Talk:0.999.../Archive 20

Numbers and numerals
I just reverted a good faith but confused edit which wanted to say that 0.9recurring and 1 are "quantitatively" equal. It seems to me that this is a variety of the confusion of numbers with numerals; another edit recently basically claimed that 0.9recurring and 1 are different because they are different numerals, with a claim that fundamentally numbers are actually (misunderstanding of) numerals. I wonder if it would be helpful to add a paragraph pointing out this distinction? Imaginatorium (talk) 13:26, 10 September 2023 (UTC)


 * The difference between a number and a numeral is a basic concept that few people seem to grasp. The new math that was common in elementary schools in the sixties presented that concept to young students and it's a simple idea. Unfortunately, it no longer seems to be part of the curriculum.
 * I think a brief treatment of that difference would be an improvement to the article. Something like
 * A numeral is a symbol that represents a number, and for any given number there are multiple ways to express it symbolically. For example: 5, V, 2+3, 101(base 2), 4.9999.... are all representations of the same number.
 * I don't think it will prevent the weekly drive-by edit from someone insisting that 0.9999... is not really equal to one, but it might be helpful to the general reader. Mr. Swordfish (talk) 14:59, 10 September 2023 (UTC)
 * The problem "numeral vs. number" is a special case of "expression vs. object represented by the expression", and even "syntax vs. semantics". For example, $$(x+y)^2=x^2+2xy+y^2$$ is false as an equality of expressions and true as an equality of polynomials. The first paragraph was confusing on this by using the nonsensical "decimal number". I boldy tried to clarify this by replacing "denotes" by "is a notation for" (this emphasizes that this is a convention), and replacing "decimal number" by "number". Possibly one could add also a sentence like "In other words, 0.999... and 1 are two diferent numerals that represent the same number". D.Lazard (talk) 16:21, 10 September 2023 (UTC)
 * I think that the problem is that merely rewording the initial statement will have no effect. Of course it would be sufficient if readers were moderately mathematically sophisticated, but if they were this page would not exist. However verbosely you word it, the ordinary readers will just pass over; so I think a separate paragraph is essential, something like Swordfish's suggestion. Imaginatorium (talk) 16:51, 10 September 2023 (UTC)
 * Agree that a simple short paragraph would be an improvement. And perhaps trim some of the wall of text that comprises the rest of the article.  I don't have specific edits in mind here, but support the basic idea of a simple treatment of number vs numeral. Mr. Swordfish (talk) 00:46, 11 September 2023 (UTC)
 * Agree that "numeral vs. number" is a special case of "expression vs. object represented by the expression", and even "syntax vs. semantics". But I don't see how that matters. An elephant is a special case of a mammal but that doesn't preclude an entire article about elephants.
 * Here, the stumbling block seems to be number vs numeral, and I don't see any reason to go into generalities about expression vs object or syntax vs semantics. Mr. Swordfish (talk) 00:44, 11 September 2023 (UTC)

How to prove
x=.999... 10x=9.999... 9x+x=9+x 9x=9 So 1=.999... JackJackRR (talk) 21:28, 4 October 2023 (UTC)


 * See the proof in the section 'Algebraic arguments'. Mind  matrix  12:59, 5 October 2023 (UTC)

Stillwell proof
I find starting off with the Stillwell proof in its current form quite counterproductive; the first, non-rigorous explanation is just as hand-wavey as the 10x - x = 9 proof, and far, far, more confusing (what does "no room" mean, informally?); and when the rigorous version is introduced, it's no more or less easy than the other rigorous proofs. I would expect the naive reader to leave this section of the article totally confused, and give up on reading the rest.

Update: I've now demoted the "elementary proof" section to "elementary demonstration", and removed the attempts at partially formalizing it that were making it confusing by smuggling in the concepts of least upper bound and limits without introducing them first. By avoiding premature formalization, I think this now flows much better into the start of the formal argument section. &mdash; The Anome (talk) 12:40, 7 October 2023 (UTC)


 * I see you've reverted my careful changes, in which I've tried to keep as much of the original structure as possible. The "informal proof" is neither informal, nor a proof; it implicitly pulls in things like limits, continuity, the idea of least upper bound, and so on, probably as a result of other editors attempting to tighten up the language. I think the best we can do with this part is to let it be fuzzy and to appeal to intutitions about the number line, and not attempt to improve on it by implicity pulling in more advanced concepts without explanation or discussion. (Just a few examples of defects: the Archimedean property is pulled out of a hat; "0.999..." is not actually defined; without the idea of limits, the reader could argue "there's always a gap, it just gets smaller"; it's not intuitively obvious that two numbers without another number between them must be the same (consider, for example, the integers); and there are more...) Then when the formal concepts are introduced as a lead-up to the rigorous proof, the reader has not been confused by their premature, and unexplained, introduction earlier. Can I suggest that we move to edit this point by point, in a way that can be justified? &mdash; The Anome (talk) 13:52, 7 October 2023 (UTC)
 * Where did you see an "informal proof"? The word "informal" does not appear in the article outside section and your edits. One of your main changes consists of replacing "proof" a word that has a precise meaning by "demonstration", a word without real meaning. Another of your changes introduces a blatant mathematical error: you wrote "if 1 were greater than all of 0.9, 0.99, 0.999, etc.," when 1 is effectively greater than all these numbers. More generally, while the article is carefully written to distinguish between intuitive explanations and mathematical proofs, most your edits amount to confuse them.
 * However, I have just remarked that you may have been confused by the fact the the proof refered to by the heading is not in the introductive paragraph of the section, but in subsection . I fixed this ambiguity. D.Lazard (talk) 14:48, 7 October 2023 (UTC)


 * Restructuring the sections like that helps a lot. It's still not elementary, as it still relies on the introduction of the idea of least upper bound and implicitly the notion of limits, and also the Archimedean property, which is not obvious at all, and are pulled out of thin air. Nor do even those suffice; there's a gaping hole in the assumption that two numbers without another "between them" must be the same; we know this is true for the reals, but this is not an elementary properly, see for example the integers, where 2 and 3 manage to be different without another integer between them. Given all this, why not just introduce Stillwell's informal argument about "not enough space" (which is fine, because it gets the feels right) and then go directly for the Dedekind cut approach, which is both rigorous and explicit? Oh, and just to nitpick your nitpick, when I wrote "if 1 were greater than all of 0.9, 0.99, 0.999, etc.," I had in mind the idea of the least upper bound of the infinite sequence ie. "if 1 were greater than (all of 0.9, 0.99, 0.999, etc.,)" (which it isn't); not "if 1 were greater than (every one of of 0.9, 0.99, 0.999, etc.,)" (which it is). I'm sorry if you didn't understand my careful wording; I should have been more careful. &mdash; The Anome (talk) 10:55, 8 October 2023 (UTC)

Terminating decimals
"Terminating decimal" is a technical term that must be linked in such an elementary article. Previously, it was linked to Repeating decimal. I agree that this is not a convenient target. I have created a redirect, and linked it to an anchor in the lead of Decimal. If this link is not correct, this is not a problem of 0.999... but a problem of the redirect page or of the target page. In any case, the link Terminating decimal must be kept. D.Lazard (talk) 15:09, 27 October 2023 (UTC)


 * Agree. Since this term may be unfamiliar to some of our readers a link is necessary. If  the article that it the target of the link needs improvement that should be discussed/implemented there, not here. Mr. Swordfish (talk) 16:18, 27 October 2023 (UTC)


 * I am a native speaker of English, which is fundamentally the target of WP:en. "Terminating decimal" is not a technical term, at all, it is simply the participle adjective "terminating", which means "it stops", qualifying "decimal". If there really were an article "terminating decimal", a link would be unnecessary, IMO, but not confusing. The "repeating decimal" article is not very good, since the first paragraph tells us that a terminating decimal is not a "repeating decimal", then the second paragraph backtracks, and says that a "terminating decimal" is one where the repeating sequence is just zeros. It cannot help to link a self-explanatory term to this. Imaginatorium (talk) 19:09, 27 October 2023 (UTC)
 * Adjectives need precise definitions in math texts and this is one instance. A decimal expansion is not an event in time or place in space so the English definition does not apply, and is in any case too imprecise. It is very unlikely you are going to get consensus in favor of your view.--Jasper Deng (talk) 19:27, 27 October 2023 (UTC)
 * It seems also that did not notice that "terminating decimal" does not link anymore to Repeating decimal; the target is an anchor in Decimal . D.Lazard (talk) 19:38, 27 October 2023 (UTC)

One more proof
According to the formation rule, the reciprocal part of the number is 9 and the non-revolving part is zero. Accordingly (9-0)/9=1. Please add this.

Bera678 (talk) 19:14, 15 December 2023 (UTC)


 * This is not a proof. Moreover, for being added here, a proof requires to be published in a textbook, and you do not provide any source. D.Lazard (talk) 09:36, 16 December 2023 (UTC)
 * Maybe I didn't fully express what I meant. But I'm sure it's proof. Although this is based on personal research, we can find a reference. Bera678 (talk) 09:44, 16 December 2023 (UTC)
 * It certainly is not a proof. It is merely quoting a rule of thumb for obtaining the value, but the rule of thumb is valid only because there is a proof of it. JBW (talk) 18:30, 16 December 2023 (UTC)
 * Did you look at the formation rule in the 'in compressed form' section of the Repeating decimal article? If you looked you can see that our number is equal to 9/9 to 1. Moreover, this evidence may be more understandable to readers. Bera678 (talk) 12:26, 17 December 2023 (UTC)
 * Yes, that may be the most convincing "evidence" for many readers, but it is not proof, since it hinges on arithmetic algorithms that first should be proven to be valid for infinite decimals. Nø (talk) 13:27, 17 December 2023 (UTC)
 * OK Bera678 (talk) 13:51, 17 December 2023 (UTC)

FAR notice
This is an FA from 2006 that underwent FAR in 2010 and was kept. This article does not currently meet the featured article criteria: Pinging @JBL; I saw your recent FA and hoped you might be able to take a look. voorts (talk/contributions) 02:23, 18 January 2024 (UTC)
 * It uses a mixture of parenthetical referencing, which is deprecated, and inline references, failing 2.c.
 * The "Elementary proof" section is entirely unreferenced, and many other sections have unreferenced paragraphs, some of which appears to contain OR (see, e.g., "Impossibility of unique representation"), failing 1.c.
 * There are weasel words and editorializing throughout and the writing style is at times casual, failing 2.


 * @Voorts I'll deal the citation style. I'm changing to sfnp for all short citations. Dedhert.Jr (talk) 02:31, 18 January 2024 (UTC)
 * The rest of the cites need work too; many of them don't use any citation formats and some of them are ref tags with harvs inside them. Since there are variations in citation style, I think they can all be changed to sfnp for conformity. voorts (talk/contributions) 02:36, 18 January 2024 (UTC)
 * One additional thing: I don't see any kind of thorough source checking in either the FA or FAR discussions. voorts (talk/contributions) 02:40, 18 January 2024 (UTC)
 * @Voorts An additional thing but optional likely. I do think that this article uses many types of math templates, math in TeX, and by simply just using HTML code. So I prefer to use Tex instead, right after completing the citations format problems. Dedhert.Jr (talk) 05:45, 19 January 2024 (UTC)
 * I'm not really well-versed in math templates on Wikipedia, so I can't really opine on what to use, but I agree that using plain html code is not the best. voorts (talk/contributions) 05:56, 19 January 2024 (UTC)
 * Thanks for the ping, . Unfortunately I've discovered about myself that I'm good at starting something more or less from scratch, and good at local spot-checking, but not very good at the kind of work needed here.  I'll try to take a look, though.  --JBL (talk) 20:56, 21 January 2024 (UTC)
 * I am willing to help with this one. Ping me if want help with anything. I will conduct a source check. For the record: I do not see any problem with the casual writing style, given the readership of this article. Hawkeye7   (discuss)  21:20, 21 January 2024 (UTC)
 * "we should convert this into the book being used as a reference (but that would require access to it to see how to use it)" Fortunately, I do. Which is why I said I would look at the sources. Hawkeye7   (discuss)  23:36, 21 January 2024 (UTC)


 * Can you explain what "weasel words" means in this context? An example or two would help... Imaginatorium (talk) 09:38, 22 January 2024 (UTC)
 * Sure, here's a couple:
 * "While most authors choose to define"
 * "Many algebraic arguments have been provided"
 * voorts (talk/contributions) 21:19, 22 January 2024 (UTC)
 * See also WP:WEASEL. voorts (talk/contributions) 21:19, 22 January 2024 (UTC)
 * I don't understand the assertion that either of these is weasel-y. These assertions might or might not be adequately sourced (to be clear: I haven't checked), but if they reflect the sources I don't see what's objectionable about them.  --JBL (talk) 23:10, 22 January 2024 (UTC)
 * "While most authors choose to define" is not in the source, so I have removed it. I'm not seeing support for the assertion "Division by zero occurs in some popular discussions of 0.999..." either. Unless someone can find one, I suggest we remove the entire bullet point. Apart from that sentence though, it is correctly sourced. Hawkeye7   (discuss)  02:10, 23 January 2024 (UTC)
 * @Hawkeye7 @JayBeeEll @Dedhert.Jr: Where are we on this? Has enough been done to fix this, or should this proceed to FAR? voorts (talk/contributions) 22:45, 21 February 2024 (UTC)
 * @Voorts I'm replying. Will trying to convert again as soon as possible, and copyedit; trying my best. Dedhert.Jr (talk) 13:37, 22 February 2024 (UTC)
 * I have converted the format footnotes into sfnp and harvtxt, and all math format in Tex. Dedhert.Jr (talk) 13:36, 23 February 2024 (UTC)
 * I have moved unused references to the Further reading section. Hawkeye7   (discuss)  19:16, 23 February 2024 (UTC)
 * The "Division by zero occurs in some popular discussions..." reads like WP:SYNTH to me (that is, WP:SYNTH dressed up with citations to the background topics being synthesized). There's maybe something to be said about how understanding limits can give a precise meaning to the intuitive idea of "division by zero" (or "division by infinity"), and limits are also important here, but without a source explicitly drawing that connection, we shouldn't include it. XOR&#39;easter (talk) 01:42, 27 February 2024 (UTC)
 * Unless there are any objections, I plan on bringing this to FAR one week from now. voorts (talk/contributions) 03:06, 26 March 2024 (UTC)

Root of student confusion?
Is there any elementary education literature on confusion caused by teaching real numbers in terms of decimal expansions instead of axiomatically or geometrically? I believe that if such an RS exits then the article should discuss the issue. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:55, 14 April 2024 (UTC)

Elementary proof supported by Stillwell?
It strikes me that the Stillwell reference for the section on the Elementary proof is not ideal. Can anyone find a better reference? Tito Omburo (talk) 22:20, 11 April 2024 (UTC)


 * I looked once but didn't have any luck finding a source that spells it out with all the steps that this subsection does. On the other hand, I'm not sure that subsection adds more clarity than it does notation. XOR&#39;easter (talk) 17:29, 15 April 2024 (UTC)
 * Are you satisfied that the section as a whole is well-supported? When I last checked, Stillwell was the only cited source, a situation you have now significantly improved. I'm less worried about whether all the steps are explicitly referenced, and we can cut the last section out if necessary. Tito Omburo (talk) 18:16, 15 April 2024 (UTC)
 * I'm significantly happier with that section than I was. The more I look at the "Rigorous proof" subsection, the more I think we could remove it without loss of clarity (perhaps even of rigor!). One thing that bothers me: the section heading "Elementary proof" is not very illuminating, and the proof is only "elementary" in a rather technical sense. The only argument for $$0.999\ldots = 1$$ that I can recall being explicitly called an "elementary proof" is this one, in the Peressini and Peressini reference. XOR&#39;easter (talk) 21:49, 15 April 2024 (UTC)
 * It is stated in the linked section that Peressini and Peressini wrote that transforming this argument into a proof "would likely involve concepts of infinity and completeness". This is far from being elementary. On the other hand the proof given here is really elementary in the sense that it uses only elementary manipulation of (finite) decimal numbers and the Archimedean property, and it shows that the latter is unavoidable.
 * Section must be removed or moved elsewhere, since completeness is not involved in the proof considered in this section.
 * This section "Elementary proof" was introduced by this edit, in view of closing lenghty discussions on the talk page (see Talk:0.999.../Archive 18 and more specially Talk:0.999.../Archive 18. The subsections and  have been introduced by this edit (the second heading has been improved since this edit).
 * I am strongly against the removal of . Instead, we could reduce to its first paragraph, since, all what follows "More precisely" is repeated in . The reason for keeping both subsections is that the common confusion about $0.999... = 1$ results from a bad understanding of the difference between an intuitive explanation and a true proof. Since this article is aimed for young students, the distinction must be kept as clear as possible. Fortunately, with this proof, we have not to say them "wait to have learnt more mathematics for having a true proof", as it is the case with the other proofs given in this article. D.Lazard (talk) 10:21, 16 April 2024 (UTC)
 * But if no one other than us calls the proof in this section "elementary", then doing so violates WP:NOR. It's not our job to compare the existing arguments and proofs, evaluate the features that they each contain, and crown one of them as the most "elementary". And to a reader not familiar with how mathematicians use the word "elementary", applying it to a proof that invokes something called "the Archimedean property" is just confusing. (It's easy to forget that the average person probably only knows that the rationals are dense in the reals.) Right now, our use of the term "Elementary proof" here is bad from the standpoint of policy (it's WP:SYNTH until we find a source saying so), and it's not great from the standpoint of pedagogy either. I moved the "Discussion on completeness" subsection to the end of the section, since it didn't really belong where it was. XOR&#39;easter (talk) 17:33, 16 April 2024 (UTC)

I think the term "elementary" is a bad one. Perhaps something indicating that the proof uses decimal representations? I think the rigorous proof should stay, and the new arrangement of content makes this clearer to me. Tito Omburo (talk) 18:19, 16 April 2024 (UTC)


 * I changed the section heading to "Proof by adding and comparing decimal numbers", which gets away from the term "Elementary" while still, I think, making it sound fairly easy. XOR&#39;easter (talk) 19:04, 17 April 2024 (UTC)

Elementary "proof"?
The article has
 * It is possible to prove the equation $$ 0.999\ldots = 1 $$ using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, or the formal construction of real numbers.

I've changed this, but was reverted. I believe it makes no sense to talk about an elementary proof avoiding any formalism like limits or the construction of the real numbers; without these, the notation 0.999... has no meaning, and there is no such thing as a proof. Thoughts? Any good sources? Nø (talk) 19:20, 29 May 2024 (UTC)


 * Chapter 1 of Apostol defines decimal expansions with no reference to limits. (Just the completeness axiom.) Tito Omburo (talk) 21:08, 29 May 2024 (UTC)
 * Read the proof: except some elementary manipulations of finite decimal numbers, the only tool that is used is that, if a real number $x$ is smaller than $1$, then there is a positive integer such that This does no involve any notion of limit or series. More, it does not involve the fact that a upper bounded set of real numbers admits a least upper bound. D.Lazard (talk) 21:17, 29 May 2024 (UTC)
 * I agree with this assessment. As for sources, a pretty clear version of this appears in Bartle and Sherbert. Basically, only existence of a real number with a given decimal expansion uses completeness.  But here, of course, existence is not an issue. Tito Omburo (talk) 21:20, 29 May 2024 (UTC)
 * While the completeness theorem (involved in the so-called rigorous proof in the statement "This point would be at a positive distance from 1") intuitively makes sense (at least to anyone who has been used to real numbers, decimal notation, and the number line for a while), to call it an elementary topic (as opposed to an advanced one) seems quite a stretch to me. Am I missing something here? Nø (talk) 07:17, 30 May 2024 (UTC)
 * That is only the Archimedean property. Completeness in not needed.  Tito Omburo (talk) 09:23, 30 May 2024 (UTC)
 * When I wrote "read the proof", I did not read it again. Indeed, numerous edits done since I introduced it several years ago made it confusing and much less elementary than needed. In particular, the proof was given twice and used the concept of number line and distance that may be useful in the explanation, but not in a rigourous proof. Also it was a proof by contradiction that I consider as not very elementary. I have fixed these issues, and restored the heading . D.Lazard (talk) 11:17, 30 May 2024 (UTC)
 * This is definitely an improvement, however "elementary" is an adjective I suggest we avoid. Classically (according to Hardy), elementary means that it does not use complex variables. Tito Omburo (talk) 11:43, 30 May 2024 (UTC)
 * "Elementary" refers also to elementary school, elementary arithmetic, elementary algebra. This is this meaning that is intended here. On the other hand, I never heard of the use of "elementary" as a synonym of "real context". D.Lazard (talk) 12:17, 30 May 2024 (UTC)
 * It's rather common, in my experience; see, e.g., . I'm not a fan of using "Elementary" in the section heading here for WP:NOR reasons, as mentioned a few sections up. XOR&#39;easter (talk) 20:31, 30 May 2024 (UTC)
 * I suppose we agree that "advanced" essentially means the same as "not elementary" (however we delineate that).
 * The point I - perhaps inadequately - tried to make with my original post above (and with the edit that was reverted, diff) is that there is no way to settle the question about the meaning of 0.999... that is entirely elementary. Nø (talk) 07:06, 31 May 2024 (UTC)
 * Here's an elementary "proof" why 0.999... is less than 1:
 * 0<1
 * 0.9<1
 * 0.99<1
 * 0.999<1
 * Hence, 0.999...<1
 * To prove me wrong, I believe you need something that is not elementary. Nø (talk) 08:42, 31 May 2024 (UTC)
 * You need the archimedean property. You do not, in fact, need completeness or limits however. Tito Omburo (talk) 09:18, 31 May 2024 (UTC)
 * {ec}If you read the proof, you will see that the only non-elementary step is the use of the Archimedean property that asserts that there is no positive real number that is less than all inverses of natural numbers, or, equivalently, that there is no real number that is greater than all integers. This is an axiom of the real numbers exactly as the parallel postulate is an axiom of geometry. Both cannot be proved, but both are easy to explain experimentally. If you consider this proof as non-elementary, you should consider also as non-elementary all proofs and constructions that use the parallel postulate and are taught in elementary geometry.
 * By the way, there is something non-elementary here. This is the notation 0.999... and more generally the concept of infinite decimals. They are very non-elementary, since they use the concept of actual infinity whose existence was refused by most mathematicians until the end of the 19th century. My opinion is that infinite decimals should never be taught in elementary classes. D.Lazard (talk) 09:44, 31 May 2024 (UTC)
 * It seem we totally agree. There is no such thing as an elementary proof. Nø (talk) 09:10, 1 June 2024 (UTC)
 * No. This is an elementary proof of a result expressed with a non-elementary notation, namely that the least number greater than all $$0.(9)_n$$ is denoted with an infinite number of 9. D.Lazard (talk) 10:55, 1 June 2024 (UTC)
 * The least number (if one exists), and it is also an elementary proof of existence. Tito Omburo (talk) 16:26, 1 June 2024 (UTC)
 * Are you claiming one can give an elementary proof of someting that doesn't have an elementary definition? Nø (talk) 18:17, 2 June 2024 (UTC)
 * The least number greater than all $$0.(9)_n$$ is an elementary concept, but the notation $$0.999...$$ is not elementary, since it involves an actual infinity of 9. D.Lazard (talk) 19:30, 2 June 2024 (UTC)
 * I would not consider the existence of a least number greater than all numbers in an infinite sequence an elementary concept. I do not consider the meaning (definition) of 0.999... an elementary concept, and thus I think an argument avoiding advanced topics cannot be a proof. Nø (talk) 15:31, 5 June 2024 (UTC)
 * I would not consider the existence of a least number greater than all numbers in an infinite sequence an elementary concept. I do not consider the meaning (definition) of 0.999... an elementary concept, and thus I think an argument avoiding advanced topics cannot be a proof. Nø (talk) 15:31, 5 June 2024 (UTC)

And yet, the proof is elementary, which suggests you should revisit some preconceptions. Tito Omburo (talk) 17:50, 5 June 2024 (UTC)


 * The proof has this sentence:
 * Let x be the smallest number greater than 0.9, 0.99, 0.999, etc.
 * This presupposes the existence of such a number. As I said, I do not consider this elementary, but I acknowledge that we don't seem to have a clear and unambiguous consensus on what "elementary"/"advanced" really means. Nø (talk) 13:04, 6 June 2024 (UTC)
 * I changed recently the sentence for avoiding a proof by contradiction. The resulting proof, as stated, supposed the existence of a least upper bound, but it was easy to fix this. So, I edited the article for clarifying the proof, and making clear that it includes the proof that the numbers greater than all $$0.(9)_n$$ have a least element. By the way, this clarification simplifies the proof further. D.Lazard (talk) 14:23, 6 June 2024 (UTC)