Talk:0.999.../Archive 20

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, ${\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}}$ 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)

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'. Mindmatrix 12:59, 5 October 2023 (UTC)

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. — The Anome (talk) 12:40, 7 October 2023 (UTC)

@D.Lazard: 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? — 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 § See also 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 § Elementary proof is not in the introductive paragraph of the section, but in subsection § Rigorous proof. 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. — The Anome (talk) 10:55, 8 October 2023 (UTC)

"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)

@Imaginatorium: 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 Imaginatorium 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)

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)

This is an FA from 2006 that underwent FAR in 2010 and was kept. This article does not currently meet the featured article criteria:

• 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.

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)

@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 {{harv}}s 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, voorts. 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'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)

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)

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'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 ${\displaystyle 0.999\ldots =1}$ that I can recall being explicitly called an "elementary proof" is this one, in the Peressini and Peressini reference. XOR'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 § Discussion on completeness 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#Elementary proof. The subsections § Intuitive explanation and § Rigorous proof have been introduced by this edit (the second heading has been improved since this edit).

I am strongly against the removal of § Rigorous proof. Instead, we could reduce § Intuitive explanation to its first paragraph, since, all what follows "More precisely" is repeated in § Rigorous proof. 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'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'easter (talk) 19:04, 17 April 2024 (UTC)

The article has

It is possible to prove the equation ${\displaystyle 0.999\ldots =1}$ 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)

(edit conflict) 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 ${\textstyle x<1-{\frac {1}{n}}<1.}$ 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 § Rigorous proof. 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'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 ${\displaystyle 0.(9)_{n}}$ 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 ${\displaystyle 0.(9)_{n}}$ is an elementary concept, but the notation ${\displaystyle 0.999...}$ 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)

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 ${\displaystyle 0.(9)_{n}}$ have a least element. By the way, this clarification simplifies the proof further. D.Lazard (talk) 14:23, 6 June 2024 (UTC)

Right now, we have footnotes that are references and footnotes that are explanatory notes or asides, the former using {{sfnp}} and <ref> tags, the latter using <ref> tags. I propose wrapping the second kind in {{efn}} instead, which has what I consider the advantage of distinguishing between the two types of notes (efn get labeled [a], [b], etc. instead of [1], [2]). One disadvantage is that there are clearly some judgement calls to be made. How do other people feel about this? (Obviously this is not urgent, am happy to have "I'm busy trying to preserve featured status and don't want to think about/deal with this" as an answer.) --JBL (talk) 21:54, 17 April 2024 (UTC)

I'd be fine with that. XOR'easter (talk) 22:15, 17 April 2024 (UTC)

Fine with me. Hawkeye7 (discuss) 22:29, 17 April 2024 (UTC)

Strongly support using efn. --Trovatore (talk) 22:35, 17 April 2024 (UTC)

OK, I've made a stab at dividing them up. XOR'easter (talk) 23:31, 17 April 2024 (UTC)

I think it's fine to add efn. Additionally, maybe both the notes and references sections should be merged into one section, containing three different lists (notes, footnotes, works cited)? Dedhert.Jr (talk) 04:04, 19 April 2024 (UTC)

I assume that your goal is to eliminate footnotes that are in fact citations? For example,

• {{efn|{{harvtxt|Bunch|1982}}, p. 119; {{harvtxt|Tall|Schwarzenberger|1978}}, p. 6. The last suggestion is due to {{harvtxt|Burrell|1998}}, p. 28: "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."}}

These could be just ref tags with rp templates for page numbers and quotes, but I don't know if that is the style you want. Johnjbarton (talk) 01:11, 27 June 2024 (UTC)

Gah, no. {{rp}} tags are ugly enough when used in isolation. Stacking three in a row and then trying to fit in a quote as well would be a mess. We handled the concerns in this section back in April; nothing more in this regard needs to be done. XOR'easter (talk) 21:16, 27 June 2024 (UTC)

I realize the mathematicians love precision and thus those special words that have meaning in math, but this article has an important point for a broader audience. I change the intro yesterday to concentrate the ideas that "It is the number one!" into the first paragraph and move the two (or is it three or maybe one) definitions to a separate section. The waffle-worded, footnoted definition will be completely opaque to naive readers. They will stop reading and never discover "This number is equal to 1.". Unfortunately my change was reverted by @Tito Omburo with an edit summary, "Restored old lede. It is important that the lede refer to an actual number, not merely some notation.", which I do not understand. Note that my lede was

• In mathematics, 0.999... (also written as 0.9, 0..9 or 0.(9)) is a notation for the number "1" represented as a repeating decimal consisting of an unending sequence of 9s after the decimal point.

In my opinion we should change the content back towards the version I suggested. Johnjbarton (talk) 14:51, 27 June 2024 (UTC)

It's misleading to say that "0.999..." is notation referring to the number 1. The notation refers to a real number, namely the least real number greater than every truncation of the decimal. The fact is that this real number is equal to one. Tito Omburo (talk) 14:58, 27 June 2024 (UTC)

Ok so how about

• In mathematics, 0.999... (also written as 0.9, 0..9 or 0.(9)) is a notation a real number equal to "1". The real number is represented as a repeating decimal consisting of an unending sequence of 9s after the decimal point.

and restoring the Definition section? Johnjbarton (talk) 15:09, 27 June 2024 (UTC)

This still boils the subject of the article down to a tautology, which it is not. 0.999... definitionally means something. It is not the same thing as the numeral 1. Tito Omburo (talk) 15:26, 27 June 2024 (UTC)

Sorry I mis-edited. I know you disagreed with "notation" as it means definitional equivalence, but I accidentally left the word. Here is the alternative I should have written:

• In mathematics, 0.999... (also written as 0.9, 0..9 or 0.(9)) is a real number equal to "1". The real number is represented as a repeating decimal consisting of an unending sequence of 9s after the decimal point.

Johnjbarton (talk) 15:52, 27 June 2024 (UTC)

Agree that the simpler get-to-the-point jargon-free lede is better. The intended audience here is not mathematicians, it's lay people who likely are not familiar with the idea that the decimal representation of a real number is not unique in all cases (ie a "terminating" decimal that repeats zeros always has another representation that repeats nines).

In particular, it's too early in the article to assume that the reader knows anything about infinite sequences and convergence. Statement like "The notation refers to a real number, namely the least real number greater than every truncation of the decimal." will be lost on the average reader.

Similarly, it's not appropriate to assume that the reader knows the difference between a numeral and a number. We can explain all this later in the article.

Agree with removing the technical details to a definition section. Mr. Swordfish (talk) 15:31, 27 June 2024 (UTC)

The problem is this is like telling lies to children. An unprepared reader has no idea what the notation 0.999... refers to. The current lede makes clear what that is. The proposed lede is actively misleading, in the name of being more accessible. The problem is that the subject of this article is not accessible to someone unwilling to grasp in some way with the concept of infinity. But this important aspect cannot be written out of the intro. Tito Omburo (talk) 15:39, 27 June 2024 (UTC)

@Tito Omburo I gather your primary concern is the lede. My primary concern is the definition sentences. I think we should move that out of the intro.

I agree that the concept of infinity is core to the article. How about a sentence in the first paragraph that explicitly calls out the concept of infinity? Johnjbarton (talk) 15:59, 27 June 2024 (UTC)

One thing to consider is that opening sentences don't always have to have the form "foo is a bar"; when that's awkward, it's fine to pick a different structure. In this case, maybe something along the lines of

In mathematics, the notation 0.999..., with the digit 9 repeating endlessly, represents exactly the number 1.

just as the first sentence, then we can continue on with elaborations. This way we can (as Johnjbarton put it) "get to the point" in the first sentence, and we haven't told any lies-to-children. By not insisting on including the phrase 0.999... is, we

• don't have to say that it "is a notation"; we just put that part before the 0.999...
• don't have to say that it's a notation for a different (infinitely long) notation, which is true if you're super literal-minded, but is extremely confusing in the first sentence, and
• don't have to talk about least upper bounds before we give the punch line

I think this small tweak could open up a lot of possibilities for making the opening sentence (at least) more understandable to non-mathematicians, without saying anything false. --Trovatore (talk) 16:13, 27 June 2024 (UTC)

I support this change. Tito Omburo (talk) 16:18, 27 June 2024 (UTC)

Also support this change. Mr. Swordfish (talk) 16:22, 27 June 2024 (UTC)

Great, I made that change. Johnjbarton (talk) 16:35, 27 June 2024 (UTC)

I reverted the change by @D.Lazard but I think it may have some things that are helpful. Unfortunately the comment by D.Lazard was added but not signed nor set as a Reply. Johnjbarton (talk) 17:41, 27 June 2024 (UTC)

Yeah, I didn't like the new version as a whole. Tito Omburo (talk) 17:44, 27 June 2024 (UTC)

Independently from Trovatore's post, I have rewritten the lead for removing jargon (in particular "denotes" is less jargonny/pedantic than "is a notation for") and unneeded technicalities from the beginning. This required a complete restructuration. By the way, I have removed some editorial considerations that do not belong here. By doing this, I deleted the last Johnjbarton's edit, but I think that my version is better for the intended audience. — Preceding unsigned comment added by D.Lazard (talk • contribs) 19:23, 27 June 2024 (UTC)

Sorry to have forgotten to sign. I did not use the "reply" button, because this is an answer to the opening post, and, as such, should not be indented.

I rewrote the first paragraph of the lead that used many terms (technical or jargon) that are useless for people that know infinite decimals and are confusing for others. Also, I added that the equality can be proved, and therefore that is is not a convention that may be rejected by people who do not like it. D.Lazard (talk) 10:37, 28 June 2024 (UTC)

Looks fine to me. Tito Omburo (talk) 10:41, 28 June 2024 (UTC)

Glad to see the work done to make this article more readable to a general audience. I hope "The utilitarian preference for the terminating decimal representation..." (last para of lede) can also be simplified, as I see what it means but as written it's well above most of the population's reading level. I'm confused by the use of the {{spaces}} template before the 1 in the first paragraph: it looks like a formatting error. What's the point of that big space? MartinPoulter (talk) 11:34, 28 June 2024 (UTC)

I have removed this sentence per WP:NPOV: authors in mathematical education have different explanation on the difficulties of the students with the equality; this is not to Wikipedia to select one amongst several. I did several other edits that are explained in the edit summaries. D.Lazard (talk) 14:21, 28 June 2024 (UTC)

The article contains the statement

... every nonzero terminating decimal has two equal representations ... all positional numeral systems have this property.

Every positional numeral system has two representations for certain numbers, but is this necessarily true of terminating representations? A counterexample would seem to be balanced ternary: the numbers that have two representations seem to be nonterminating, e.g. 1 = 1.000..._bal3 has no other representation, but 1/2 = 0.111..._bal3 = 1.TTT..._bal3 (where T = −1) has two. Or maybe I need some coffee? —Quondum 01:56, 30 June 2024 (UTC)

Well, the trouble is that "balanced ternary" is not a "usual" positional numeral system. Perhaps it might be better to write, "and this is true of all bases, not just decimal". In the end it depends whether you think Wikipedia is here for the benefit of lawyers, or just to help people understand things. Imaginatorium (talk) 04:07, 30 June 2024 (UTC)

A better viewpoint is that such systems have no "terminating" representations at all, but only ones that eventually repeat the digit 0. That's beyond the scope of this article. Still, I think we should de-emphasize the notion of "terminating" representations. We don't really need to talk about them. We can say, for example, that 3.4999... is the same as 3.5000.... --Trovatore (talk) 05:36, 30 June 2024 (UTC)

While I agree that "terminating representations" are a little peripheral, including that to answer the immediate question "How easy is it to find such values?" seems reasonable, although extrapolating from the example would seem obvious to us, and the phrase is adequately defined as linked. I don't feel strongly about keeping "terminating representations" or any other specific description of the class, though. I have clarified the statement in a way that fits the section 0.999... § Generalizations.

That aside, it is interesting that having multiple representations depends on the definition of a positional numeral system as having position weights and values associated with symbols that do not depend on the value of other digits; I say this, because Gray codes are remarkably close to being a positional system and (extended to a fractional part) they evidently have a unique representation for each real number. —Quondum 15:11, 30 June 2024 (UTC)

Not sure quite what you mean about the Gray codes. The key point here is that representations of this sort are zero-dimensional in the product topology, which seems to be the natural one to put on them, whereas the reals are one-dimensional. So a continuous surjection from the representations to the real numbers cannot be an injection with a continuous inverse, because otherwise it would be a homeomorphism, contradicting the previous observation about dimension. Therefore there must be reals with non-unique representations.

Maybe the representation by Gray codes you're talking about isn't continuous (or its inverse is not); would need to see what you mean. --Trovatore (talk) 18:30, 30 June 2024 (UTC)

I intended that a digit incrementing reverses the interpretation of all subsequent digits – in my quest to get rid of the infinite number of digits 'rolling over' (e.g. 0.999... to 1.000...). Looking more closely, I see that my supposition was incorrect: you still get two representations for a (presumably the same) set of real values, but now a pair of equal representations differs in one digit instead of infinitely many. This leads me to wonder whether a representation composed of an infinite sequence of discrete symbols could avoid doubles, once we have the the restriction that every real number must have a representation. —Quondum 19:12, 30 June 2024 (UTC)

If you put no restrictions at all on what you mean by a "representation", the answer is clearly yes, you can avoid duplicates. For example the set of all countably infinite decimal strings and the set of all reals have the same cardinality, so there's a bijection between them. You can even make that bijection pretty explicit, by playing games with (the proof of) the Schröder–Bernstein theorem.

However, if you care at all about continuity, you're going to need to deal with the dimension issue I mentioned in my previous comment. --Trovatore (talk) 19:52, 30 June 2024 (UTC)

I was having difficulty framing the issue properly, and I thought about the cardinality argument after my post. I this context, all I would care about in this context is finding a simple constructive definition of a class of surjective maps from sequences of symbols to the reals for which multiple representations occur, ideally independent of the axiom of choice and it would be nice if it gave a sense of the lengths one might have to go to to avoid these multiples. The class of standard positional systems suffices, but clearly the class for which this is true is larger, and this gives room for possibly finding a class that is simpler to define. Improving on just copying the second sentence in Positional notation seems to me to be challenge. —Quondum 21:40, 30 June 2024 (UTC)

In this context, a nice way to think of continuity is that, if you want finitely much information about the answer, you need only finitely much information about the input. On the "real" side, finitely much information means an open interval. On the "representation" side, it means finitely many digits. If that's true in both directions, then there must be duplicates.

I'm a little skeptical that this can (or should) be worked into this article, but it would be satisfying if it could be. --Trovatore (talk) 22:13, 30 June 2024 (UTC)

I'm not of the opinion that anything more complicated than a sentence defining 'positional numeral system' is called for here to address the concerns that have been expressed, and it seems that my hope that a broader class could be defined more easily was too much. I have to confess that I do not know what you mean by 'finitely many digits': we are dealing with a countably infinite sequence of digits, each of which is an element of a finite (or possibly infinite) set of symbols. I'm afraid educating me on this is, for the moment, a lost cause, so I suggest that we just focus on language to include in the article. —Quondum 23:35, 30 June 2024 (UTC)

I tend to agree that this is getting off-scope for the talk page. I'll drop a note on your talk page. I don't think it's a lost cause; I probably just haven't found the right way of explaining myself.

But as long as we're here, I do want to correct the record for the benefit of any lurkers. Turns out my maunderings about the continuity of the inverse mapping were unnecessary. As long as

• The alphabet is finite (or at least there are only finitely many choices for a digit at any given position),
• The mapping is continuous, and
• The mapping is injective

you get continuity of the inverse mapping, and therefore a contradiction, for free. That's because the representation space is compact (by Tychonoff's theorem), so any closed set is compact. Then the continuous image of a closed set is compact and therefore closed, which in the injective case implies that the inverse map is continuous.

So if we can source it, we could say that any continuous interpretation of the representations would have to have duplicates. Is that appropriate for this article? I doubt it. This article ought to be pitched considerably lower. Anyone who understands the above argument isn't looking to understand 0.999.... --Trovatore (talk) 03:06, 1 July 2024 (UTC)

Very nice. This also in some sense explains how p-adic integers can have unique expansions: the p-adic integers are compact. Tito Omburo (talk) 10:59, 1 July 2024 (UTC)

Or more importantly, the p-adic integers are zero-dimensional, or it might be easier to think of it in terms of the p-adic integers are totally disconnected (not quite equivalent but it gets at the same point for our purposes).

The representations are totally disconnected whereas the reals are connected, so intuitively, to map the representations to the reals, you have to "connect something", which is where the duplicates come from. The p-adics are already totally disconnected, so the problem doesn't come up. --Trovatore (talk) 20:05, 1 July 2024 (UTC)

Also, the product space ${\displaystyle \{0,...,9\}^{\mathbb {N} }}$ is totally disconnected, and so is not the continuous image of a real interval. Tito Omburo (talk) 23:16, 1 July 2024 (UTC)

As user:Imaginatorium point out, the statement is valid in usual positional numeral systems, but not in all non-standard positional numeral systems. Luckily, positional numeral systems redirects to a section of List of numeral systems on "Standard positional numeral systems", so I suggest we simply use that wikilink. Nø (talk) 15:32, 30 June 2024 (UTC)

We should never have a redirect and its plural linking to different places, so that is not a solution. —Quondum 15:54, 30 June 2024 (UTC)

I have changed that redirect to be consistent with the singular form, after verifying that there are no mainspace uses. In any event, the definition at Positional notation (essentially a weighted sum) is precisely correct for the statement as it now stands (i.e. including all nonstandard positional systems that meet this definition, with the proviso that they can represent all real numbers), and as supported by the text of the article. —Quondum 16:43, 30 June 2024 (UTC)

As far as I understand, this section discusses supposed properties of all positional numeral systems. But this supposes a precise definition of a positional numeral system, and of a positional numeral system that accepts infinite strings. Without such a definition, everything is original research.

As an example, the standard p-adic representation of p-adic numbers is an example of a positional numeral system such that there is always a unique representation.

By the way it is astonishing that nobody mention what is, in my opinion, the main reason for which there is so much confusion with the subject of the article: it is that "infinite decimals" make a systematic use of actual infinity, a concept that is so counterintuitive that, before the 20th century, it was refused by most mathematicians. It seems that some teachers hope that kids could understand easily concept that were refused by mathematicians and philosophers a century ago. D.Lazard (talk) 16:44, 30 June 2024 (UTC)

I think you're a bit off on that point, Prof. Lazard. My impression is that the rejection of the actual infinite was more in theory than in practice, and its systematic use considerably predated the 20th century, since real analysis was developing in the mid-19th century and used the actual infinite implicitly. It took Cantor to make it explicit, but the ideas of Bolzano and Cauchy and Weierstrass and Dedekind were already laying the groundwork.

That said, sure, it's a key psychological point.

ObSMBC. --Trovatore (talk) 18:53, 30 June 2024 (UTC)

This is interesting, but does it apply? As I understand it (and admittedly this is outside my area of knowledge), p-adic numbers do not embed the reals. The ability to represent all reals is core to the statement that there are necessarily multiple representations. —Quondum 16:57, 30 June 2024 (UTC)

Interesting point about p-adic numbers. I think the lead should mention infinity somewhere. I think the issues are resolved if the article is clear on what a "positional number system" is. I am unclear exactly what is meant. Tito Omburo (talk) 17:01, 30 June 2024 (UTC)

So we have two people saying that a clear definition of a 'positional number system' is needed in the article, and I tend to agree in the context of this claim. I imagine that this can be omitted from the lead, but it might make sense in 0.999... § Generalizations. —Quondum 18:09, 30 June 2024 (UTC)

Does the argument in 0.999...#Impossibility of unique representation come from somewhere? Other than that, the sourcing seems OK. XOR'easter (talk) 05:10, 29 June 2024 (UTC)

The first and the second, as well as the bullet list, remain unsourced. Dedhert.Jr (talk) 06:18, 29 June 2024 (UTC)

I have provisionally trimmed the passage I couldn't find support for. It wasn't technically wrong, as far as I could tell, but we aren't a repository for everything that could be said about a math topic. XOR'easter (talk) 17:47, 29 June 2024 (UTC)

Probably a correct removal, but sort of a pity, since it's the only bit of actual mathematical interest.

No matter. This isn't really a math article, or shouldn't be. Mathematicians are unlikely to care about 0.999... per se. We should keep that in mind when thinking about how to present the material. I'm totally against lies to children, but I also don't see the point in making this an article about real analysis. If you understand real analysis you don't need this article. --Trovatore (talk) 21:50, 29 June 2024 (UTC)

Are any of the concerns in the FA review still outstanding, then? XOR'easter (talk) 02:30, 23 July 2024 (UTC)

There seems to be an error in the intuitive explanation:

For any number x that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than x⁠⁠.

If we set x = 0.̅9 then the sequence will never reach a number larger than x. 2A01:799:39E:1300:F896:4392:8DAA:D475 (talk) 12:16, 4 October 2024 (UTC)

If x = 0.̅9 then x is not less than 1, so the conditional statement is true. What is the error? MartinPoulter (talk) 12:50, 4 October 2024 (UTC)

If you presuppose that 0.̅9 is less than one, the argument that should prove you wrong may apprear to be sort of circular. Would it be better to say "to the left of 1 on the number line" instead of "less than 1"? I know it's the same, but then the person believing 0.̅9 to be less than one would have to place it on the number line! Nø (talk) 14:47, 4 October 2024 (UTC)

What does the notation 0.̅9 mean? Johnjbarton (talk) 15:43, 4 October 2024 (UTC)

It means zero followed by the decimal point, followed by an infinite sequence of 9s. Mr. Swordfish (talk) 00:24, 5 October 2024 (UTC)

Thanks! Seems a bit odd that this is curious combination of characters (which I don't know how to type) is not listed in the article on 0.999... Johnjbarton (talk) 01:47, 5 October 2024 (UTC)

@Tito Omburo. There are other unsourced facts in the given sections. For example:

• There is no source mentions about "Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as..." in Dedekind cuts.
• There is no source mentions about "Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits ⁠b₁, b₂⁠⁠, b₃, ..., and one writes..." in Nested intervals and least upper bounds. This is just one of them.

Dedhert.Jr (talk) 11:00, 30 October 2024 (UTC)

The section on Dedekind cuts is sourced to Richman throughout. The paragraph on nested intervals has three different sources attached to it. Tito Omburo (talk) 11:35, 30 October 2024 (UTC)

Are you saying that citations in the latter paragraph supports the previous paragraphs? If that's the case, I prefer to attach the same citations into those previous ones. Dedhert.Jr (talk) 12:52, 30 October 2024 (UTC)

Not sure what you mean. Both paragraphs have citations. Tito Omburo (talk) 13:09, 30 October 2024 (UTC)

The logic in the so-called intuitive proofs (rather: naïve arguments) relies on extending known properties and algorithms for finite decimals to infinite decimals, without formal definitions or formal proof. Along the same lines:

• 0.9 < 1
• 0.99 < 1
• 0.999 < 1
• ...
• hence 0.999... < 1.

I think this fallacious intuitive argument is at the core of students' misgivings about 0.999... = 1, and I think this should be in the article - but that's just me ... I know I'd need a source. I have not perused the literature, but isn't there a good source saying something like this anywhere? Nø (talk) 08:50, 29 November 2024 (UTC)

I inserted "or equal to" in the lead, thus:

In mathematics, 0.999... (also written as 0.9, 0..9, or 0.(9)) denotes the smallest number greater than or equal to every number in the sequence (0.9, 0.99, 0.999, ...). It can be proved that this number is 1; that is,

${\displaystyle 0.999...=1.}$

(I did not emphasize the words as shown here.) But it was reverted by user:Tito Omburo. Let me argue why I think it was an improvement, while both versions are correct. First, "my" version it s correct because it is true: 1 is greater than or equal to every number in the sequence, and any number less than 1 is not. Secondly, if a reader has the misconception that 0.999... is slightly less than 1, they may oppose the idea that the value must be strictly greater than alle numbers in the sequence - and they would be right in opposing that, if not in this case, then in other cases. E.g., 0.9000... is not greater than every number in the corresponding sequence, 0.9, 0.90, 0.900, ...; it is in fact equal to all of them. Nø (talk) 12:07, 29 November 2024 (UTC)

I think it's confusing because 1 doesn't belong to the sequence, so "or equal" are unnecessary extra words. A reader might wonder why those extra words are there at all, and the lead doesnt seem like the place to flesh this out. Tito Omburo (talk) 13:40, 29 November 2024 (UTC)

Certainly, both fomulations are correct. This sentence is here for recalling the definition of the notation in this specific case, and must be kept as simple as possible. Therefore, I agree with Tito. The only case for which this definition of ellipsis notation is incorrect is when the ellipsis replaces an infinite sequence of zeros, that is when the notation is useful only for emphasizing that finite decimals are a special case of infinite decimals. Otherwise, notation 0.100... is very rarely used. For people for which this notation of finite decimals has been taught, one could add a footnote such as 'For taking into account the case of an infinity of trailing zeros, one replaces often "greater" with "greater or equal"; the two definitions of the notation are equivalent in all other cases'. I am not sure that this is really needed. D.Lazard (talk) 14:46, 29 November 2024 (UTC)

Could you point to where the values of decimals are defined in this way - in wikipedia, or a good source? I can eassily find definitions in terms of limits, but not so easily with inequality signs (strict or not).

I think the version with strict inequality signs is weaker in terms of stating the case clearly for a skeptic. Nø (talk) 17:45, 30 November 2024 (UTC)

Agree that both versions are correct. My inclination from years of mathematical training is to use the simplest, most succinct statement rather than a more complicated one that adds nothing. So, I'm with Tito and D. here. Mr. Swordfish (talk) 18:24, 30 November 2024 (UTC)

I think many mathematicians feel that "greater than or equal to" is the primitive notion and "strictly greater than" is the derived notion, notwithstanding that the former has more words. Therefore it's not at all clear that the "greater than" version is "simpler". --Trovatore (talk) 03:13, 1 December 2024 (UTC)

The general case is "greater than or equal to", and I would support phrasing it that way. I think we don't need to explain why we say "or equal to"; just put it there without belaboring it. --Trovatore (talk) 03:06, 1 December 2024 (UTC)

The redirect 0.999999999999999999999999999999999999999 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 March 18 § 0.99999.... until a consensus is reached. Rusalkii (talk) 03:47, 18 March 2025 (UTC)

0.999... is not equal to 1 in the surreal number system. This article is spreading misinformation.

https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-to-1-surreally-this-is-not-so/ https://mindyourdecisions.com/blog/2014/09/17/surreal-0-999-repeating-is-not-equal-to-1-video/ 172.13.197.175 (talk) 18:11, 2 June 2025 (UTC)

The article mentions the surreal number system once and does not claim that they are equal. EvergreenFir (talk) 18:14, 2 June 2025 (UTC)

A search of the archives turns up a number of hits, but this one seems to have directly addressed it: Talk:0.999.../Arguments/Archive_3#Reals_vs._Surreals. —Locke Cole • t • c 18:30, 2 June 2025 (UTC)

The bulk of this article is about the real numbers, or at least that's my reading of it and I would imagine that most of our readers are making that assumption too. There's a later section on alternative_number_systems, which contains the one brief mention of the surreal numbers but the rest of the article is about the reals (Although it also applies to subsets such as the rationals and and supersets such as the complex numbers, discussing that would probably just be a distraction for most readers)

That said, I don't think it hurts anything to make it clear at the outset that we're talking about the reals and not some other system, so I'll implement that minor change. Mr. Swordfish (talk) 23:51, 21 June 2025 (UTC)

I'm a bit concerned that the opening paragraphs of the "Rigorous proof" subsection aren't supported by any sources. The "Intuitive explanation" given in the preceding subsection points to a book and a paper, both of which do cover the topic, but neither of which give the argument with detailed notation like "0.(9)_n" or an explicit connection to the Archimedean property of the reals. The other sources in that section don't give the "Rigorous proof" either, as far as I can tell. For example, the Meier and Smith book defines the Archimedean property to be that for every real number, there is a natural number that is bigger. Then it shows as a corollary that for every real number ${\displaystyle r}$, there is a natural number ${\displaystyle n}$ such that ${\displaystyle 1/n<r}$. But the "0.(9)_n" argument isn't given anywhere. As it stands, the rigorized version of the intuitive argument doesn't come from anywhere. Stepwise Continuous Dysfunction (talk) 23:45, 25 June 2025 (UTC)

This is pretty well the same proof: Stewart & Tall (2015) The Foundations of Mathematics, 2nd ed. pp. 38–39. Here's the 1977 1st ed. –jacobolus (t) 00:36, 26 June 2025 (UTC)

Thanks! That does look to be what we needed. I have added this source to the article. Stepwise Continuous Dysfunction (talk) 00:53, 26 June 2025 (UTC)

The current version of the article contains the following phrase:

...the Archimedean property, a defining axiom of the real number system

I am unaware of any source that states the Archimedean property as an axiom of the real numbers. Conventionally, the axiom expressing Completeness of the real numbers is either Dedekind completeness or the Least-upper-bound property, although there are other equivalent statements. While the Archimedean property follows from any of these axioms, I don't think that it is equivalent and therefore could be substituted as an axiom.

If not, then we need to correct the article. If so, then we'd need a source and we're still faced with the issue that the article is at variance with the "usual" literature. Or at least with the ones I'm familiar with.

I'm correcting the language pending further discussion here. Mr. Swordfish (talk) 23:43, 21 June 2025 (UTC)

It is sometimes described as an axiom (the "axiom of Archimedes"), and I think axiom works better in this context than property. E.g., it is one of Hilbert's axioms. Tito Omburo (talk) 11:55, 22 June 2025 (UTC)

Many things are called "axioms" that are not a "defining axiom of the real numbers". This appears to be an example since although the Archimedean property can be proved from the Dedekind completeness or the Least-upper-bound property axioms, those do not follow from the Archimedean property. i.e. The Archimedean property is not strong enough to serve as an axiom of the real numbers. It's not, so let's not say that it is. Mr. Swordfish (talk) 12:59, 22 June 2025 (UTC)

I mean, by itself completeness is also "not strong enough to serve as an axiom of the real number system", viewed in isolation. Completeness axiom + ordered axiom. Or completion of rationals + Archimedean axiom. Both give you the real numbers. (The latter was Hilbert's viewpoint. And if we're worried about independence of the axioms defining the rals, something rarely discussed, here is a list of axioms, proven independent, which includes a property equivalent to the Archimedean property.) But my point is that the Archimedean property is an axiomatic property, in the colloquial sense: something assumed to be true of the structure. If some other set of axioms didnt imply the Archimedean property, we wouldn't accept them as defining a "real number". It was foundational, and preceded all other attempts to systematize the real number system. Tito Omburo (talk) 16:16, 22 June 2025 (UTC)

The usual axiomatic treatment of the reals is to list the field axioms, the order axioms that make it an ordered field, and some version of completeness. The three together characterize the reals up to isomorphism.

If there's a source that develops real analysis from the Archimedian property as an axiom we might be technically correct calling it an an "axiom of the real numbers", but we'd be promoting a viewpoint that sufficiently unusual that it is possibly WP:FRINGE. Mr. Swordfish (talk) 16:39, 22 June 2025 (UTC)

I would encourage you to retract your viewpoint that this is WP:FRINGE. That is simply not tenable. It was one of the first axioms in history! (And the "usual axiomatic treatment" is a bit of a lie, isn’t it. The "usual axioms" that appear in many real analysis texts are not independent! Nor are they usually referred yo as "axioms".) Anyway, would you settle for "a defining property" rather than "axiom"? Tito Omburo (talk) 16:45, 22 June 2025 (UTC)

I would suggest "a foundational property": Although the Archimedean axiom is not always used in formal definitions of the real numbers, it is an axiom in all axiomatic definitions of the geometric line (the continuum, in the teminology of the 19th century, the real line in modern language). It is clear that Dedekind had this axiom in mind when defining the real numbers for modeling the continuum. If you do not like "foundational property", one may use also "fundamental property" D.Lazard (talk) 17:04, 22 June 2025 (UTC)

Foundational property is good, but I still think defining property would be more suitable for the basic audience (with axiom even better). (Although I'm also quite happy with Swordfish's latest, with my small copy edit.) Tito Omburo (talk) 17:45, 22 June 2025 (UTC)

My take is that I don't think we need to add flowery language like "fundamental property" or "foundational property", but I'm not going to object to either of those. Mr. Swordfish (talk) 18:27, 22 June 2025 (UTC)

IMO, this is precisely why axiom is better. "Foundational property" feels "flowery", but isn't. Axiom is correct and immediately understandable. Tito Omburo (talk) 18:48, 22 June 2025 (UTC)

The problem with "axiom" is that it assumes a particular axiomatic setup (or at least one of a particular proper subset of possible axiomatic setups) for describing the same thing (the real numbers). The issues being discussed are not specific to the axioms used. Even "foundational property" suggests that you're coming at the question from a foundationalist perspective. I would just go with "property". --Trovatore (talk) 19:14, 22 June 2025 (UTC)

"Axiom" is an ordinary English word. It means, specifically, something whose truth is taken to be self-evident and is not questioned. The word "axiom" does not mean "axiom of set theory". It certainly is an axiom of the real numbers. It does not assume any particular setup. Indeed, as an axiom it is more primitive than even completeness (see Hilbert). If a model of the "real numbers" didnt satisfy Archimedes axiom, it wouldn't be considered the real numbers! Tito Omburo (talk) 21:38, 22 June 2025 (UTC)

It's true that there is a sense of "axiom" that is an ordinary English word, but using that sense in a mathematics article invites confusion, and anyway it's not totally clear that the Archimedean property is an axiom in that sense.

Linguistic aside: Generally after "model of" you name a theory, not a structure. The real numbers are not a theory but a structure. So "model of the real numbers" doesn't exactly make sense in the usual usage, though there are available meanings that I would let you get away with. --Trovatore (talk) 22:21, 22 June 2025 (UTC)

Except that the real numbers existed before anyone formalized them. When they were formalized, if they did not satisfy the Archimedean axiom, that formalization would have been rejected. Thus Archimedes is an axiom. (See the article axiom.) Tito Omburo (talk) 09:35, 23 June 2025 (UTC)

I would say it was a discovered truth about them rather than an axiom. --Trovatore (talk) 17:29, 23 June 2025 (UTC)

It is explicitly on of Hilbert's axioms. Tito Omburo (talk) 17:48, 23 June 2025 (UTC)

Sure. There are probably lots of axiomatizations of the theory of the reals that include it as an axiom. That's a fact about those axiom systems, not a fact about the reals. The property itself is a fact about the reals, but calling it an axiom is not. --Trovatore (talk) 17:50, 23 June 2025 (UTC)

This is an argument that the reals have no axioms then, just properties. Do I understand correctly? Tito Omburo (talk) 17:55, 23 June 2025 (UTC)

Correct. There is no such thing as an "axiom of the reals" separate from a particular axiomatization. --Trovatore (talk) 19:12, 23 June 2025 (UTC)

Whether the real numbers "exist" is a controversial philosophical question. –jacobolus (t) 17:57, 23 June 2025 (UTC)

When I started this discussion I was not aware of any axiomatic presentation of the real numbers that included Archimedes property. This paper provides an axiomatic presentation of the real numbers that does include Archimedes property as an axiom. So, I've learned something here.

But this presentation is a rather unusual approach that leaves out much of the field axioms, replacing them with axioms based on ordering that then imply the usual field axioms. While it's not WP:FRINGE it's not how the axioms defining the reals are conventionally presented.

My take is that referencing such an unusual and unconventional approach in the lead of a general-interest article is not ideal. Whether the real numbers actually exist, or which set of axioms to use to define them is outside the scope of the lead section of this article.

With that said, I'm now of the opinion that referencing the Archimedes property in the lead is an unnecessary complication, and would propose the following third paragraph for the lead:

An elementary proof is given below that involves only elementary arithmetic and the Archimedean property: that there is no positive real number less than the reciprocal of every natural number. There are many other ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. Other proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

— Preceding unsigned comment added by Mr swordfish (talk • contribs) 18:51, 23 June 2025 (UTC)

I would skip the wikilink to mathematical rigor, which seems more distracting than helpful if we are linking mathematical proof immediately afterward. –jacobolus (t) 19:06, 23 June 2025 (UTC)

I have no objections to that. Mr. Swordfish (talk) 19:15, 23 June 2025 (UTC)

Any objections to the proposal above with the mod suggested by –jacobolus ? If none, I'll go ahead and make the change. Mr. Swordfish (talk) 15:19, 25 June 2025 (UTC)

Also "intuitive arguments" should link to the title mathematical intuition (which currently redirects to logical intuition). A link to intuition in general is not that useful. (Or we could skip wikilinking anything from there.) –jacobolus (t) 16:43, 25 June 2025 (UTC)

I oppose strongly to the removal of the sentence that is struck out in the above quotation. The Archimedean property is a fundamental and very elementary; no reason for hiding it. And this is true that the given proof is elementary. As quoted above, the paragraph suggest that readers must learn calculus before having a rigorous proof. This is a form of unacceptable pedantry: "if you do not know much mathematics, you cannot have a rigorous proof".

This being said, I don't oppose to replace the statement of the Archimedean property by its original formulation: "for every real number, there are integers that are greater". This would need to add in the proof a short paragraph showing the equivalence of the two formulations (to take the multiplicative inverses). D.Lazard (talk) 17:30, 25 June 2025 (UTC)

Agree with all of Lazard's points. Tito Omburo (talk) 17:48, 25 June 2025 (UTC)

If we're going to mention the Archimedes property in the lead, then the simpler version as stated by D.Lazard is preferable to what is there now.

The recent edit is an improvement, and perhaps better than what I proposed above. Combining that with D.Lazard's and jacobolus' suggestions gives:

There are many ways of showing this equality, from intuitive arguments to rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only elementary arithmetic and the Archimedean property: for every real number, there are natural numbers that are greater. Other proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

Opinions? Mr. Swordfish (talk) 00:17, 26 June 2025 (UTC)

Agree, I thanked the editor who made the change, and did not yet have a chance to opine here. Tito Omburo (talk) 00:19, 26 June 2025 (UTC)

Is that in fact the usual statement of the Archimedean property? I always thought of it as "for any real positive ε and any natural number N, if you add ε to itself enough times, you get something bigger than N". Certainly this is equivalent given the field properties for the reals, but that requires getting into algebra for something that seems more order-theoretic. --Trovatore (talk) 00:34, 26 June 2025 (UTC)

I don't know what the usual statement is, but it's the version that is given in the book we cite, and I think it's the most convenient for the purposes of an intro paragraph, where we want a statement whose meaning is clear and whose truth seems as obvious as possible. Stepwise Continuous Dysfunction (talk) 00:43, 26 June 2025 (UTC)

I changed the link at "intuitive" to point to mathematical intuition and swapped out the explanation of the Archimedean property, since D.Lazard's suggestion is simpler and makes for a more digestible introduction. I think the longer discussion of the Archimedean property below may need some modifications. For one thing, the current text says "there is no positive number that is less than ${\textstyle {\frac {1}{10^{n}}}}$ for all ⁠${\displaystyle n}$⁠" and calls this "a version of" the Archimedean property. Bringing in the specific number ${\textstyle {\frac {1}{10^{n}}}}$ here makes this seem more like a consequence of the property than a "version" of it. I am not aware of a source that defines the Archimedean property using a specific base like that. Stepwise Continuous Dysfunction (talk) 00:32, 26 June 2025 (UTC)

I have done a little rewriting to try and address this. Stepwise Continuous Dysfunction (talk) 00:54, 26 June 2025 (UTC)

In the section "Intuitive explanation", we say:

Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines.

From this we conclude that 0.999... cannot equal a number larger than 1. However, one could replace "number larger than 1" with "1", and thus seemingly conclude that 0.999... cannot equal 1. Can we strengthen the wording to get rid of this problem - without making it too complex to warrant the heading? Nø (talk) 09:35, 14 August 2025 (UTC)

We don't need to consider numbers larger than 1. The argument shows that 1 is the least upper bound of the sequence, since it is an upper bound and no number smaller is. Since this is an intuitive argument, I think we could just gloss this point without trying to be too precise. No one would argue that 0.999... could be greater than one. Tito Omburo (talk) 10:01, 14 August 2025 (UTC)

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)

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)

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)

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)

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

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)

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)

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)

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)

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

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)

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)

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)

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

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)

(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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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

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)

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)