Talk:0.999...

Yet another anon

Moved to Arguments subpage

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:

• 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)

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'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'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)

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)

efn?

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)

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)

(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., [1]. 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 $0.(9)_{n}$ 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)

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)

lede that gets to the point without jargon.

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)