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view · edit Frequently asked questions Q: Are you positive that 0.999... equals 1 exactly, not approximately? A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research should never be added to a Misplaced Pages article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so. Q: Can't "1 - 0.999..." be expressed as "0.000...1"? A: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10 toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of ${\displaystyle {\frac {0.000\dots 1}{10}}}$ would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0. Q: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1? A: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit ${\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}$ is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1. Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1. A: No. By this logic, 0.9 < 0.999...; 0.99 < 0.999... and so forth. Therefore 0.999... < 0.999..., which is absurd. Something that holds for various values need not hold for the limit of those values. For example, f (x)=x/x is positive (>0) for all values in its implied domain (x ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that ${\displaystyle 0.x_{1}x_{2}x_{3}...}$ must be less than ${\displaystyle 1.y_{1}y_{2}y_{3}...}$ for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation. Q: 0.999... is written differently from 1, so it can't be equal. A: 1 can be written many ways: 1/1, 2/2, cos 0, ln e, i, 2 - 1, 1e0, 12, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a bijection from decimal representations to real numbers. Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount? A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; differentials, which are not numbers at all; differential forms, which are not real numbers and have anticommutativity; 0, which is not a number, but rather part of the expression ${\displaystyle \lim _{x\rightarrow 0^{+}}f(x)}$, the right limit of x (which can also be expressed without the "+" as ${\displaystyle \lim _{x\downarrow 0}f(x)}$); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application. Q: Are you sure 0.999... equals 1 in hyperreals? A: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given here, the problem is that almost same sequences give different hyperreal numbers, ${\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}$, and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...). Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly? A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a WP:POV desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system. Q: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right. A: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on real analysis, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in engineering, mathematics, statistics, computer science, or a natural science, it would probably help you in the future anyway. Q: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article? A: The criteria for inclusion in Misplaced Pages is for information to be attributable to a reliable published source, not an editor's opinion. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Misplaced Pages policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Misplaced Pages policy on original research.

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Yet another anon

Moved to Arguments subpage

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 ${\displaystyle 0.999\ldots =1}$ 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)

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 , , etc. instead of , ). 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 ${\displaystyle 0.999\ldots =1}$ 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)

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)

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 Misplaced Pages 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)

Sourcing question

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)

Two representations in every positional numeral system with one terminating?

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 Misplaced Pages 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)

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