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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.

Yet another anon

Moved to Arguments subpage

Intuitive explanation

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)[reply]

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)[reply]

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)[reply]

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

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

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)[reply]

B and C

@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 ⁠ $b 1, b 2 ⁠⁠, b 3, ...$ , 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)[reply]

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)[reply]

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)[reply]

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

Intuitive counterproof

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)[reply]

Greater than or equal to

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,

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Image

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

There is no consensus to remove the image, and a rough consensus to keep it. Mr. Swordfish (talk) 21:42, 10 January 2025 (UTC)[reply]

The image included at the top of this article is confusing. Some readers may interpret the image to mean that 0.999... represents a sequence of digits that grows over time as nines are added, and never stops growing. To make this article less confusing I suggest that we explicitly state that 0.999... is not used in that sense, and remove the image. Kevincook13 (talk) 17:31, 1 January 2025 (UTC)[reply]

I do not see how this is confusing. The caption reads: "Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely" - nothing remotely like "sequence... that grows over time". I cannot see how one could meaningfully add a comment that "0.999..." is not used in a sense that has not even been mentioned. Of course lots of people are confused: that is the reason for the article, which in an ideal world would not be needed. Imaginatorium (talk) 04:29, 2 January 2025 (UTC)[reply]

If a sequence of digits grows over time as nines are added, and never stops growing, it is reasonable to conclude that the digit nine is repeating infinitely. Kevincook13 (talk) 18:14, 2 January 2025 (UTC)[reply]

Yes, notation 0.999... means that the digit nine is repeating infinitely. So, the figure and its caption reflect accurately the content of the article. D.Lazard (talk) 18:28, 2 January 2025 (UTC)[reply]

When we use the word repeating we should expect that some people will think we are referring to a process which occurs over time, like the operation of a Repeating firearm. Kevincook13 (talk) 22:03, 2 January 2025 (UTC)[reply]

You can think of this as a "process" if you like. 0.9999... means the limit of the sequence [0.9, 0.99, 0.999, 0.9999, ...]. Of course in mathematics nothing ever really "occurs over time", though I suppose you could consider it a kind of algorithm which if implemented on an idealization of a physical computer with infinite memory capacity might indefinitely produce nearer and nearer approximations. –jacobolus (t) 22:20, 2 January 2025 (UTC)[reply]

I think you are going in a very productive direction. We should explain to readers how what they might think we mean, "occurring over time", relates to what we actually mean. Kevincook13 (talk) 00:43, 3 January 2025 (UTC)[reply]

I personally think that would be distracting and not particularly helpful in the lead section. There is further discussion of this in § Infinite series and sequences, though perhaps it could be made more accessible. –jacobolus (t) 03:42, 3 January 2025 (UTC)[reply]

Yes, I agree that detailed discussion does not belong in the lead section. I personally think that the image is distracting and not helpful. In the lead section we can simply state that in mathematics the term 0.999... is used to denote the number one. We can use the rest of the article to explain why. Kevincook13 (talk) 16:23, 3 January 2025 (UTC)[reply]

Except that it's not true that 0.999... denotes the number one. It denotes the least number greater than every element of the sequence 0.9, 0.99, 0.999,... It's then a theorem that the number denoted in this way is equal to one. Tito Omburo (talk) 16:31, 3 January 2025 (UTC)[reply]

It also denotes the least number greater than every number which is less than one, just as 0.333...denotes the least number greater than every number which is less than one-third. That's why we say it denotes 1/3, and why we also say that the one with 9s denotes 1. Imaginatorium (talk) 17:39, 3 January 2025 (UTC)[reply]

@Tito Omburo, notice that @Imaginatorium just wrote above "we also say that the one with 9's denotes 1". The description "the least number greater than every element of the sequence 0.9, 0.99, 0.999,..." does describe the number one, just as does "the integer greater than zero and less than two". Kevincook13 (talk) 18:21, 3 January 2025 (UTC)[reply]

This is an incorrect use of the word "denotes". Denotes an equality by definition, whereas one instead has that 0.999... and 1 are judgementally equal. For example, does "All zeros of the Riemann zeta function inside the critical strip have real part 1/2" denote True or False? Tito Omburo (talk) 18:56, 3 January 2025 (UTC)[reply]

I think you are inventing this - please find reliable sources (dictionaries and things) to back up your claimed meaning of "denote". Imaginatorium (talk) 04:55, 9 January 2025 (UTC)[reply]

I agree that it is better to write that the term is used to denote the number one, rather than that the term denotes the number one. Kevincook13 (talk) 20:06, 3 January 2025 (UTC)[reply]

Its not "used to denote". It is a mathematical theorem that the two terms are equal. Tito Omburo (talk) 20:46, 3 January 2025 (UTC)[reply]

I think we can make this issue very clear. Assume that x equals the least number greater than every element of the sequence 0.9, 0.99, 0.999,... . Applying the theorem we learn that x = 1. Substituting 1 for x in the opening sentence of this article we have: In mathematics 0.999... denotes 1. If we also insist that 0.999... does not denote 1, we have a contradiction. Kevincook13 (talk) 18:45, 4 January 2025 (UTC)[reply]

You have redefined the word "denote" to mean precisely the same as "is equal to", which is confusing and unnecessary. It's better to just say "is equal to" when that's what you mean, so that readers are not confused. –jacobolus (t) 18:56, 4 January 2025 (UTC)[reply]

I agree that redefining the word denote would be confusing and unnecessary. I simply defined a variable x to be equal to a number, the least number. Kevincook13 (talk) 20:04, 4 January 2025 (UTC)[reply]

I'm in agreement with @Imaginatorium and @D.Lazard on this. The image does not suggest a process extended over time, and it correctly reflects the (correct) content of the article, so there is no need to remove it. I'm not persuaded that people will interpret "repeating" as purely temporal rather than spatial. If I say my wallpaper has a repeating pattern, does this confuse people who expect the wallpaper to be a process extended over time? (Are there people who think purely in firearm metaphors?) MartinPoulter (talk) 17:30, 3 January 2025 (UTC)[reply]

Consider the number 999. Like the wallpaper, it contains a repeating pattern. That pattern could be defined over time, one nine at a time. Or it could be defined at one time, using three nines. Kevincook13 (talk) 18:27, 3 January 2025 (UTC)[reply]

Is it OK if I go ahead and edit the article, keeping in mind all the concerns which have been raised with my proposed changes? Kevincook13 (talk) 17:56, 8 January 2025 (UTC)[reply]

Can you be more specific about which changes you want to implement? MartinPoulter (talk) 20:32, 8 January 2025 (UTC)[reply]

The first change would be to remove the image. Kevincook13 (talk) 15:06, 9 January 2025 (UTC)[reply]

I'm confused, @Kevincook13. Where in the above discussion do you see a consensus to remove the image? You have twice said the image should be removed, and I have said it should stay. No matter how many times you express it, your opinion only counts once. Other users have addressed other aspects of your proposal. Do you sincerely think the discussion has come to a decision about the image? MartinPoulter (talk) 13:47, 10 January 2025 (UTC)[reply]

No. I do not think there is agreement on removing the image. (I don't personally think it is spectacularly good, but the argument for removing it appears to me to be completely bogus.) Imaginatorium (talk) 04:57, 9 January 2025 (UTC)[reply]

The term 0.999... is literally a sequence of eight characters, just as y3.p05&9 is. Yet, the term itself implies meaning. I think confusion about the term can be reduced simply by acknowledging different meanings the term might imply. It does imply different meanings to different people. We can respect everyone, including children who are not willing to simply accept everything a teacher tells them. We can do our best to help everyone understand what we mean when we use the term. Kevincook13 (talk) 15:32, 9 January 2025 (UTC)[reply]

For example, if a child thinks that by 0.999... we mean a sequence of digits growing over time, and the child objects when told that the sequence of digits is equal to one, we can respond by saying something like the following: You are correct that a growing sequence of digits does not represent one, or any number, because the sequence is changing. We don't mean that 0.999... represents a changing or growing sequence of digits. Kevincook13 (talk) 16:12, 9 January 2025 (UTC)[reply]

We don't mean a changing or growing sequence of digits. That is what it is confusing to say that we mean a repeating sequence of digits. Kevincook13 (talk) 16:15, 9 January 2025 (UTC)[reply]

What we mean is a number. Kevincook13 (talk) 16:18, 9 January 2025 (UTC)[reply]

This article is about the meaning of 0.999... in mathematics not about the possible meanings that people may imagine. If people imagine another meaning, they have to read the article and to understand it (this may need some work), and they will see that their alleged meaning is not what is commonly meant. If a child objects to 0.999... = 1, it must be told to read the elementary proof given in the article and to say which part of the proof seems wrong. D.Lazard (talk) 16:58, 9 January 2025 (UTC)[reply]

What do we mean by the term number? A number is a measure, not a sequence of digits. We may denote a number using a sequence of digits, but we don't always. Sometimes we denote a number using a word, like one. Sometimes we use a phrase such as: the least number greater than any number in a certain sequence. We may use a lowercase Greek letter, or even notches in a bone. Kevincook13 (talk) 16:44, 9 January 2025 (UTC)[reply]

By the term "number", we mean a number (the word is not the thing). It is difficult to define a number, and this took several thousands years to mathematicians to find an acceptable definition. A number is certainly not a measure, since a measure requires a measurement unit and numbers are not associated with any measurement unit. The best that can be said at elementary level is something like "the natural number three is the common property of the nines in 0.999..., of the consecutive dots in the same notation, and of the letters of the word one". D.Lazard (talk) 17:20, 9 January 2025 (UTC)[reply]

I see. A number is not a measure, but it is used to measure. Thanks. Kevincook13 (talk) 17:40, 9 January 2025 (UTC)[reply]

A number is a value used to measure. Kevincook13 (talk) 17:42, 9 January 2025 (UTC)[reply]

The caption on the image is: Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely.

The caption can be understood to mean that the term 0.999... is a zero followed by a decimal point followed by the digit 9 repeating infinitely, which meaning is distinct from the meaning that 0.999... denotes the number one.

If we retain the caption, we may communicate to readers that we mean that 0.999... is a repeating sequence, which sequence denotes the number one. That doesn't work because repeating sequences themselves cannot be written completely and and therefore cannot be used to notate.

0.999... is notation. The purpose of this article should be to help others understand what it denotes. If it denotes a repeating sequence of digits, then we should say so in the lead sentence. Kevincook13 (talk) 18:32, 9 January 2025 (UTC)[reply]

How does the first sentence of the article not explain that notation? The meaning of the notation is the smallest number greater than every element of the sequence (0.9,0.99,...). Tito Omburo (talk) 18:39, 9 January 2025 (UTC)[reply]

Because it does not make sense to say that the sequence is repeating, because all the nines have not already been added, and at the same time to say that the sequence represents a number, because all the nines have already been added. It is confusing because it is contradictory.

When we say that the sequence is repeating, people who are not trained in mathematics will likely assume that we mean that all the nines have not already been added, and therefore that the sequence is changing and therefore, does not represent a number. Which, I believe, is why the subject of this article is not more widely understood. Kevincook13 (talk) 19:05, 9 January 2025 (UTC)[reply]

I think I understand part of the confusion, which I've hopefully tried to correct with an edit. The notation 0.999... refers to a repeating decimal, a concept which had not been linked. There is a way of associating to any decimal expansion a number as its value. For the repeating decimal 0.999..., that number is 1. Tito Omburo (talk) 19:09, 9 January 2025 (UTC)[reply]

I like the edits. Because the least number is one, the meaning of the lead sentence can be understood to be that 0.999... is a recurring decimal whose value is defined as one. The notation below should match. Instead of

0.999...=1

, we should write

0.999...\ {\overset {\underset {\mathrm {def} }{}}{=}}\ 1

. Kevincook13 (talk) 19:40, 9 January 2025 (UTC)[reply]

No. The point is that the notation has a definition which is a standard one for repeating decimals of this form. It is a theorem that this number is one, but that is not the definition. Tito Omburo (talk) 19:47, 9 January 2025 (UTC)[reply]

I agree. What you are saying agrees with what I am saying. It is a theorem that the least number is one, not a definition. The notation has a standard definition which defines the notation to be equal to the least number, whatever that least number is.

Given that the notation is defined to be equal to the least number
And given a theorem that the least number does equals one
Therefore the notation is defined to be equal to a number which does equal one.
Note that it does not follow from the givens that the notation is equal to one, or that the notation is equal to the least number.

Kevincook13 (talk) 20:23, 9 January 2025 (UTC)[reply]

This is not correct, but I feel like we're talking in circles here. Cf. WP:LISTEN.

Let me try one more thing though. If we wanted a more explicit definition of 0.999..., we might use mathematical notation and write something like

0.999\ldots \ {\stackrel {\text{def}}{=}}\ \sum _{k=1}^{\infty }9\cdot 10^{-k}=1.

This is discussed in the article in § Infinite series and sequences. –jacobolus (t) 02:58, 10 January 2025 (UTC)[reply]

Can you see that the summation is a process which must occur over time, and can never end? Do you notice that k cannot equal 1 and 2 at the same time? However, if we insist that the summation does occur all at once, then we affirm that k does equal 1 and 2 at the same time. We affirm that we do intend contradiction. If so, then we should clearly communicate that intention. Kevincook13 (talk) 15:14, 10 January 2025 (UTC)[reply]

Please stop misusing the word denotes when you mean "is equal to". It's incredibly confusing. –jacobolus (t) 20:57, 9 January 2025 (UTC)[reply]

I agree that the difference between the two is critical. I've tried to be very careful. Kevincook13 (talk) 21:13, 9 January 2025 (UTC)[reply]

I don't know if this will help at all, but it may. I think that we have been preoccupied with what infinity means, and have almost completely ignored what it means to be finite. We don't even have an article dedicated to the subject. So, I have begun drafting one: Draft:Finiteness. Kevincook13 (talk) 00:00, 10 January 2025 (UTC)[reply]

I think the problem here is that there are two levels of symbol/interpretation. The literal 8-byte string "0.999..." is a "symbol for a symbol", namely for the infinitely long string starting with 0 and a point and followed by infinitely many 9s. Then that infinitely long symbol, in turn, denotes the real number 1.

It's also possible that people are using "denote" differently; I had trouble following that part of the discussion. But we need to be clear first of all that when we say "0.999..." we're not usually really talking about the 8-byte string, but about the infinitely long string. --Trovatore (talk) 05:06, 10 January 2025 (UTC)[reply]

This is also a weird use of "denote", in my opinion. For me, the word denote has to do with notation, as in, symbols that can be physically written down or maybe typed into a little text box. For example, the symbol ⁠

\pi

⁠ denotes the circle constant. The symbol ⁠

1

⁠ denotes the number one. The mathematical expression ⁠

ax^{2}+bx+c=0

⁠ denotes the general quadratic equation with unknown coefficients.

An "infinitely long string" is an abstract concept, not anything physically realizable, not notation at all. From my point of view it doesn't even "exist" except as an idea in people's minds, and in my opinion it can't "denote" anything. But again, within some abstract systems this conceptual idea can be said to equal the number 1. –jacobolus (t) 07:05, 10 January 2025 (UTC)[reply]

While you can't physically use infinitely long notation, I don't see why it should be thought of as "not notation at all". Heck, this is what infinitary logic is all about. In my opinion this is the clearest way of thinking about the topic of this article — it's an infinitely long numeral, which denotes a numerical value, which happens to be the real number 1. The reason I keep writing "the real number 1" is that this is arguably a distinct object from the natural number 1, but that's a fruitless argument for another day. --Trovatore (talk) 07:15, 10 January 2025 (UTC)[reply]

The infinitely long string. The one that is not growing over time because it already has all of the nines in it, and because it is not growing can be interpreted as a number. The one that is repeating, because it does not at any specific instance in time have all the nines yet. That one? The one that is by definition a contradiction? Kevincook13 (talk) 15:56, 10 January 2025 (UTC)[reply]

"By definition a contradiction". Huh? What are you talking about? If you can find a contradiction in the notion of completed infinity, you're wasting your time editing Wikipedia. Go get famous. --Trovatore (talk) 18:25, 10 January 2025 (UTC)[reply]

Above, I just described P and not P, a contradiction. Kevincook13 (talk) 19:26, 10 January 2025 (UTC)[reply]

Um. No. You didn't. I would explain why but in my experience this sort of discussion is not productive. You're wandering dangerously close to the sorts of arguments we move to the Arguments page. --Trovatore (talk) 21:13, 10 January 2025 (UTC)[reply]

I'm not wasting my time. I believe in Wikipedia. Kevincook13 (talk) 19:33, 10 January 2025 (UTC)[reply]

We look to famous people to tell us what to understand? Kevincook13 (talk) 19:40, 10 January 2025 (UTC)[reply]

I see Wikipedia as a great place for people to learn about and evaluate the ideas of people who, over time, have become famous for their ideas. Kevincook13 (talk) 20:04, 10 January 2025 (UTC)[reply]

The fact of the matter is that if any theory logically entails a contradiction, then that theory is logically inconsistent. If we accept logical inconsistency as fact, then we can save everyone a lot of time by saying so. Kevincook13 (talk) 20:19, 10 January 2025 (UTC)[reply]

I suggest that we address each of the following in our article:

The 8-byte term
(0.9, 0.99, 0.999, ...)
The least number
The growing sequence
The contradiction

Kevincook13 (talk) 17:11, 10 January 2025 (UTC)[reply]

There is no contradiction. There is no growing sequence. 0.999... is indeed infinitely long, and = 1. Hawkeye7 (discuss) 21:14, 10 January 2025 (UTC)[reply]

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.