Talk:Proof that 0.999... equals 1/Archive07

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Archive This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

This archive page covers approximately the dates between 2006-02-23 and 2006-06-01.

Contents

Major edit

In a long-promised back-to-basics move, I have trimmed the article of accumulated cruft, and added a sentence to the opening. Specifically:

  1. The section entitled "Definitions and justifications" is no more. I discussed this with the author when it first appeared, and also proposed deleting it on this talk page without objection.
  2. The joke, which always made an awkward section, has been restored to its original and more proper form, an external link.
  3. The wikilinks on √2 and π, which make the symbols more difficult to read for no obvious benefit, have been removed.
  4. The revised wording surrounding these is no more, as it confounded the difference between number and notation.
  5. The opening sentence has been augmented to acknowledge at the outset the role of notation, representation, standard reals, and equality.

These may appear to be bold changes, but are very much in keeping with the article before dubious alterations crept in. Thanks for keeping the talk page clear for editorial discussions like this. --KSmrqT 19:54, 25 February 2006 (UTC)

I wonder if it's necessary to emphasize the real numbers so much. On the one hand, it's important to work within the confines of the real numbers and not get distracted by larger sets of numbers, which could be anything. And infinite decimal expansions do properly belong in the real numbers. But repeating decimal expansions belong in the rational numbers, and the statement 0.999... = 1 also makes sense in the topology of the rationals. Even if you want to use the reals in the end, a perfectly valid proof could go along the lines of:
  1. 0.999... is the sum of a rational sequence which happens to converge in the rationals, and it converges to the rational number 1.
  2. The rationals are continuously embedded in the reals.
People who believe in immediate predecessors in the real numbers won't like the second step, or at least they won't if they think about it, but the first step should be easier to swallow than its counterpart in the reals. This aproach could be helpful in that it isolates a couple of issues. What do you think? Melchoir 20:14, 25 February 2006 (UTC)
It is technically interesting to contemplate such an approach. Perhaps you'd like to try a sandbox version on this page. I have two concerns in advance. (1) Ultimately we do want to relate decimal expansions to reals, not just rationals. (2) The full proof is likely to be more difficult, not less, because of the extra ideas (e.g., "continuous embedding") required. --KSmrqT 00:25, 26 February 2006 (UTC)
Sure, I'll give it a shot later. Melchoir 00:51, 26 February 2006 (UTC)
The wikilinks on √2 and π, which make the symbols more difficult to read for no obvious benefit, have been removed. Similarly, I think giving the first few digits of each of those numbers makes the explanation difficult to read (especially with the confusing misuse of ellipses as I explained earlier), for no obvious benefits.
I considered that a wikilink (and I don't see how that makes it difficult to read) to the article with the digits would be preferable to unnecessarily mentioning them here. But I'm happy to do without the wikilinks either. I'm just having difficulty seeing where exactly the objections lie to using anything other than ellipses for these numbers? Mdwh 20:25, 25 February 2006 (UTC)
You state that ellipses are misused, despite my previous rebuttal: ellipsis means omission, no more and no less. The article as written is clear in distinguishing what is omitted, recurring digits or otherwise, at each use. You and other readers might as well learn to cope now, because the usage here is standard. We want to visually represent unending decimal expansions with an ellipsis, no matter whether the number is rational or irrational. The mapping from notation to meaning is uniform (as seen in the advanced proofs), and so is the intent of the ellipsis.
Wikipedia mathematicians have discussed elsewhere the issue of links on symbols; the consensus is we're opposed. I agree with that view. If you can find a graceful way to incorporate links, attaching them to neither the symbols nor the expansions, and not overly diverting the text, give it a try here and let's see what we think. I'm open to persuasion, but not optimistic. --KSmrqT 00:25, 26 February 2006 (UTC)

Title

This seems like an odd title for an encyclopedia article. Why not "The Equality of 0.999... and 1"? Then you could start off the article with something like, "The equality of 0.999... and 1 is considered a proven fact in standard mathematics..." I don't know if "fact" is the right word in math, or if "standard mathematics" is the right term, but as it stands now, this article could be more encyclopedia-like. Nareek 16:27, 20 March 2006 (UTC)

Or why not just "0.999..."? Fredrik Johansson 17:01, 20 March 2006 (UTC)
I'm not sure the names you suggest are better. The current title describes exactly what the article is about: The proof(s) of the equailty. Titles like "The equality of 0.999... and 1" or "0.999..." don't clearly describe the subject of the article. Besides, an initial sentence like the one you describe kind of belittles the truthfullness of this statement. Kind of like saying "That the sun rises every day is considered a proven fact in standard astronomy". But I would consider a grammatical change of the title, say "Proofs for the equality of 0.999... and 1". -- Meni Rosenfeld (talk) 17:34, 20 March 2006 (UTC)
An encyclopedia organizes information by subject; that's why the WP convention is to start out an article with the subject of the article in bold-face.
I was really just trying to come up with a sentence that could start with the title of the article, but the fact is that mathematicians mean something different by .999... then lay people do ("almost 1"), and it wouldn't be bad to reflect that somehow in the lead of the piece.
0.999... would not be a bad title for the article. Nareek 17:51, 20 March 2006 (UTC)
The title "Proof that 0.999... equals 1" is both correct and a common form. Other mathematics articles use exactly this form, or the similar "Proof of". We prefer not to change the title of an article if a single proof becomes multiple proofs. See List of mathematics articles (P)#Pro. I therefore strongly oppose a move. --KSmrqT 19:55, 20 March 2006 (UTC)
There's five articles (out of what looks like hundreds or possibly thousands) using the "Proof that..." form. It still sounds funny to me. Nareek 21:27, 20 March 2006 (UTC)
Ah, it's been a while since this one rose to the top of the watchlist.
There is nothing intrinsically wrong with an article whose scope and title are "Proof that 0.999... equals 1". However, this being an encyclopedia, I believe it cries out for an article on 0.999... . Looking at other examples of articles on specific mathematical statements and their proofs:
And then we have proofs of named theorems:
Then there are two outliers Proof that the sum of the reciprocals of the primes diverges and Proof that 22 over 7 exceeds π, which should probably be fixed as well. All the other "Proof(s) of..." articles are redirects to articles with more general titles.
There are two options that make sense to me. One option is simply to create a new article named 0.999... . It would discuss the number (including human and computational confusions), define it in several ways, offer brief sketches of proofs that these definitions lead to 1, and link to the present article. However, that would make it very redundant, since the present article already includes discussion, and it already offers definitions that are not found elsewhere (for example, they are not found at Recurring decimal or Construction of real numbers). Therefore, I advocate the other option: expanding the nominal scope of this article and moving it to 0.999... . Melchoir 21:37, 20 March 2006 (UTC)
We don't want or need an article specifically on 0.999… even though we do want this proof article. This is a service article that supports a variety of general articles, such as real number and recurring decimal. --KSmrqT 22:39, 20 March 2006 (UTC)
I'd say that the issues around the number and what notation such as "..." means are better placed in existing articles such as recurring decimal - after all, these concepts apply to all recurring numbers, not just 0.9... . I think this article title is fine as it is. Mdwh 22:58, 20 March 2006 (UTC)

Intermediate proofs

Sporadically some misguided folks try to add proofs to the "Advanced" section, but without any of the necessary attention to detail necessary at that level. Details of importance include definitions of standard real numbers and their equality, and definition of the recurring decimal meaning in terms of these real numbers. The usual added "proof" is something along the lines of "just use the usual (sic) limit formula for sum of a geometric series". Please, this is not helpful. --KSmrqT 20:19, 1 April 2006 (UTC)

I've made an attempt at putting such a proof, which I believe should be in the article, in the elementary section. -- Meni Rosenfeld (talk) 07:51, 2 April 2006 (UTC)
Sorry, no again. Your geometric series "proof" is not an advanced proof, and it certainly is not an elementary proof (as in pre-calculus, for example). In fact, it is not a proof at all. As I already pointed out, it essentially assumes without comment the very things that we need to explicitly consider.
In detail, here is what you inserted:
Series proof
The notation 0.9999… actually means an infinite sum:
\frac{9}{10^1} + \frac{9}{10^2} + \frac{9}{10^3} + \cdots = \sum_{n=1}^{\infty}\frac{9}{10^n}
This is an infinite geometric series, and its sum is:
\sum_{n=1}^{\infty}\frac{9}{10^n} = \sum_{n=1}^{\infty}9(0.1)^n = \frac{9(0.1)^1}{1 - 0.1} = \frac{0.9}{0.9} = 1
I should have thought it was obvious that these statements are inappropriate for youngsters first learning about decimal fractions. Please look at the structure of the existing article, which is extremely carefully about what is introduced and when. Then look at the talk archives and see the kind of damage that is caused by careless attempts to invoke infinite sums. The existing proofs scrupulously avoid that pitfall. The "limit proof" already covers the material you would have to introduce, but does so explicitly and in careful detail.
I appreciate the appeal of a quick handwave that says, "See, it's just an infinite geometric series; done". But please trust me (or better yet, study the advanced proofs and the talk archives), this "proof" is not helpful. --KSmrqT 10:03, 2 April 2006 (UTC)
In fact, it is not a proof at all. Well hang on, none of those "elementary proofs" are really proofs, they are just as non-rigorous and "hand waving". I disagree with the geometric series proof being in the Advanced section, but I don't see why it should be listed along with the other non-rigorous proofs - or else, we should remove all the elementary proofs.
In my opinion the distinction isn't (or shouldn't be) in terms of "how easy to understand", but rather in terms of "non-rigorous versus rigorous". The only reason we list non-rigorous proofs is because the rigourous ones are hard to understand.
The concept of a geometric series is fairly basic - the only thing confusing to "youngsters" is the sigma notation, not the concept (I'm pretty sure we covered series like 1 + 0.5 + 0.25 + ... long before rigorously proving things involving infinite series - similarly we have 0.9 + 0.09 + 0.009 + ...). Perhaps it could be reworded to make it more easy to understand - e.g., the way it's phrased at Geometric sequence is much simpler. And yes, it's hard to prove the formula, but then it's equally hard to rigorously prove the other elementary proofs. Mdwh 16:03, 2 April 2006 (UTC)
Well, I disagree. I fail to see how "see, it's just a geometric series" is worse than "see, it's just 3 * 0.3333...". I think there is a significant audience that will appreciate the geometric series idea more than the two other proofs in the elementary section (which really are handwaving), and yet do not seek the level of detail that is in the advanced section. If anything, the geometric series proof should be in the elementary section instead of the two others. And I also think the article is rather poorly structured. -- Meni Rosenfeld (talk) 11:24, 2 April 2006 (UTC)
This is NOT an elementary proof. This is almost identical to the limits and series proof I put together and then pulled back apart as a demonstration for my AP Calc class. At the same time, it makes assumptions about the nature of number theory that are unverifiable without higher-level college math. I'm not so sure if it belongs here. If anything, it could be considered a verification, I guess... - CorbinSimpson 15:57, 2 April 2006 (UTC)
All of the elemntary proofs make assumptions that are unverifiable without "higher-level college math".
The point about elementary proofs, I guess, is to show consistency with other things that the reader may believe. E.g., that 1/3 = 0.3..., or that the formula for geometric series is correct. So to refute the argument, they have to also say that 1/3 doesn't equal 0.3..., or that the formula is incorrect. Mdwh 16:09, 2 April 2006 (UTC)
And if we don't think these sorts of hand-wavy arguments are appropriate, then all of them should be removed. Mdwh 16:09, 2 April 2006 (UTC)
I agree that the geometric series should be included. Perhaps it and the other elementary "proofs" should be referred to as "illustrations" instead, to sharpen the distinction between the formal proofs and the handwaving.--Trystan 18:43, 2 April 2006 (UTC)
The geometric series should be included, and all of the proofs should continue to be referred to as proofs. All this talk about which arguments are more "advanced" or "rigorous" or "formal" or "sophisticated" is cosmetic, and it misleads us from the mathematical differences: the proofs assume different kinds of usable background results. At the top, we assume that long division and the decimal system work; there are theorems to that effect, and there is no shame in using them. At the bottom, we assume that the rationals are an Archimedean field; there is a theorem to that effect, too. Geometric series call on yet another proved theorem.
I think there's a danger of falling into a kind of macho attitude: that kids rely on sophisticated numeral systems but real mathematicians start from ZF every morning. In practice, real mathematicians are lazy; they avoid arguing from definitions and assume the strongest theorems their audiences will believe. On this page, we have a broad audience, so we invoke different results in different sections, and geometric series should be one of them. Melchoir 20:18, 2 April 2006 (UTC)

I'm surprised nobody took the hint from the heading of this talk section, "Intermediate proofs". If the geometric series idea were to find its way in at all, it would logically come between the "hi kids" section and the "hi colleagues" section of the article. But again, its contribution is questionable. In more detail, here's why.

Since some folks may not have the patience to review the archives, let me recap as applied to the proposal. First, it's important to realize that historically, here and across the web, this topic draws enormous attention, some unwanted. Therefore the contents have been carefully crafted to speak to three different groups. Elementary school students learn about representation and arithmetic with integers, fractions, and then decimals; they don't know series, calculus, or ZF axioms. Higher mathematics students know what a rigorous proof is, and need to learn about the construction of number systems and representations. Critics, attackers, and the great unwashed are drawn to anything that looks like suspiciously unacknowledged handwaving.

One of the greatest attack magnets has been "infinite sum". Therefore, the article has so far avoided using this term. Instead, we talk about a sequence, and whether it is Cauchy or has a limit. Note that a sequence may fail to have a limit. Also note that a sequence of rationals may not have a rational limit, and that a bounded set may have a least upper bound it does not include.

If we say "The notation 0.9999… actually means an infinite sum", then we skip over pivotal details. Are we talking about reals or rationals or even non-standard reals? What is the sequence? Does it have a limit? What do we really mean by "infinite sum"? Then, without explanation, the infinite sum transmutes into an infinite geometric series. The sum of a finite geometric series is not a source of controversy, but again we need to be careful about hidden use of limits for an infinite series.

Thus even on its own terms, this proposal is too sloppy.

Finally, who is the audience for this proposed argument? The background to properly appreciate the sum of an infinite geometric series should also be adequate to understand the existing limit proof, which is careful but honestly not very challenging. --KSmrqT 22:04, 2 April 2006 (UTC)

The audience in my mind is those who stopped taking mathematics classes somewhere between high-school precalculus and university upper-division real analysis. Among well-educated people, this category includes almost everyone. Melchoir 22:53, 2 April 2006 (UTC)
I strongly disagree that this infinite series proof should be taken off the article. I have to say, I think that people coming to this encyclopedia will be at all levels. We need to have proofs at all levels as well. This is an encyclopedia, not a textbook at one level. I, personally, think that this proof is the most clever and the easiest to understand if you can grasp the concept of infinity. I was shocked when I came to this page for the first time and it was not there. I think that the argument that it is not a beginner or an advanced proof is useless; anyone can edit this encyclopedia, just add a new intermediate proofs section. --Mets501talk 02:26, 3 April 2006 (UTC)
It's as clever as the proof in this classic Sidney Harris cartoon. Instead of being shocked, be studious. Read the archives of this talk page. (All six of them!) Add the necessary rigor to the proposal. Then try to appreciate the genuine simplicity and economy and completeness of the limit proof — without miracles. --KSmrqT 04:18, 3 April 2006 (UTC)

Entities and Unicode

I've reverted the wholesale conversion of HTML entities, like "δ", to UTF-8 encoded Unicode characters, like "δ". Actually, the one conversion to which I really object is frasl, the fraction slash, "⁄". The problems with the ⁄ conversion are that (depending on font) it can be too easily confused with a plain slash, "/", and that the kerning effect can be a nuisance in the edit window (though it's lovely in the article). --KSmrqT 09:49, 5 April 2006 (UTC)

Problematic attempt

Along with reverting the hideous overbar, I removed the following new section, entitled Algebra:

All these proofs depend on three properties of real numbers:

  • They are a field: any two real numbers can be added, subtracted, multiplied and divided, and except for division by 0, the result will be a real number.
  • They are closed, which means that every decimal represents a real number,
  • The integers are unbounded; any real number is less than some integer.

It is perfectly possible to study systems which look more or less like the real numbers, but do not have one of these properties; and mathematicians do. For example, if the integers were bounded, the Archimedean property would also be false, and there would be two real numbers with the same decimal expansion to any number of digits; in fact, infinitely many for every decimal expansion. In such systems the proofs above would not be true; but the study of real numbers is more convenient.

The statement of dependencies is flat wrong. The order proof does not use real arithmetic, nor real closure, nor the fact that any real is less than some integer. None of the proofs explicitly use the last property of reals. Worse, the second statement confounds two separate ideas: (1) the reals are closed — which is true but not used in the proofs, and (2) every decimal represents a real number — which is not a property of the reals at all. In fact, the latter statement presupposes part of what we are examining! We require a few properties of reals, rationals, integers, and sets; but we also must carefully establish a relationship between decimal expansions and reals. So, upon examination, the proposed new section does more harm than good. --KSmrqT 14:55, 18 April 2006 (UTC)

I see my draft was unclear.
  • I assert the (obviously true) statement that every decimal expansion converges to some real number; this has been read as the false claim that these real numbers are distinct.
    • I said means; I meant, and should have said, implies
  • Any statement about decimal expansions requires at least that the reals are a ring, so that the sequence of decimal approximations is defined. Septentrionalis 15:42, 18 April 2006 (UTC)
A priori, statements about reals imply nothing about a relationship of reals to decimal expansions. We must explicitly establish a relationship. Thus:
  • Convergence to a real is not obviously true, nor are claims about distinctness. Or, at least, this is an injudicious context in which to make such assertions. If we were to use non-standard reals or alternative relationships, we could (as was pointed out) get different outcomes.
  • Using "implies" instead of "means" is still a mistake. Yes, the standard reals are a real closed field, and if we use a Cauchy sequence or limit approach to relate decimal expansions and reals, then the closure (which is not mere arithmetic closure) gets us to implication. But this is, at best, a clumsy statement.
  • We could define the meaning of decimal expansions any way we like. The definition used for the order proof is based on rationals, not reals, so it can hardly be the case that real properties are required.
The required elements for an advanced proof are three-fold:
  1. Define the real values.
  2. Define equality of reals.
  3. Define the mapping from decimal expansions to reals.
It is fun, and potentially informative, to consider other systems. Some links to descriptions of such alternatives can be found in the talk page archives. In those turbulent times it seemed unwise to add them. Perhaps now it's safe. --KSmrqT 17:32, 18 April 2006 (UTc)
Most of the objections that litter this page would, if thought through, be appeals to a non-archimedean arithmetic. Acknowledging that this idea is not incoherent, and pointing out the problems with it, may actually reduce the amount of nonsense this article has to fight off. Septentrionalis 03:41, 19 April 2006 (UTC)

I have given it another draft. I have intentionally left some assumptions, chiefly about what constitutes a "decimal expansion", unexpressed. As it is, the phrase "any finite number of places" may be too pedantic for this article. Please discuss rather than reverting. Septentrionalis 04:02, 19 April 2006 (UTC)

Sorry to revert again, but after trying to amend what was added, there was almost nothing useful left. The first bullet point is vacuous, because even for standard reals and decimal expansions it is true. I'm not convinced this little proof article is the place to go into alternative reals, but if such a section is to be added it needs to be more careful and informative. There have been repeated attempts to somehow discuss sidesteps, alternatives, what can go wrong. As a rule I believe in reinforcing a theorem by showing both examples and "counterexamples", the latter in the sense of near-misses, where one or more essential prerequisites is omitted. With that in mind, some potential content that I've brought up for this article includes reals based on a topos other than the usual Set, p-adic numbers, and Hackenstrings.
The most basic sidestep is to use a different set of values, including infinitesimals. More subtle is to do so by changing the definition of equality. It's hard to see how we can change the mapping from decimal expansions to reals if we use standard reals, but perhaps someone knows of such an option.
I believe the Hackenstrings article is the closest in content, level, and spirit to this article. Perhaps a careful sentence or two and a link would suffice? --KSmrqT 18:51, 19 April 2006 (UTC

The first statement is informal, but (read as intended) it is of course not true of the standard reals; I qualified it only to avoid arguments about "decimal expansions" of transfinite ordinal length. A collegial editor would have amended it. Hackenstrings are one case of non-archimedean algebras; a link there is insufficient. Septentrionalis 19:25, 19 April 2006 (UTC)

I have consciously attempted to avoid fine-drawn but non-intuitive distinctions, in the interests of "rigor"; there is no complete rigor without the approach of Principia Mathematica, which would be disastrous for this article. For example, I have intentionally blurred the choice: either a non-archimedean ordered ring is not a field, or it contains inverses of infinitesimals. Anyone who is going to make that distinction probably has no reason to read this article. Septentrionalis 21:42, 19 April 2006 (UTC)

I have amended and extended the new section in natural ways. I can't say that I'm thrilled with such material in this little service article, and would rather it found a better home. But in the interests of being collegial… Consider it an experiment. --KSmrqT 23:09, 19 April 2006 (UTC)

If it doesn't cut down on the fluff, it can go out again. Septentrionalis 00:48, 20 April 2006 (UTC)

Unless you are dealing with a truly strange structure, the following holds, where ε is a positive infinitesimal:

1 > 1 - ε > 1 - 10 ε ≥ 0.9999.... > 0.9999999 for every length of 9's.

Therefore the Archimedean property is necessary. Septentrionalis 00:48, 20 April 2006 (UTC)

Well actually, in the hyperreals (using the natural embedding of decimal expansions into the hyperreals), we have
1 = 0.999... > 1 - ε > 1 - 10 ε > 0.999...9
It is true, however, that without the Archimedean property a number might possess the properties we expect of 0.999... (0.999...9 < 0.999... ≤ 1) yet be different from 1. We could then choose to name one of those other numbers 0.999..., but it would be a very arbitrary choice. Rasmus (talk) 06:48, 20 April 2006 (UTC)
Rasmus (talk) 06:48, 20 April 2006 (UTC)
  • I would consider any atructure in which 0.999.... was both defined and strictly greater than an upper bound for the set {{0.9, 0.99, 0.999,...} to be a "truly strange structure".
  • If 0.999.... is not defined, the theorem discussed in this article is ( vacuously) false. Septentrionalis 18:19, 20 April 2006 (UTC)
In the reals, you usually derive the Archimedean property from the least upper bound property, so viewed in that light, it should not be a big surprise that 0.999... won't be the least upper bound for (0.9, 0.99, 0.999, ...} in a non-archimedean field. I would be surprised to find a non-archimedean ordered field in which (0.9, 0.99, 0.999, ...} had a least upper bound (let \epsilon; be a positive infinitesimal and assume x is a upper bound, then x-\epsilon; should be, too).
In the hyperreals (which I would argue is the least strange of the non-archimedean structures), you might make a case for defining 0.999... as (0.9, 0.99, 0.999, ....). Then 1-0.999... is an infinitesimal. But even then (0.9, 0.99, 0.999, ....)>(0.89, 0.989, 0.9989, ....)>(0.999...9, 0.999...9, 0.999...9, ...). (The embedding of 0.999...9 in the hyperreals is (0.999...9, 0.999...9, 0.999...9, ...). Rasmus (talk) 20:40, 20 April 2006 (UTC)
I agree; IIRC a universal least upper bound property combined with the field and order axioms implies the axiom of Archimedes, so l. u. b.'s should be rare or non-existent in non-archimedean arithmetics. Septentrionalis 17:32, 21 April 2006 (UTC)

Archimedean property

I reverted the changes of "reals" to "real numbers" as being unnecessarily wordy, and also inconsistent. For example one sentence reads

  • "The step from rationals to reals is a huge extension,…"

and "rationals" is just as informal as "reals". But frankly, both are used at least as often and as formally as "rational numbers" and "real numbers" throughout mathematics.

But in reverting, some collateral damage was the loss of a fine point about whether any advanced proof, and our limit proof in particular, implicitly depends on the Archimedean property. The limit proof certainly does not do so explicitly. It does depend on the fact that every Cauchy sequence defines a real number, and that one very specific sequence has the limit zero. But I'm not sure I would want to say these are consequences of the Archimedean property of the reals. The difference in wording used in the article is so small it makes little difference, but I'm curious. --KSmrqT 01:07, 20 April 2006 (UTC)

That depends on the definition of Cauchy sequence in the non-archimedean arithmetic:
  • If Cauchy sequence means "partial sums are eventually less than any positive element", the sequence is not Cauchy.
  • If it means "eventually less than 1/N" for any integer N", a Cauchy sequence need not define a unique element, for two elements differing by an infinitesimal will be equally valid limits.
  • Many non-archimedean arithmetics are not complete. Septentrionalis 18:26, 20 April 2006 (UTC)
Our limit proof only compares rationals. A sequence is Cauchy if the differences are eventually less than any positive rational, which works even in the presence of infinitesimals. And two reals-as-sequences are considered equal if their difference sequence has a limit of zero, again meaning less than any positive rational; this treats infinitesimally different sequences as equal. Essentially, infinitesimals are defined out of existence. --KSmrqT 20:52, 20 April 2006 (UTC)
Precisely. If Cauchy series (in this sense) converge, there are no infinitesimals. Conversely, if there are infinitesimals, series which are Cauchy relative to the rationals do not converge to any single value. Septentrionalis 22:53, 20 April 2006 (UTC)

Simple but effective

Noe substituted this:

  • And like integers, for most numbers a different series of digits means a different number. The notable exception is numbers that can be represented as a terminating decimal, e.g. 1. Without changing the value, one may either add trailing zeros (as in 1.00 or 1.000…), or decrease the last non-zero digit by one and add recurring trailing nines (0.999…).

for this:

  • And like integers, for most numbers a different series of digits means a different number (ignoring trailing zeros as in 0.250 and 0.2500). The one notable class of exceptions is numbers with trailing repeating 9s.

It incidentally raises a curious point: Why is it that nobody complains about 0.25 and 0.250 and 0.2500 representing the same real number, but the unending 9s provoke such a reaction?

Technically, the amendment is correct (including, subtly, the fact that zero has no 9s version). Rhetorically, it is less effective at drawing attention to the psychologically salient point, the theme of the article, trailing 9s. Thus I prefer to keep the original. If others have a strong preference for the extended language, please note that it's better to avoid Latin abbreviations such as "e.g." — exempli gratia, and that an example like 0.25 (with internal decimal separator) is clearer than 1. In the latter case the decimal point and end of sentence coincide, or else the decimal separator is omitted, mixing notations. (The European version would be either "1." or "1,." — which highlights the difference. Notice that the article always says "decimal separator", never "decimal point", to facilitate translation.) And we don't want to introduce 0.24999… until the end of the article, to keep focus on the title proof. --KSmrqT 17:10, 21 April 2006 (UTC)

Well, I still like my version better. I think the arguments for using 0.25 rather than 1 here, while absolutely not mentioning 0.24999…, are weak. And (being a little polemical), as the text now stands, first we are told to ignore numbers that can be written as a terminanting decimal, then we are told that the notable exception is the numbers that end in repating 9's, as if that is a completely different thing - while in fact it is exacly the same numbers. - By the way, the reason nobody objects to 0.25 and 0.2500 being equal is that everyone understands (or have been indoctrinated into believing) that zero is nothing.--Niels Ø 21:59, 2 May 2006 (UTC)

user:KSmrq recently reverted my change of "(ignoring trailing zeros as in 0.250 and 0.2500)" into "(ignoring trailing zeros as in 0.250 and 0.25000…)". I do not understand why. There is really no need for two examples with a finite number of zeroes, but 0.25 with infinitely many trailing zeroes is a slightly different thing so it might warrant an explicit mention. Thus, I think "(ignoring trailing zeros as in 0.2500)" or "(ignoring trailing zeros as in 0.250 and 0.25000…)" are fine, but I think "(ignoring trailing zeros as in 0.250 and 0.2500)" is inferior.--Niels Ø 15:18, 1 June 2006 (UTC)

(I've been enjoying the use of popups for rolling back, but they provide no opportunity to comment the change. Sorry to leave you wondering.) With 0.25 versus 0.24999… the equality requires an endless sequence of 9s. The point about trailing zeros is distinct from the issue of endlessness, and we don't want to confuse the two. So it actually matters that 0.250 and 0.2500 are both finite, both distinct from 0.25 and from each other, and yet represent the same real value. It also matters as psychological preparation. Many discussions about 0.999… act as if having two distinct representations is unusual and suspicious, and the use of endless repetition is even more suspicious. Having two perfectly innocuous examples of added zeros with no repetition reinforces the point that we use multiple representations all the time, and never fret about it. I strongly prefer the existing wording for all these reasons. --KSmrqT 23:03, 1 June 2006 (UTC)

Unencyclopedic article:liable for deletion

This article is extremely unencyclopedic, Wikipedia is not a repository for proofs of mathematical theorems. I think it can be claimed with reasonable conviction that this article wouldn't survive the merciless deletionists at AfD. However it contains some very good content. I think the only solution is to move relevant portions of the article to different articles such as Recurring decimal or other articles where the information will be relevant. This should be done soon for I will file AfD in a week or so. Of course if you think that this article can survive an AfD without at least being transwikied, you are welcome to stay inactive. Loom91 06:45, 5 May 2006 (UTC)

If you honestly think nothing can be done, why wait? AfD lasts five days, does it not? Why don't you nominate it right now? Melchoir 07:01, 5 May 2006 (UTC)
Now that I've got that out of my system, let me explain. There are at least ten other proof articles currently on Wikipedia; see Category:Proofs, List of mathematical proofs, and the "Title" section of this very talk page. Of these, only one has undergone AfD: Wikipedia:Articles for deletion/Proof that 22 over 7 exceeds π. A couple of people wanted to transwiki or merge, and a couple thought that such a numerical comparison was undeserving of proof, but the clear consensus was to keep. Some went out of their way to point out that the article does not violate WP:NOT, among other policies.
And that is, in fact, the only AfD on record of its kind. Searching VfD instead, we find only Wikipedia:Votes for deletion/Proof that 1 = 2, which was not a proof at all, and Wikipedia:Votes for deletion/Proof that 0.999... equals 1, which never happened. Perhaps you think it's time for a new test? If so, I'm serious: nominate this article now. Melchoir 07:58, 5 May 2006 (UTC)
Apologies for stating the obvious, but... Contrary to Loom91's pessimism, the article does just fine on AFD. -- Meni Rosenfeld (talk) 15:46, 5 May 2006 (UTC)
Actually, this would be notable for inclusion as a meme. - Corbin ∫ 1 ɱ p s ɔ ♫ Rock on, dude! 14:18, 16 May 2006 (UTC)

A slight problem

Obviously I don't understand this but Fred Richman is a real proffessor as far as I can tell.Geni 19:00, 5 May 2006 (UTC)

  • Largely a linguistic argument, although the objection to the multiplication of Dedekind cuts which represent negative numbers is an actual difficulty. The crux of the argument is that we should say that the limit of 0.999... is 1, or 0.999... converges to 1. I think this is purely verbal, and the ideas of limit and convergence are already present in the "..." but perhaps we should say something about this, heading towards Philosophy of mathematics. Septentrionalis 19:24, 5 May 2006 (UTC)
(edit conflict) Good find! That webpage was first discussed at Talk:Proof that 0.999... equals 1/Archive02#Mathematicians who think 0.999... and 1 are different numbers; it also appears later in that archive and once in Talk:Proof that 0.999... equals 1/Archive05. My recap is that the author is purposefully constructing a set of numbers, the "decimal numbers", where the analogue of 0.999... is unequal to the analogue of 1. This is a novel definition, and "decimal numbers" usually means something else -- see Decimal -- so really one should call his numbers "Richman numbers" or something. The author goes on to describe what can only be called flaws in his own Richman numbers, such as the nonexistence of analogues for 1/3 or −1, the impossibility of doing subtraction, and the presence of two different kinds of numbers "living uneasily together". You can't even do algebra on the Richman numbers, or get calculus off the ground. This is why everyone uses the real numbers instead, whether they realize it or not, and why expressions like −1, 1/3, and "0.999..." are interpreted within the real numbers by default.
Of course, you might want to open a discussion on whether this information should be included in the article. That idea, too, has already been discussed, although its proponent at the time (me) was acting a little unreasonably. Well, enough history. Melchoir 19:26, 5 May 2006 (UTC)
Not strictly novel; it's the intuitionist version of the standard reals, with the usual intuitionist constraints. On that level, my claim above that this is a purely verbal argument is PoV, although (as a good Hilbertian) I hold it true. Septentrionalis 19:57, 5 May 2006 (UTC)
I am unfamiliar with intuitionist mathematics. Is it really standard procedure to separate 0.999... and 1? Pardon me for my skepticism, but it seems like it would be more well-known. Melchoir 20:19, 5 May 2006 (UTC)
I am not qualified to describe at length a position with which I disagree; but I think there are three personae in Richman's article: the believer, the skeptic, and Richman himself. The believer holds that 0.999... = 1; the skeptic that 0.999... |= 1; Richman that 0.999... is a process, which is only defined insofar as it is finitely constructible. (It is at this point that my POV is that intuitionism stops being fully coherent; follow the link for something on theirs.) Thus the business about the sum of 0.555... and 0.444... being possibly <,>, and = 1; statements that involve actually realized infinite collections, like the digits in the two strings, are not meaningful in intuitionism. Septentrionalis 22:07, 5 May 2006 (UTC)

Move to "Multiple decimal representations"

I've now moved this article to "Multiple decimal representations", as per a suggestion made by someone else in the AfD discussion. -- The Anome 21:43, 5 May 2006 (UTC)

Oh, come on -- if you are just going to revert the move, at least please discuss it here. The edit comment for the reverse move was "moved Talk:Multiple decimal representations to Talk:Proof that 0.999... equals 1: Rvt unilateral move disapproved on talk." Which talk page, where, please? Perhaps I haven't looked hard enough, but I can't find any reference to this on this talk page or its archives. -- The Anome 21:44, 5 May 2006 (UTC)
  • Oh ,feel free to be bold until reverted; but the subject of this article is this proof. A general and rigorous discussion could well be put under Multiple representations; but it wouldn't be this one. Septentrionalis 21:52, 5 May 2006 (UTC)
  • I saw it just today; ask KSmrq. My apologies for assuming it was this talk page. Put this on WP:RM if you like; I will limit myself to opposing. Septentrionalis 21:52, 5 May 2006 (UTC)
  • I would oppose such a move. The article Decimal does not adequately describe multiple decimal representations, and it should be expanded to do so. This article, however, is simpler, clearer, and more relevant to most readers if it focuses on the canonical example. Melchoir 22:18, 5 May 2006 (UTC)
  • This article is not about "multiple decimal representations". Paul August ☎ 23:15, 5 May 2006 (UTC)

factuality disputes

"3 × 0.3333… equals 0.9999…; "??? the only way you could come upon that is with round-off error. 71.213.29.198 04:02, 6 May 2006 (UTC)

Seems obvious enough to me; even my mother buys that argument. In any case, the method you refer to is preceded by the caveat, "Elementary proofs assume that manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition." Now, if you'd like to dispute those assumptions, perhaps you'll be more interested in Talk:Proof that 0.999... equals 1/Arguments. Melchoir 04:12, 6 May 2006 (UTC)
I suppose we may need a caveat about intuitionism here <sigh>: Not all mathematicians agree that 0.999... is the limit of the sequence {0.9, 0.99, 0.999...}. Will this do? Septentrionalis 16:00, 7 May 2006 (UTC)
No, I find that misleading. A mathematician who defines "0.999..." as anything other than the limit (Richman again?) must be aware that he is doing a different mathematics from everyone else, whether it's intuitionism or what. Even Richman admits that the usual definitions lead to the usual result. Melchoir 20:01, 7 May 2006 (UTC)
As I understand intuitionism, it argues (more or less) that statements about limits that (in principle) involve infinite calculation are not well-defined, which seems to be what Richman is saying. But I would be perfectly happy to keep it out of the article, unless mentioning it averts edit wars. Septentrionalis 23:08, 8 May 2006 (UTC)
No, we do not need to mention this or all the other variations on the standard reals, because the article explicitly begins with assuming standard reals and because we discuss some alternatives at the end. Imagine the effect on all our articles if we started down that path! Gruesome. So relax. --KSmrqT 20:06, 7 May 2006 (UTC)

Thought on the discussion

Just out of curiosity, for those disputing this article, assuming we knew nothing of the rigorous proofs given here, does it really matter if 0.999...=1 or not? It's a widely accepted idea that this is so, and in the real world, being right is only worth one point out of ten. It's like when your dad's fixing the door because it's too tall. You try to tell him that he's sawing off the wrong end, but when he doesn't want to hear it, being right doesn't do you a lick of good, it won't change what he's doing. I remember a girl at a physics symposium who had a chance to discuss limits of functions versus their values at a given point. She very correctly stated that \lim_{x\to\infty}{1\over x}=0 and that one over infinity was not really zero. She pushed herself on anyone who would debate this topic with her and left a lasting impression of stubborness in their minds. People that could have made her life easier in a couple of years and from whom she could have learned many things, now only remember they never want to see her again. What I'm trying to say is, being right won't change anything, it just makes for a lot of unnecessary stress. Guardian of Light 03:34, 9 May 2006 (UTC)

Yes, it does matter. If you're doing mathematics and you assume one inconsistency, you can prove anything, including 1=2. Being vague about limits, in particular, kept mathematicians producing nonsense (of the form -1 > infinity) throughout the eighteenth century. Septentrionalis 05:10, 9 May 2006 (UTC)
You raise one of the most important philosophical questions for a human being. Is it necessary to do the right thing or only the easy thing. There is no right or wrong answer. Loom91 07:05, 9 May 2006 (UTC)
Well, Guardian of Light, assuming that your audience isn't going to show up and answer your question, maybe I can. Historically, the deniers on this talk page have had axes to grind with mathematicians and educators, and they exult in being smarter than all us sheep. So, there's the personal side. Then, some of them also seem to think that all of analysis is wrong, so they imagine they're fighting a larger battle than just this one point. And finally, at least one user has been confused but actually willing to learn. Melchoir 07:19, 9 May 2006 (UTC)
Ah, then you would love the Indiana Pi Bill. Assistance like this we do not need, thanks just the same. While it is certainly true in human affairs that being right is not the same as being persuasive, mathematics demands both. It can be a stressful discipline. But here's a little insight, courtesy of Richard Feynman: “For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.” He was investigating the Space Shuttle Challenger disaster. If you think being right can cause stress, try being the decision-maker who was wrong. --KSmrqT 08:15, 9 May 2006 (UTC)

Comment

Perhaps the article should go into some of the reasons that people intuitively have a hard time accepting that 0.999... = 1 not approximately but exactly.

Also I'm not entirely satisfied with the title. It is certainly descriptive but it seems like an odd format for the title of an encyclopedia article. Is there a short formal name for this set of proofs or for the question that they settle? If not I'd suggest something like "Equality of 0.999… and 1" Dv82matt 21:19, 11 May 2006 (UTC)

I think perhaps this bit could be expanded on a little more, as it seems to describe the logical error that people fall back on when trying demonstrate the supposed inequality. "The 9s case does surprise, perhaps because any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1. Thus infinity, a sometimes mysterious concept, plays an important role behind the scenes." --Trystan 00:35, 12 May 2006 (UTC)
We've discussed and discarded this more than once; see #Where we go wrong, for example. --KSmrqT 00:43, 12 May 2006 (UTC)
For the record, I still support this idea. Melchoir 01:37, 12 May 2006 (UTC)
Much of the interest in this article is generated by the feeling of incredulity the concept generates. I think it would be worthwhile to address it as it would be of interest to many people. Dv82matt 02:34, 12 May 2006 (UTC)
There's a relatively common error made in induction, where instead of concluding that the result applies for all finite values of n, i.e. for any finite number of 9s following a decimal, the erroneous leap is made to assume the result applies for the case of n = infinity. That seems to be analagous to the predominant line of faulty reasoning that makes people insist that 0.999... must be less than 1.--Trystan 05:27, 12 May 2006 (UTC)
For example, the post below. It may be worth stating explicitly that 0.999... is not 0.999 or 0.9999 or 0.999...999 for any finite number of 9's. Septentrionalis 18:30, 12 May 2006 (UTC)
Thanks, I think much of the incredulity about the concept does stem (in one way or another) from that faulty induction step you mention. Another possible source of confusion might go something like this, if 0.999... = 1 then why doesn't 0.888... = .9. Obviously it is trivial to see the misunderstanding here but it may appeal on an intuitive level where it is not made explicit. --Dv82matt 21:28, 12 May 2006 (UTC)
That's not a good example because it makes no sense. It's obvious that 8/9 is not equal to 9/10. Rishodi 00:29, 13 May 2006 (UTC)
You are right, it is obvious. I was intending to show why it might likewise seem obvious that 0.999... is not equal to 1. The mistaken impression is that any number with an infinite series of nines to the right is somehow being treated differently from other comparable numbers.--Dv82matt 01:24, 13 May 2006 (UTC)

I think unless we can find a reliable source which says why people have difficulty with this idea, any reasons we might give would be speculative and hence in violation of WP:NOR. Paul August ☎ 20:48, 17 May 2006 (UTC)

I would definitely agree with that. The new Pop Culture section already references the work of David Tall and "some of the misunderstandings he has encountered in his college students". The specifics of some of those misunderstandings are described in his paper Cognitive Development In Advanced Mathematics Using Technology, with the most relevant section beginning on page 12. The upshot of the part dealing specifically with 0.999... is:
Interviews revealed that students continued to conceive of 0 ⋅ 9˙ as a sequence of numbers getting closer and closer to 1 and not a fixed value, because "you haven’t specified how many places there are" or "it is the nearest possible decimal below 1".
Another paper, Mathematical Beliefs and Conceptual Understanding of the Limit of a Function by J.E. Szydlik, isn't freely available in full text without access to JSTOR, but echoes the "incomplete process" and "infinitesimally smaller than 1" views described in the first.
The above is the result of a quick Google Scholar search, so there could easily be more out there. It's somewhat difficult to determine what search terms would capture works on the subject.--Trystan 03:42, 18 May 2006 (UTC)

Corbin's Proof

I have a more rigorous proof grounded in series, but this one will do for now. All you have to do to demonstrate that .9~ =/= 1 is refute the following: All repeating decimals have a corresponding rational representation. What is the rational representation of .9~ if it is not 9/9? - Corbin ∫ 1 ɱ p s ɔ ♫ Rock on, dude! 14:14, 16 May 2006 (UTC)

Ah, but if you are doing your arithmetic correctly, 9/9 will always produce 1, and never 0.99999.... This 0.99999.... is a silly way to show 1 in the first place as no ratio of integers will produce it when you do your division properly. (Anon. post)

That is exactly the point. Why on Earth write one as an infinite digit string when you can just write "1"? However, the meaning of 0.999... can only be 1, as numerous different arguments show. I think it is important to understand that the meaning of decimal numbers is not God given, but defined by Man. Mathematicians could let decimals ending in 999... be undefined, thereby avoiding some problems:

  • All reals would have exactly one representation (apart from the ambiguity involved in 1, 1., 1.0, 1.00, and 1.000..., and in 0, -0).
  • Any number beginning 0.9 would be strictly smaller than 1.

But by choosing to live with these problems, mathematicians avoid other problems:

  • All digit strings correspond to exactly one real number.

There is just no way of getting a one-to-one correspondence between digit strings and real numbers.--Niels Ø 08:26, 26 May 2006 (UTC)

That is like making a law that pi is exactly equal to three. All real numbers have more than one representation, and that has been defined in mathematics for a really long time. —Mets501talk 11:08, 26 May 2006 (UTC)
Now that comparison is not correct. The ratio of the circumference of a circle to its diameter is what it is; "defining" it to have a different value would introduce a contradiction. We (mankind) have no choice here - though of course, one may define the letter π to represent any value one likes, and then call the ratio something else. With the decimals, the situation is different: Disallowing 0.999..., we would be contradicting the vast majority of mathematicians, but the system would still work. Here, mankind had a choice, though we wikipedians do not: We have to report what mathematicians do. If we fail to understand such differences, and fail to express ourselves clearly about them, our articles will be open to attack, and, given the open nature of wikipedia, they will be attacked. Let's aim at an article that is irrefutably correct, and that answers the most common objections.--Niels Ø 14:09, 26 May 2006 (UTC)
Exactly. Paul August ☎ 17:29, 26 May 2006 (UTC)

An addition to the elementary proofs section:

Ladies and gentlemen, I propose an addition to the elementary proofs section. The proof I would like to add is as follows:

===Another fraction proof===
Any repeating single digit is equal to that digit over nine; for example, 0.444... is equal to 4⁄9. (This can be proved by dividing 4 by 9; the result is 0.444...) Therefore, 9⁄9 must be equal to 0.999..., but any number over itself equals one, so 9⁄9 = 0.999… = 1.

If anyone sees a reason why not to add it or if we should change it, please leave comments. Otherwise, it makes sense and is a fairly simple explanation. The only argument against it I can see is that the article has already made the point; but I think that even if we can never have enough proof to satisfy the nonbelievers, we might as well put more.

(Actually, I already added that section to the article, but then removed it once I saw the extensive talk section -- I figured I'd better put it on here first, or else I'd end up getting a bunch of angry mails ;-) .) Bobburito 04:40, 4 June 2006 (UTC)

I think that it's too redundant with the 1/3, 2/3, 3/3 thing to be included. Maybe both in the same paragraph would work though. —Mets501talk 05:40, 4 June 2006 (UTC)
Please do not add this. Experience shows that it raises more questions than it answers. A typical objection is that a proper computation of 9 divided by 9 will never give the repeating decimal. An advantage of the 1⁄3 argument already in the article is that it shows how repeating 9s can naturally occur in a routine computation, and that the result needs to be 1 for consistency. (Thanks very much for asking first!) --KSmrqT 08:45, 4 June 2006 (UTC)
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