opinion, that belongs in the lead -- but does not need elaboration. (Alternativelt, in the lead, we could merely state (and wikilink) the context in which it is true, and leave the exotic contexts as a footnote or e.g. as a section near the end.) Nø (talk) 07:49, 24 August 2026 (UTC)

I'm not that opposed to a brief mention of other number systems, but we don't need the level of detail provided in the "in other number systems" section, which repeats much of the material at the main articles.

Further, the final paragraph contains two distinct ideas which are not closely related: other number systems and positional notation always has multiple representations. Since these are separate ideas, I'd suggest moving the latter to the second paragraph and condense it to a single sentence.

As for "other number systems", we don't explicitly state that the context here is the real numbers. We probably should, so the reader can understand what "other" refers to. My preference would be to state that the article is about the real numbers and leave "other number systems" out, but that's not the current consensus. Mr. Swordfish (talk) 15:04, 24 August 2026 (UTC)

I edited the lead for

• Mentioning that everything remains true in other bases with 9 replaced with the largest digit (this was lacking, although I guess that every reader should know the binary number system).
• Removing the last sentence, which is wrong: the decimal number system with infinite sequences of 9 excluded is a valid positional number system in which every real number has a unique representation.
• Reordering some sentences.

I did not read recently § In other number systems. So, for the moment, I have no opinion on this section. D.Lazard (talk) 17:14, 24 August 2026 (UTC)

Per above discussion, I agree to remove § In other number systems and to replacethe current last sentence of the lead with

There are ordered number systems, such as surreal numbers, that contain the real numbers and in which the least upper bound represented by 0.999... either does not exist or differs from 1.

D.Lazard (talk) 18:36, 24 August 2026 (UTC)

> Per above discussion, I agree to remove § In other number systems

Are there any objections to this? Mr. Swordfish (talk) 15:12, 27 August 2026 (UTC)

I don't think the whole top-level section should be removed. Richman (1999) "Is 0.999... = 1?" is explicitly on topic, as is Katz & Katz (2010) "Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era". Some discussion of hyperreal numbers and surreal numbers seems valuable to include in this article. I'm not sure about p-adics. The whole section could probably be shortened somewhat though. We should try to stay both comprehensible to readers unfamiliar with the topics discussed and also on topic.

Did you try to figure out who wrote the text of these various sections, and ping them explicitly? –jacobolus (t) 16:15, 27 August 2026 (UTC)

Looks like @Melchoir wrote a bunch of this material in 2006. They might have thoughts about this, or know which other authors were contributing at that time and might be worth pinging. –jacobolus (t) 18:03, 27 August 2026 (UTC)

Yes, thanks for pinging me, since I don't have the article on my watchlist! I did write the section on other number systems. In brief, I don't support removing it. I think it serves an important function for readers with mathematical curiosity. And of course, it serves the encyclopedic purpose of summarizing what other authors have written about the topic. I mean, I might have written the section, but isn't my personal WP:OR flight of fancy! It's genuinely what a certain kind of mathematician writes about when they write about 0.999...

I don't necessarily have an objection to tightening the prose, where possible. For example, the second paragraph of the p-adic subsection is just a recap of what the p-adics are. It's nice to draw a parallel with the discussion of Cauchy sequences earlier in the article, but it's not strictly necessary to introduce the topic. So that paragraph could probably be cut, trusting the reader to follow the links if they want to learn more. (On the other hand, the other paragraphs should not be cut, because they cover material that isn't available in other articles.) Melchoir (talk) 20:44, 27 August 2026 (UTC)

It looks to me that § In other number systems could perhaps be condensed, but we shouldn't cut the whole thing. And if we did cut the whole thing, we should remove the corresponding sentence from the intro, since there would be nothing for it to summarize. So, I disagree with this proposal in both directions. Stepwise Continuous Dysfunction (talk) 17:48, 27 August 2026 (UTC)

I think the padic numbers should remain, as it shows a related phenomenon in other number systems, clearly germane for the topic of the article. Sławomir Biały (talk) 17:52, 27 August 2026 (UTC)

Historical Perspective: 0.999... as a variable expression

While there is no doubt that today 0.999... (and other representations of the recurring decimal) is a notation that represents a fixed real number, which can also be represented simply as 1, I don't believe this was always the case. In early (late 19th, early 20th century) algebra textbooks, you see examples when introducing the concept of limit that imply that recurring decimal notation was, at that time, a representation of variable expression, not a constant one.

Concrete links to follow, but a prototypical example would be a definition of a limit as, "if a variable becomes and remains as close we please to a fixed value L, we say that the limit of the variable is L and write $\lim x=L$", followed by a number of examples among which might be: $\lim 0.333...={\frac {1}{3}}$.

With the definitions and notations established in such texts, this only really makes sense if we interpret the recurring decimal as a variable expression. One could argue that a constant is a special type of variable that only takes one value, and so also has a limit equal to that constant value, but I don't think textbooks would use such a redundant case in an example. And besides, there are sometimes further discussion which lend credence to this interpretation: talk of expressions "never equalling" their limiting value, but instead differing by "an infinitesimal" (which is defined as a variable whose limit is zero).

This does in fact conform to an interpretation of recurring decimal notation as a "process" rather than a representation of a fixed number. Mathematically, this is just a matter of choice of notation, but pedagogically it is different. And more significantly, it seems to have been a historical fact. I would like to edit the article to reflect this but don't want to either a) be mistaken for someone arguing that the modern convention interprets 0.999... as anything other than 1, and b) muddy the waters on an article that is already the subject of much controversy, so would like opinions to get a consensus. I also am not sure whether this belongs on this article or the one on repeating decimals, which seems to avoid technical discussion of things like limits.

Here are some examples:

• https://archive.org/details/firstcourseinhig00merrrich/page/54/mode/2up (clearest example at the bottom of p55. Over the page the argument continues: "$\textstyle {\frac {1}{3}}-x<k$ can be satisfied by taking a sufficient number of decimal places for $x$" implying that the expression may take on the value of any finite decimal approximation.
• https://archive.org/details/advancedalgebra01collgoog/page/n208/mode/2up?q=repeating
• https://files.eric.ed.gov/fulltext/ED622306.pdf p484 Exercise 7, 8

Fish-Face (talk) 13:45, 6 May 2026 (UTC)

This is very interesting. If these examples are typical of the period then it probably does deserve some coverage in the article. As I've mentioned, I think the article is already too long for the rather trivial subject matter, so it needs to be done carefully and reservedly.

It's interesting to see the way the nomenclature used in old texts reflects foundational conceptions. I learned a lot of math ahead of my school schedule from my dad's old texts. I remember it would describe, say, ellipses, as "the locus of a point" that satisfied a certain equation. I never quite knew why they used "locus" in this way. I found out only fairly recently that apparently curves didn't use to be thought of as sets of points at all. Exactly what a "locus" was in their view I'm still not sure, but this did give some insight into the choice of wording. --Trovatore (talk) 19:52, 7 May 2026 (UTC)

I can't see that this would help with this article; after all, it is a curious thing, not 100% genuinely encyclopedic, because it is really about remedial maths education. FWIW, I did maths in the 1960s, both school and university in England, and find "locus" absolutely normal. I think I thought of it as a "path", rather than the "position(s)" which might be a direct translation of the Latin. But in any event, the locus is obviously the (set) of points that satisfy the equation, whether (set) is just the ordinary word referring to a collection of things, or the "set" of set theory.

But back to the reference (I'm looking at Merrill and Smith): it seems to me they are simply identifying the unending decimal expansion with the sequence of initial parts thereof, as in (0.9, 0.99, 0.999, 0.9999, ...). But some of what they write seems to me to be not very good: the definition of "limit" talks about x "taking on a series of values", rather than talking about the limit of f(x) as x goes to [something]. So some of the wording comes over oddly: saying that if x = .333..., then lim x = 1/3. Could one say that lim 0.5 = 0.5, or only lim 0.5 = 1/2? Anyway, these are terminological oddities from the distant past; if they belong anywhere, it is in an article on the development of calculus, and you would need sources, where people have studied the history, and analysed the progression of ideas and terminology. I fished out my analysis textbook (Tom M. Apostol) from a mere 40 years after M&S, and Chapter 4, The Limit Concept mentions the limit of a sequence, and the limit of a function. There is no "limit of a variable". And even more so, it is crucial to avoid the idea that any of this is a "process". Imaginatorium (talk) 07:17, 8 May 2026 (UTC)

I would add that it's not really clear from the excerpts what the meaning of the sequence of symbols "0.999..." was. If there is a clear statement somewhere, then maybe it could be added, but I am reluctant to infer meaning from a (potentially sloppy) use of notation only. Sławomir Biały (talk) 07:20, 8 May 2026 (UTC)

@Sławomir Biały I have not been able to find any *formal* treatment of recurring decimals other than those we are familiar with. Unfortunately this does leave some room for interpretation; indeed one can read in older texts some disturbing things, like "0.111... = 111.../1000..." in Robinson's "The Theory of Circulating Decimal Fractions". Before the 19th century we just don't have the same level of formalisation. But this could be useful for describing how informal intuition evolved to become rigorous. Fish-Face (talk) 08:49, 8 May 2026 (UTC)

@Imaginatorium if I think back to my own schooling, in fact the concept of a function was secondary to that of a variable: we were first taught what it means for y to equal 3x, for example, before carrying this as an example of a function and using functional notation. I believe we are seeing this in these textbooks. If you read Cauchy, you will also see similar language (he defines "infinitesimal" in the exact same way, as a variable whose limit is zero)

To my modern eyes, this treatment seems sloppy, too, and raises questions like you have asked. I suspect high school textbooks were just behind the times compared to undergraduate texts like Apostol.

The attraction of inclusion is that it could show a development of mathematics in which, formally, recurring decimals are treated differently than their "limits" but unlike non-standard analysis, actually seems to have been used and taught.

As for the article, is it your view that the textbooks alone are insufficient as sources and you'd prefer a source that discusses the historical progression of teaching? Unfortunately I wouldn't know where to start there. Is it your view that inclusion would be harmful to the overall point, even if justified with sources? Fish-Face (talk) 08:43, 8 May 2026 (UTC)

My point is very narrow: only that I think this would not improve this (really quite odd) article. There is some discussion above as to whether it is too long - after all it is the recitation and proof of a humdrum simple fact, and is long precisely because it will inevitably attract crank arguments. ("Crank" is a technical term, not just abuse.) More general material like this would not help the cranks, and other people would mostly never read it because they assume the article is addressed to cranks. Imaginatorium (talk) 12:17, 8 May 2026 (UTC)

I do see the point, but I think it's not really possible to make hay out of it without understanding what people were doing (or thought they were doing). Maybe these algebra texts provide some clue earlier-on in the text. Just at a glance, they seem to have just put "lim" in there because they knew a limit had to be involved somehow, but just kind of copped out. They might be useful as a kind of missing link between old and modern views; Goursat had just been published a few years prior. Euler seemed quite happy to just sum a geometric series and regard it as self-evident what it meant. The (amusing) example of Robinson is incomprehensible in my opinion. Secondary sources at least tracing how the notion of limit influenced the understanding of decimals (or geometric series) might be helpful. Sławomir Biały (talk) 12:37, 8 May 2026 (UTC)

I think the example from Robinson is rather instructive, in that the only way it makes sense to me is if you view all three components as being variable expressions which take on the "obvious" values in succession.

There is an earlier confirmatory statement in the Merrill book: https://archive.org/details/firstcourseinhig00merrrich/page/8/mode/2up?q=decimal (quote: "the [repeating] decimal is never exactly equal to the common fraction") which I think makes it clearer. Use of temporal language ("never") indicates an understanding of the notation as representing some kind of varying quantity. Fish-Face (talk) 13:47, 8 May 2026 (UTC)

Never is not necessarily "temporal language". It can also be used in the sense of "under no circumstances", e.g., "A bachelor is never married." Or, "You will never find a more wretched hive of scum and villainy." Stepwise Continuous Dysfunction (talk) 17:16, 8 May 2026 (UTC)

If some curve is a "locus of points" or "locus of a point" satisfying some property, that means that points satisfying the property are on the curve, and every point on the curve satisfies the property. There's sometimes a bit of hint of motion to the concept, as the basic tools of geometry, starting with the compasses and straightedge, draw a circle or a line segment (i.e. pull a marking tool along the curve) rather than just plotting down one point at a time. –jacobolus (t) 07:36, 8 May 2026 (UTC)

Certainly true, but I have heard the same thing as @Trovatore with regard to loci, and that an even simpler example is that of a line: classical Euclidean geometry is two-typed: there are points, and there are lines. We can speak of a point being on a line, and so we can speak of finitely many points being on a line, but there is no language in the classical theory to express that a line is made of points. I think Greek geometers, and mathematicians for centuries since then, thought of lines and points as wholly distinct.

Indeed, Euclid's Element's doesn't talk about numbers per se, and so areas and lengths are not measured by a common abstract entity, so there is a difficulty in comparing them to each other. (But no difficulty in comparing length to length). To compute an area was to produce a square of the same area. I don't know if this is getting too far from the topic but it may help illustrate the evolution of mathematical thought in other areas. Fish-Face (talk) 12:06, 8 May 2026 (UTC)

I would not be inclined to read too much into the details of the phrasing in these examples. Inferring what people actually think, based on language that may itself be poorly considered and not crafted with regard to its implications, is not easy. It is also outside the scope of Wikipedia. We could only write about this if some secondary source like an article in a peer-reviewed mathematics education journal looked at old textbooks and evaluated what their language implies. Stepwise Continuous Dysfunction (talk) 17:14, 8 May 2026 (UTC)

Surely a textbook is unlikely to be a primary source; it's not that textbook authors were developing these definitions themselves. But I take your point that it would be better to have a historical survey.

Digging through a few more chapters of textbooks, I found a very clear statement on p176 of https://upload.wikimedia.org/wikipedia/commons/f/f6/Second_course_in_algebra_%28IA_cu31924002937906%29.pdf "The numerical value V of the recurring decimal .666... is a variable depending on the number of 6's annexed on the right." Fish-Face (talk) 19:09, 8 May 2026 (UTC)

Textbooks are certainly not primary sources. They're more likely to be tertiary sources, which are also not ideal for WP sourcing. In this case, though, I think the textbooks could count as adequate sourcing for a passing mention, if we can agree on whether it's appropriate. --Trovatore (talk) 19:19, 8 May 2026 (UTC)

Using a textbook's phrasing to make inferences about the author's mental model is using it as a primary source for the claims being made. A secondary source in this context would be something like a scholarly review that did a systematic analysis of old textbooks' treatment of a topic. –jacobolus (t) 19:37, 8 May 2026 (UTC)

It's not just "better" to have a historical survey, it's obligatory here. The discussion here isn't just using textbooks as references in the ordinary way. Instead, it is doing an exegesis of them. It's more akin to sifting through old homework problems for indications of sexism: turns of phrase are being scrutinized for what they might imply and the connotations they potentially carry. This could be raw material for a journal publication on math pedagogy and its history; maybe there's some consistent pattern of old books talking about "the limit of a variable" in ways that people don't now. But that would be WP:SYNTH. Likewise, looking at what an old book says about the notation ".666···" means and using that to infer what the same authors would have said about the notation ".999···" is drawing a conclusion that the source does not itself give, and so not within the remit of what Wikipedia can do. Stepwise Continuous Dysfunction (talk) 19:33, 8 May 2026 (UTC)

This argument strikes me as...technically correct, but a bit legalistic. However I also don't think the value-added from discussing the interpretation from these old textbooks is large, so I'm not inclined to argue the point. --Trovatore (talk) 20:18, 8 May 2026 (UTC)

For the record, I think interpreting 0.999... as a sequence (0.9, 0.99, 0.999, 0.9999, 0.99999) that has a limit (1), rather than as a notation representing one number, is close to the thinking of many students who have issues with "0.999 = 1". We - or rather, some mathematicians in the not so distant past - have decided that 0.333... = 1/3 (i.e., that 0.333... represents the limit of the implied sequence, not the sequence itself), and hence 0.999... = 1. Nø (talk) 07:57, 24 August 2026 (UTC)

i dont care about the 0.999 = 1 arguments but this article is written terribly

it sounds like a mildly passive agressive, half-angry teacher making a lecture on 2010 wikipedia, like this literally reminds me EXACTLY of how it looks when you look at old wikipedia archives, the article it very argumentative rather than academic, personally i dont think that it should be written in the exact way that it is, however regardless of the whole qutoe on quote "debate" or debate about it being a debate, or debate about having a debate about it being a debate and so on, i think that then if we consider that it is written as concrete 100% truth that 0.999 = 1, then it should at least be done in an encyclopedic, educational and proper tone. LoganStylus (talk) 03:12, 23 August 2026 (UTC)

The proper academic article on the subject would be Repeating decimal. This article is a slightly oddly named repository of arguments, about misunderstandings, and how to counter them. I personally think that is OK - but if you think the tone is bad, change it, or give specific examples you'd like the community to work on. Nø (talk) 10:42, 23 August 2026 (UTC)

see i feel like then the article should be moved to Counterarguments against 0.999 = 1 or something like that LoganStylus (talk) 17:13, 23 August 2026 (UTC)

Why? It has many proofs of the identity, and no counterarguments. Sławomir Biały (talk) 17:14, 23 August 2026 (UTC)

"0.999. . . = 1" is the statement the article is about, apparently because it is a statement discussed in reliable sources. So, what would it be, besides examinations and analysis of that statement and arguments concerning that statement? A "tone" objection is not very well made as a general objection, because it fails to identify which wordings are being objected to. Such generalized tone arguments read more like objection to the topic entirely, and if your objection is to the topic, "0.999. . . = 1", see WP:AFD. Alanscottwalker (talk) 12:46, 23 August 2026 (UTC)

I feel the urge to point out here that no one claims 0.999 is equal to 1. Well, almost no one, anyway. --Trovatore (talk) 21:26, 23 August 2026 (UTC)