Talk:0.999...

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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 Wikipedia 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^-k 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

{\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

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)=x³/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

0.x_{1}x_{2}x_{3}...

must be less than

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, i⁴, 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

\lim _{x\rightarrow 0^{+}}f(x)

, the right limit of x (which can also be expressed without the "+" as

\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,

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 Wikipedia 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 Wikipedia policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Wikipedia policy on original research.

0.999... is a featured article; it (or a previous version of it) has been identified as one of the best articles produced by the Wikipedia community. Even so, if you can update or improve it, please do so.

This article appeared on Wikipedia's Main Page as Today's featured article on October 25, 2006.

Article milestones
Date	Process	Result
May 5, 2006	Articles for deletion	Kept
October 10, 2006	Featured article candidate	Promoted
August 31, 2010	Featured article review	Kept

Current status: Featured article

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Arguments Archives: Index, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Yet another anon[edit]

Moved to Arguments subpage

FAR notice[edit]

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)[reply]

@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)[reply]

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)[reply]

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)[reply]

@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)[reply]

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)[reply]

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)[reply]

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)[reply]

"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)[reply]

Can you explain what "weasel words" means in this context? An example or two would help... Imaginatorium (talk) 09:38, 22 January 2024 (UTC)[reply]

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)[reply]

See also WP:WEASEL. voorts (talk/contributions) 21:19, 22 January 2024 (UTC)[reply]

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)[reply]

"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)[reply]

@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)[reply]

@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)[reply]

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)[reply]

I have moved unused references to the Further reading section. Hawkeye7 (discuss) 19:16, 23 February 2024 (UTC)[reply]

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)[reply]

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)[reply]

Elementary proof supported by Stillwell?[edit]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Root of student confusion?[edit]

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)[reply]

efn?[edit]

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)[reply]

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

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

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

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

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)[reply]