Talk:Proof that 0.999... equals 1/Archive10

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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.

Contents

Move from Talk:Proof that 0.999... equals_1

Should the archives be moved, as well?

Talk:Proof that 0.999... equals 1/Archive01
Talk:Proof that 0.999... equals 1/Archive02
Talk:Proof that 0.999... equals 1/Archive03
Talk:Proof that 0.999... equals 1/Archive04
Talk:Proof that 0.999... equals 1/Archive05
Talk:Proof that 0.999... equals 1/Archive06
Talk:Proof that 0.999... equals 1/Archive07
Talk:Proof that 0.999... equals 1/Archive08
Talk:Proof that 0.999... equals 1/Arguments/Archive 1

— Arthur Rubin | (talk) 22:49, 14 September 2006 (UTC)

Meh. I guess it couldn't hurt. Melchoir 22:52, 14 September 2006 (UTC)
There's no need, as long as the links from this page are ok. Mariano(t/c) 09:51, 15 September 2006 (UTC)

Great Article

I liked this article, it made me think. For what it's worth I think this equation should make everything clear:

0.999... + (⅓ - 0.333...) = 1

Or in other words the assertion that 0.333... = ⅓ is also wrong and hence the fractional proof is rubbish.

simonthebold 23:48, 25 October 2006 (UTC)

Exactly?

I agree with Savidan's recent edit, and this is actually something that has bothered me for a while. The reason is that people have a tendency to use strong words when they don't actually mean them. For example, someone might say "I have measured the length of this rod, and it is exactly 24cm". The length is obviously not exactly 24cm, and it is unlikely that the confidence is more than 0.1mm. What he meant to say is that the amount of error is extremely small by his standards. Likewise, one might interpret "0.999... is exactly 1" as "the error in approximating 0.999... as 1 is extremely small". In this scenario, using the word "exactly" has failed to fulfill its purpose. Sticking to just "0.999... is equal to 1" clarifies that we are talking strictly about the mathematical notion of equality, and not about some vague notion of precision. This mathematical notion is clear enough; two things are either equal, or not. Anyone with any doubts about what equality really means can always follow the link. -- Meni Rosenfeld (talk) 11:22, 20 September 2006 (UTC)

Personally, I disagree. The fact that people misuse exactly does not mean that it now means the opposite thing. "Exactly" still means "in an exact fashion", the fact that people sometimes use "exactly" when they should use "with high accuracy" does not mean the two things are now synonymous. In fact, I think the word "exactly" stops people using a retort I've witnessed, saying, "well, they're very close, so they're close enough to be called equal". -Maelin 13:21, 20 September 2006 (UTC)

Of course, "exactly" and "with high accuracy" are not synonymous. All I'm saying is that people may interpret it this way. I have little doubt that the word "exactly" will not eliminate the confusion, and I tend to believe it will increase it. I don't see anything stopping people from saying, "well, they're very close, so they're close enough to be called exactly equal". -- Meni Rosenfeld (talk) 13:54, 20 September 2006 (UTC)

It's true many people do not understand the word "exact" (or "exactly"). But there is no doubt what the correct meaning of this word is, and the argument against using it here turns back on itself. Most of us have made math or physics shcool work where we've written awful things like "h=24.0 m", meaning the height of the flag pole (as it most likely would have been) is 24 m plus/minus 0.05 m. So when talking about exact equality, I think the claim that "equal" is less likely to be misunderstood than "exactly equal" is wrong.--Niels Ø 15:31, 20 September 2006 (UTC)

Well, if you all think that adding "exactly" reduces the confusion, then by all means add it. Perhaps we should add a clarifying note like "That is, 0.999... and 1 are two symbols for the same thing"? -- Meni Rosenfeld (talk) 16:32, 20 September 2006 (UTC)

I've added "exactly" plus a clarification. What do you think? —Mets501 (talk) 18:53, 20 September 2006 (UTC)

Perfect. I didn't really like Supadawg's version, though, so I reverted to yours. -- Meni Rosenfeld (talk) 08:25, 21 September 2006 (UTC)

I don't feel strongly about "exactly", so I'm going to take this issue and run with it. Among non-mathematicians, real numbers often are decimals, in which case the equality of numbers 0.999... = 1 is something that has to be enforced by hand. Of course, mathematicians know that there is a kosher way to force things to be equal: passing to the quotient by some equivalence relation. Where does the equivalence come from, and what does it mean? It's just the realization that the decimals 0.999... and 1 are very close -- close enough to be called equal. So we take two symbols for unequal things, and we "call them equal" by first calling them equivalent and then allowing them to represent their own equivalence classes. This convenient, notation-abusing sleight-of-hand goes unnoticed by the laity, who rightly suspect that we're attempting to force almost-equal things to be truly equal, and who decline to assume that we know what we're doing.

My point is that we shouldn't worry too much about finding ways to dispel popular myths, either by wording or by content. Today it's just the word "exactly", but tomorrow...? That way lies POV and OR, and more importantly this particular myth is largely true. What we really need is a couple of references that build the real numbers from decimal expansions, so that we can expose that truth. Melchoir 19:06, 20 September 2006 (UTC)

I disagree with much of what you write here, for the simple reason that the decimal expansion is not the primary meaning of a real number, it's just a convenient mechanism which turns out to exist (both conceptually and historically - people have discussed √2 way before the decimal expansion was invented). The primary meaning is a minimal set of numbers which extends the rationals and has no holes (in the sense of, say, the Intermediate value theorem). Just because some non-mathemticians think otherwise does not make "0.999... < 1" any more correct. -- Meni Rosenfeld (talk) 08:23, 21 September 2006 (UTC)
Whoa there! I don't claim that decimals are the primary meaning of the reals, nor would I base any argument on such a claim. For that matter, I don't admit the existence of a "primary meaning" at all. But decimals are historically the first definition of a real number (√2 is interesting but it doesn't get you the set of irrationals); educationally they are the only definition that gets taught to non-specialists; and popularly they are the only definition most people know, if they know one at all. And they work. Just because some mathematicians shun them doesn't make decimals any less correct!
If you're worried that I'm about to insert a section claiming that 0.999... < 1 for some reason connected to decimals underlying the reals, fear not. I don't think that way, and even if I did, I'm not stupid enough to try it. Unfortunately, without a crystal ball, I can't describe in detail the hypothetical section we should have. I just know it's missing. Melchoir 10:07, 21 September 2006 (UTC)
(Decimal) Dedekind cuts are just a mathematically precise way of saying that a decimal expansion *is* the meaning of a real number. A decimal expansion is a sequence of rational numbers less than the the real number it represents (except for the last digit in the expansion, if there is one). Since there is an ordering on the rational numbers, this is equivalent to a Dedekind cut.35.11.50.219 16:36, 20 October 2006 (UTC)
No, not really. Of course, Dedekind cuts and decimal expansions are ultimately equivalent to each other, and to every other definition of real numbers, but they are quite different at the outset (there's more to the decimal expansion than the mere fact that it is a sequence, and Dedekind cuts are about sets anyway). Also, if I were to choose a digit expansion as the fundumental definition of a real number, I would have obviously chosen binary, not decimal. I repeat my statement: The decimal expansion is just something that happens to exist, and which can be (easily, perhaps) shown to be equivalent to the other constructions. -- Meni Rosenfeld (talk) 13:51, 22 October 2006 (UTC)
I'm glad we've cleared that up. I'm still not convinced that this "hypothetical section" is necessary, but we'll get to that when you have a better idea of what you think should be added. -- Meni Rosenfeld (talk) 10:21, 21 September 2006 (UTC)
Well, it might be some time. Currently this material isn't in the cards... and by the cards I mean the books. Melchoir 10:35, 21 September 2006 (UTC)

Pictures

Hey WTF is with the stupid pictures. They make no sense and add nothing. —The preceding unsigned comment was added by 82.93.10.238 (talk • contribs).

The pictures are a visual representation of a difficult-to-visualize mathematical concept. If you're unable to understand them, it is not because they "make no sense", it is because you haven't been taught how to interpret them. Melchoir was kind enough to explain it to me here. By the way, you can sign your comments using four tildes (~~~). Also, additions to talk pages are supposed be appended at the bottom of the page. Thanks. Supadawg (talk • contribs) 21:59, 28 September 2006 (UTC)
I'm not _quite_ as dumb as that. I meant the other two specifically, in fact, the pathetic "artist's impression" (WHY?) and the jocular(??) and pointless calculator image.82.93.10.238 22:09, 28 September 2006 (UTC)Flutterby
Hmm, didn't see those; thought you meant the other picture. It might have helped, though, if you had provided specific examples and reasons why you thought the pictures should be removed. Having said that, I completely agree: they add no information whatsoever to the article. The artist's interpretation of 0.999... does not help one grasp the concept of 0.999... equaling 1, and the calculator image is of a syntax error, not a problem with the calculator. Supadawg (talk • contribs) 22:16, 28 September 2006 (UTC)
I've never been sure what to do with Image:999 Perspective.png, and I didn't add it to the article, but I'll defend it anyway: It is the only depiction of the infinite decimal itself, and the clearest indication that the 9s repeat infinitely. The calculator image accompanies text about the usefulness of calculators. What's wrong with images that illustrate the topic without trying to make some point? Melchoir 22:25, 28 September 2006 (UTC)
I think that if the article was still "Proof that ...", then the artist's impression wouldn't be much use. Now that the article is on 0.999... itself, I would say that we need some picture of the number, and just pointing to "1" on the number line wouldn't really cut it. I actually think it's a rather neat way of showing the infinity of nines without resorting to the ellipsis notation. As for the calculator one, put me down as neutral. Confusing Manifestation 04:12, 29 September 2006 (UTC)
Yeah, I like the artistic impression image (in fact, while this is the first time I've seen such an image, I've been thinking about how interesting it could be many years ago), and completely agree with everything ConMan said. I think it should stay. About the "Calc thingy" image, I am inclined to exclude it. -- Meni Rosenfeld (talk) 07:19, 29 September 2006 (UTC)
Some exquisitely moronic comments here. Why is some picture the "clearest indication that the 9s repeat infinitely"? That's what we have the definition for. Pictures lie. Why do we _need_ "a picture of the number"? There is not even such a thing. It's at best a picture of this particular representation of 1. Whoop. All in all, it's at best an obfuscation. And the syntax error is too silly for words and really indefensible in anything that takes itself seriously. But hey, it's your Wikipedia.82.93.10.238 21:28, 30 September 2006 (UTC)Flutterby
It is an artistic representation. If it is free, it is welcomed. -- ReyBrujo 21:43, 30 September 2006 (UTC)
There is no place for personal attacks on Wikipedia or in any intelligent discussion. If you don't agree with someone, fine, but that does not make them a "moron". The picture of 0.999... is not necessary; it clarifies that 0.999... recurs without end. The calculator screenshot, though I think it should be excluded, illustrates the point that many people disagree with the assertion that 0.999... equals 1 because technology is incapable of assisting them. Also, it's everyone's Wikipedia, and that includes you. Supadawg (talk • contribs) 23:39, 30 September 2006 (UTC)
Technically, he was not making a personal attack, since he was only saying that the comments (not the people who made them) are moronic. In any case, of course the image does not add anything mathematically, but this is not a mathematical textbook here. This is an encyclopedia entry, and there's no reason we shouldn't add images which illustrate the concept - the same way articles about people have pictures \ drawings of them, and articles about geometric figures have images of them. -- Meni Rosenfeld (talk) 09:30, 1 October 2006 (UTC)
Splitting hairs over the definition of a personal attack shouldn't protect him. Check the section on wikilawyering. Supadawg (talk • contribs) 19:33, 1 October 2006 (UTC)
Still, I'm glad the picture of the calculator went. I do like the artist's impression of 0.999 -- but 'it's free' is not an argument for putting it in. If an article does not need pictures, leave them out! -- Cugel 07:38, 25 October 2006 (UTC)

Fractals

"Applications of these identities have been used in patterns in decimal expansions of fractions and fractals." I'm particularly curious as to what fractals make use of this property. Do any even exist? —The preceding unsigned comment was added by 82.93.10.238 (talk • contribs).

I've tweaked that sentence to be more specific; the application itself is in 0.999...#Applications. Melchoir 22:31, 28 September 2006 (UTC)

Lead section, again

For the FAC, I've expanded the intro, hopefully in a different way from before. Supadawg, I hope this isn't too detailed for your tastes. Melchoir 22:18, 29 September 2006 (UTC)

Hm? I don't remember speaking out against detailed introductions; I just remember moving parts of the old introduction to their proper place as section introductions. I have no problem with the new version, as it does a great job of touching on each subject presented in the article. I just think it's a little esoteric, that's all. Supadawg (talk • contribs) 22:40, 29 September 2006 (UTC)
Thanks! A little esoteric I can live with. The details I guess I meant were the listed descriptions of the proof methods. Melchoir 22:43, 29 September 2006 (UTC)

Ordering

I'm going to try re-ordering the sections (again) in response to criticism in the FAC. Here's my thought process:

  • Given that the material is broken up into sections by topic, it would be impossible to order it chronologically or by difficulty.
  • And ordering the material by strict implication would make it unreadable and unmotivated.
  • Therefore, the sections should be ordered by prerequisites and by motivation.
  • Applications does not inform any other section, and Generalizations informs only Applications and a single subsection buried deeply within Other number systems. Neither section motivates anything else. So these two should come last.
  • The first two proof sections inform Skepticism, while Skepticism motivates the third proof section along with Other number systems. So Skepticism should be inserted into the break, or at least the Education part; Popular culture should stay at the end.

Advantages:

  • Motivation.
  • The most well-known material occurs in what are now the first three sections, while obscure stuff appears later.
  • Independent of logical issues, it just makes sense to follow the section on real numbers with the section on not-real numbers. It may also be more neutral.

Disadvantages:

  • The proofs that 0.999… = 1 are interrupted. (But it is now more clear why the process of finding more proofs is extended so long, whereas before the process was carried out as a monolith, without motivation or breathing room.)
  • The proofs - skepticism - more proofs layout may appear to be a contrived debate-of-the-strawmen, which is usually a sign of POV and OR. (But here there is no POV, since only verifiable opinions are described. And the skepticism/education angle is apparent in the references that support the final proof section, so the establishment of motivation isn't OR either.)
  • The sections following "Other number systems" are easier to understand than it is, not harder. (This is bound to happen unless one places that section at the end, which would be logically unsatisfactory and too pro-real.)

Whew! Melchoir 16:50, 2 October 2006 (UTC)

Oh, and I realize that we'll (I'll) have to rewrite some of the glue prose. But a lot of it needed rewriting anyway. Melchoir 16:55, 2 October 2006 (UTC)

Psychological issues?

"Many students reject the equality and become vocal with their objections on the Internet, revealing a variety of psychological and mathematical issues" (2nd paragraph, emphasis added) Personally, I think that's very funny, and I therefor hesitate to say that I don't know that it's quite appropriate. Or maybe it just needs a citation :) By the way - there's some nice work that's been done on this page. I'd been given the informal "0.999... = 1 because you can't fit another number between them", but a more rigorous proof was nice to see. --Badger151 18:40, 7 October 2006 (UTC)

Hmm... by "psychological" I was trying to refer to the intuitive process vs. object issue that gets some play in the Skepticism section. I can certainly expand on that area if you like; it's the subject of serious research, and there are untapped resources to bring in.
"Psychological" might be poor wording, but I can't think of a decent replacement. "Educational" would be kind of clumsy, and "developmental" is kind of jargonish... Melchoir 19:29, 7 October 2006 (UTC)
I don't know, I could see some real psychological issues coming up, though in all seriousness, I think I see what you mean. Cognitive seems very clinical. Perceptual, perhaps? That's the word that comes to mind in skimming David Tall's article from the reference list.
I haven't looked through many of the mathemetical articles here, but I'm guessing that the difficulties faced by students in understanding this concept also interferes with their understanding other concepts. Better than expanding on the subject here might be on a page on learning theory(?) --Badger151 00:38, 8 October 2006 (UTC)
I like cognitive better. A broad article on advanced mathematics education would be really hard to write. Some of the examples I have in mind speak specifically to 0.999…, so they belong here better anyway. I don't have much to do today, so perhaps I'll show you what I mean. Melchoir 22:08, 8 October 2006 (UTC)
I like that paragraph you added. As I may have said, I'm not schooled in advanced mathematics (I took AB calculus, but setting up the equations baffled me) but the 3 * (0.333... = 1 / 3) = 0.999... = 1 proof works well for me. The difficulties you describe later in the paragraph remind me of the more universal ability people often have for selectively disbelieving things that they find uncomfortable to believe, be it information in mathematics, politics, or religion. There might be a term for it; I'll take a look.
For the initial discussion (in the header) of the difficulties students face, I wonder if it might make sense to directly link to the "Skepticism" section: "...revealing a variety of psychological and mathematical issues (see here)..." enabling the curious reader to link directly to the explanation. --Badger151 06:02, 9 October 2006 (UTC)
Thanks, and there's more; I just have to marshal it all together. I prefer not to have section links within the lead section, since theoretically it should describe the entire article anyway, and the table of contents should be enough. Melchoir 16:57, 9 October 2006 (UTC)

trivial question...

I randomly stumbled across this article, and now I'm curious, what would be the correct way to represent "the greatest possible value that is less than 1"? Or is such a concept undefined, and, if so, does it have a representation outside of mathematics? 130.111.246.77 03:11, 13 October 2006 (UTC)

Well, there's no such real number. But you don't have to go outside of mathematics to get outside of the real numbers; it's fairly routine in mathematics to augment standard structures with new elements, if they have desirable properties. Like adding i and getting the complex numbers.
There are ways to enlarge the real numbers and produce a set where 1 does have an immediate predecessor. What that new number would be called is left to be determined by whoever popularizes the notion, but "1−" would be a good choice. It may even coincide with 0.999…, as in Richman's "decimal numbers" from the 0.999...#Breaking subtraction section. I'd have to ask, though, why would you want to add such a number? Would it be good for anything? Melchoir 03:28, 13 October 2006 (UTC)
There is no such number. There are an infinite amount of numbers less than 1. Every time you choose a number x where x < 1, there will always be a value y such that x < y < 1. Falsedef 05:35, 26 October 2006 (UTC)
Oops, I've seen that you've already talked about number density below Melchoir. Can you explain the following further: "There are ways to enlarge the real numbers and produce a set where 1 does have an immediate predecessor." As I believe any set containing a predecessor would not be discrete, and therefore inapplicable. Falsedef 05:43, 26 October 2006 (UTC)
Well, you could take the ordered product of the reals with the integers R x Z, identifying the real number r with the pair (r, 0). In the new set, every number (r, n) has an immediate predecessor, namely (r, n - 1). So the immediate predecessor of 1 would be (1, -1). This set is discrete, whereas the real line isn't discrete. Melchoir 06:05, 26 October 2006 (UTC)
The system is quite interesting, especially concerning infinite series, but seems to be quite pointless. Other than limits where n approaches infinity, n and r values always seem to be independent of eachother. Perhaps it could be used as representation of a hierarchy system? Falsedef 18:49, 26 October 2006 (UTC)

Impressed!

It's been several weeks since I've looked at this article. The lead is SO much better. Reading the lead as a lay person to mathematics now tells me why this article is important and that it is reported established fact (before it seemed to suggest that it was instead establishing that fact). The general shape of the article is much improved, especially the additions of the Skepticism section and the sections about application and related questions. This is now an article that I can access, find intersting, and use. Well done and thank you to everyone. Chadbald 01:31, 14 October 2006 (UTC)

Thank Melchoir, he's done most of the work. Well, there is at least one person, whose opinion I value too highly to disregard, who thinks the article is terrible in its current state - but he seems to be the only one. We're glad you like it. -- Meni Rosenfeld (talk) 21:38, 14 October 2006 (UTC)
I realise that this might not be the place, but I wanted to let those responsible for this article know that I found it a good read. Quite good how you included information on the issue of intuition. Was wondering, is 0.9999...8 the same as 0.999.... ? Cheers. ShaiM 12:41, 22 October 2006 (UTC)
No. 0.999...8 isn't defined, since you'd need to find the end of an infinite (i.e. endless) string of nines on which to stick an 8. And since the string of nines is endless, that end doesn't exist. Many people propose a similar number of 0.000...1 as a potential answer to the question "What is 1 - 0.999...?", but such a numeral is meaningless, at least in the real numbers (the correct answer is, of course, zero). The very fundamental concept of a recurring decimal is that its decimal expansion does not end - there is no last place which you can change. -Maelin 13:24, 22 October 2006 (UTC)

Simiplification

"This can be simplified by saying that: given any real number over 9, the number repeats itself infinitely after the decimal point, then 9⁄9 equals 0.999... which equals 1."

Why was this deleted? any number divided by nine euqals the number reapeating itself infinitely after the decimal point. For example, 1/9 = .111..., 5/9 = .555..., 12/99 = .121212..., 123/999 = .123123123..., so what does 9/9 equal to?

The statement is correct, but it's clumsier than what is said above, and is not really a proof. The fact that 1/9, 2/9, 3/9 etc. equal 0.111..., 0.222..., 0.333... etc. does not prove that 9/9 = 0.999.... The same way that the fact that 02 + 0 + 41, 12 + 1 + 41, 22 + 2 + 41, ..., 392 + 39 + 41 are all primes does not prove that 402 + 40 + 41 is prime (and it isn't). -- Meni Rosenfeld (talk) 21:29, 14 October 2006 (UTC)

Perhaps this idea belongs more in the Generalizations section towards the bottom? Melchoir 05:45, 15 October 2006 (UTC)

...I don't understand the point of this addition to 0.999...#Algebra proof either. The pattern can be proved by a generalization of any of the first three proofs: taking multiples of 0.111… = 1/9, multiplying by 10, or summing a geometric series. In fact, I'm pretty sure that all of these methods occur in elementary textbooks. So what's the meaning of "this proof does not work", or for that matter "the original premise"? And why "zero cannot repeat into infinity"? We should just delete it and add a short note to 0.999...#Generalizations. Anyone agree? Melchoir 22:10, 18 October 2006 (UTC)

Move request

It seems that it hasn't been proposed yet, but could you move the article to 0.999... = 1? I think it's more relevant and descriptive. CG 14:37, 15 October 2006 (UTC)

This article was recently moved to the present title. I suggest we stop moving artiles around, just because they could be called something else.--Niels Ø 15:24, 15 October 2006 (UTC)
It's just a suggestion. CG 17:10, 15 October 2006 (UTC)
I don't think titles should even be descriptive. See, for example, the redirect from September 2006 Thailand military coup d'état to 2006 Thailand coup d'état, which is the simplest title that gets the job done. "0.999..." also avoids making a statement. You wouldn't move the other articles to Negative zero equals zero, Division by zero is undefined, or Zeno's paradoxes are resolved by modern calculus, would you? Melchoir 20:00, 15 October 2006 (UTC)
If we want the simplest title then surely it should be simply 1, since 0.999... = 1 (duck). I'm generally against the move, as the current article works well under its current name, a page describing the decimal expansion 0.999... Also as its on WP:FAC at the moment it causes problems moving at this time. --Salix alba (talk) 20:22, 15 October 2006 (UTC)


Couldn't it be that 1/9 = .111..., 8/9 = .888... so if you add, 9/9 = .999... = 1 Xenero 01:07, 30 October 2006 (UTC)

Better explanation of base 2 and 3?

Referring to the "Generalizations" section, the article says "For example, in base 2 (the binary numeral system) 0.111… equals 1, and in base 3 (the ternary numeral system) 0.222… equals 1." Would it be easier to understand if the article said "For example, in base 2 (the binary numeral system) 0.111… equals 1 (in base 10), and in base 3 (the ternary numeral system) 0.222… equals 1 (in base 10)."? That part just seemed a bit ambiguous to me, so it's just a suggestion. Brix. 17:36, 20 October 2006 (UTC)

Since 1 means the same thing in binary, ternary and decimal notations, I fail to see the point of your suggestion. -- Meni Rosenfeld (talk) 18:58, 20 October 2006 (UTC)

Tone?

Does anyone else think this article has a horrible condescending tone?

Where? Melchoir 20:46, 24 October 2006 (UTC)
Throughout the whole piece, but especially in the 'Skepticism in Education' section. It doesn't seem very Wiki to me, and certainly not featured article...Ravenmasterq 23:46, 24 October 2006 (UTC)
Well, could you point out a sentence that has a horrible condescending tone? Melchoir 00:00, 25 October 2006 (UTC)

Yes. Yes it's condescending, that is.WikiManGreen 00:35, 25 October 2006 (UTC)

Best not to get too stressed over what gets featured. Part of why I left for a bit was that the featured articles seemed rather poor to me so I thought "if this is the best they can do, why bother?" However featured isn't so much "the best they can do" as "this is what several editors have agreed to call the best at this particular moment" which is a much different thing. Featured articles have been removed. In any event the tone is "some people don't accept this, those people are idiots and here's why." That may or may not be true, but the article indicates non-zero infinitesimals are used in some systems. In real number terms this is correct, but I'm not sure that mean it's true at the level implied. In addition to that I always though .333... being a third is because we're base-ten so some approximation has to be used not because it truly is a third.--T. Anthony 01:50, 25 October 2006 (UTC)

Hang on, this article doesn't call anyone idiots. If you read the mathematics education research literature, many of the relevant authors actually express awe at the intuitive leaps that untrained students are capable of, and they search hard for systematic problems and solutions, rather than crying "stupid students, bad teachers" and giving up. On the other hand, they also make it clear that even when students' intuitive expectations happen to align with some mathematical structure, it's inappropriate to assume that they're mentally working within that structure.
If this article doesn't reflect the respectful, curious attitude taken in the literature, then that's something we have to correct. But you have to help out by pinpointing where the problem is, rather than make broad accusations.
Oh, and .333... = 1/3, exactly. Melchoir 01:59, 25 October 2006 (UTC)
Here is the statement that I found condescending: As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". ...as well as the rest of the paragraph. It kind of comes off as "If you don't agree with me then your brain isn't big enough." Unfortunately, the whole thing is a quote, so I'm not sure what can be done about it. It is common that people who are in a field too long start loosing sight of that field's assumptions, and can't respond well when those assumptions are challenged. Think of LP purists objecting to CDs. Some of them had quite convincing proofs that digital sound could never be as good as analog. Algr 05:58, 25 October 2006 (UTC)
Well, it's just a description of Dubinsky's theory. The authors aren't saying that the students' brains aren't big enough -- can you imagine trying to get such an inflammatory and unscientific statement published in a scholarly journal named Educational Studies in Mathematics? They are proposing the existence of critical cognitive steps on the way to understanding concepts like 0.999…. And that's what our article says. Melchoir 06:09, 25 October 2006 (UTC)
Why don't we teach Calculus to first-graders? Why not astrophysics? Because their education is not yet complete. Advancing as a student is, by definition, the completion of intermediate steps. Jboyler 23:28, 25 October 2006 (UTC)

Congratulations

What a wonderful article. Maths is something I have never claimed to understand but I find it fascinating. I only got as far as the fractional proof, but to there it is well written, clear, inciteful and informative. Thank you to all involved in making this a featured article. It's great exercise for the brain. It's beaut mate and a marvellous featured article! Maustrauser 00:27, 25 October 2006 (UTC)

Ambiguous statement

"...researchers of mathematics education have studied the reception of this equation among students, who often vocally reject the equality. Their reasoning is often based..." Whose reasoning: The researchers', or the students'? I'd clear up the sentence myself, but I don't which way is correct. Thanks. Dmp348 00:45, 25 October 2006 (UTC)

Based on the context, it looks like it's the students' reasoning. Fixed it, thanks. Supadawg (talk • contribs) 00:49, 25 October 2006 (UTC)

What a tragedy!

Suffice to say, if such a trivial mathematical quandary as whether 0.999... equals 1 becomes irrelevant by 2100, I will consider mankind to have collectively unplugged its head from its ass, and made some actual progress at the business of living. Chris 00:56, 25 October 2006 (UTC)

Spoken like someone who truly does not understand the applications of mathematics. The fact that 0.999... equals 1 is a basic principle of the real number system. How about math for math's sake? Next time, keep your two cents to yourself, and don't ridicule the hard work the other editors have put into this article. Supadawg (talk • contribs) 01:06, 25 October 2006 (UTC)

Wow, clearly not a mathematician. Engineer perhaps? *checks* oh i thought computer scientists had more sense. The result that 0.9 recurring is another way of writing 1 is fairly important to our understanding of the leap between Q and R. Even if you disagree, the amibiguity of our system of real numbers is an interesting point, and can help our understanding of the concept of infinity, as 0.9 recurring is infinitely close to 1. Triangl 01:06, 25 October 2006 (UTC)

We have 1,450,000 articles here at last count. There's no requirement that a user be interested in every one of them. But I submit it might be more useful for our critic Chris to look in on and edit one of the other 1,449,999 that's more to his liking as opposed to making a gratuitous comment like this one. Newyorkbrad 01:10, 25 October 2006 (UTC)
Calm down Chris, I am no mathmatician & really don't understand the significance of this, but we really should respect other people's work.Cameron Nedland 02:31, 25 October 2006 (UTC)

My baby!

Oh, the little article that could. You've grown up so well. *sniff* But seriously, congrats to all the editors that brought this page up to featured article status. Awesome work. --Brad Beattie (talk) 00:59, 25 October 2006 (UTC)

Would this be in Applications?

I'm abit surprised that the article doesn't give credit to Bullion, and it's use of .999/.9999 purity notations. Why the blantant omission? --293.xx.xxx.xx 01:02, 25 October 2006 (UTC)

What exactly does a purity notation very close to 1 have to do with the theorem that 0.999... equals 1? Supadawg (talk • contribs) 01:10, 25 October 2006 (UTC)
From said article:
Note that a 100% pure bullion is not possible, as absolute purity in extracted and refined metals can only be asymptotically approached.--293.xx.xxx.xx 02:39, 25 October 2006 (UTC)
That's a consequence of the nature of bullion, not an error in the math of it. Producing pure bullion would be like reaching absolute zero: something will always disturb it. Also, I never saw 0.999... as a repeating decimal, I only saw 99.99%, which is less than 1. Supadawg (talk • contribs) 20:58, 25 October 2006 (UTC)
Why is it that standardized testing will refer to a 99th percentile but never a 100?Jboyler 07:50, 26 October 2006 (UTC)

Notation

I've seen 0.9 recurring written as 0.9999... written with a dot over, but never witha dash, although I'm not saying that notation doesn't exist, merely that I see it more frequently represented with what looks like an Umlaut. Triangl 01:11, 25 October 2006 (UTC)

Were you by any chance educated in the UK? Melchoir 01:15, 25 October 2006 (UTC)
Personally I have never seen the dot, but I see the dash all the time, perhaps it's a national thing (I'm American).Cameron Nedland 02:32, 25 October 2006 (UTC)
At least in Canada, we use the dash. It's a regional thing, I gather. --Brad Beattie (talk) 12:15, 25 October 2006 (UTC)
Yes appears to split along Commonwealth/North American lines I'm assuming from the above. We used the dot in Malaysia IIRC Nil Einne 16:10, 25 October 2006 (UTC)
I'm from Sweden; and here I've mainly seen and used the dash. JoergenB 20:02, 27 October 2006 (UTC)

So what do you do for 71/99? Do you put dots over both numbers?Flarity 21:53, 28 October 2006 (UTC)

Yes you write it as 0.71 with a dot over the 7 and the 1. For a number like 6/7 you write it as 0.857142 with a dot over the 8 and the 2 to show the start and end of the repeating pattern. Theresa Knott | Taste the Korn 22:01, 28 October 2006 (UTC)

Intuitive counter-proof

An intuitive counter to the assertion that 1 = 0.999... might hinge on wether 10 * 0.999... = 9.999... . For example, 10 * 0.99 = 9.9, not 9.99, so what does that mean for an infinite series like 0.999...?

A student might intuitively think of 0.999... as representing 1 minus an infinitesimally small number, or
0.\bar{9} = 1 - 0.\bar{0}1


Considering this intuitive assumption, the algebraic proof might go something like this:
c = 0.\bar{9} = 1 - 0.\bar{0}1
10c = 9.\bar{9} = 10 - 10*0.\bar{0}1
10c - c = 10 - 10*0.\bar{0}1 -(1 - 0.\bar{0}1)
9c = 9 - 10*0.\bar{0}1 -(-0.\bar{0}1)
9c = 9 - 10*0.\bar{0}1 + 0.\bar{0}1
9c = 9 - 9*0.\bar{0}1
c = 1 - 1*0.\bar{0}1 = 1 - 0.\bar{0}1 = 0.\bar{9}
c = 0.\bar{9}

I've removed the above from the article, for a few reasons. First, there's no such thing as a counter-proof. Second, it's original research because no known source actually operates with this intuitive assumption. Third, even in its own schema it proves nothing; it's just a manipulation that begins and ends with the same statement. Melchoir 01:35, 25 October 2006 (UTC)

Oh, and I reverted [1] because the first sentence is dealt with in a later section, and the second sentence is just wrong. Melchoir 01:37, 25 October 2006 (UTC)

I agree with both these edits as a correct application of the WP:NOR policy. --Ryan Delaney talk 01:40, 25 October 2006 (UTC)
and has been mentioned before...there is no such thing as an infinite expansion with a ending digit. Brentt 02:15, 25 October 2006 (UTC)

Time to trim the fat at the math faculties

If professors in math faculties have time to waste "proving" this nonsense there are too many of them and we could save the country money by firing some. Golfcam 01:43, 25 October 2006 (UTC)

Rest assured, mathematics professors do not waste their time or money on such trivial matters. See http://arxiv.org/list/math/recent. Melchoir 01:46, 25 October 2006 (UTC)
What's with all the trolling on this talk page? First of all, this is to talk about the article not the subject of the article. Second, this article is relevant, notable, and verifiable. There are a whole bunch of other articles on this site that do not meet those criteria, so why don't these editors go to those articles and deal with those. Gdo01 01:46, 25 October 2006 (UTC)
My old high school math teacher proved it in under a minute. Don't worry. Supadawg (talk • contribs) 02:02, 25 October 2006 (UTC)
I find these trolls amusing. Its comedic. To humor the trolls: This topic is more relavant to math education than what working mathematicians actually do. Its a topic that highlights the clash between initial intuitions about numbers and sound theory. The reason why this topic is ever discussed, usually by math educators, has little to do with what .9999... equals (its indisputably 1), its about the reluctance of students to let go of intuitive concepts which don't work in theory. It is important that students grasp what .9999... because if they can't grasp that, then they won't be able to grasp limits. And if you can't grasp limits, then your pretty much not going to be able to grasp a good portion of math and science. Brentt 02:11, 25 October 2006 (UTC)
Word. Melchoir 02:13, 25 October 2006 (UTC)
Some of the trolling going on here seems to be good old fashioned anti-intellectualism, based on the idea that careful thought about esoteric matters is a waste of time. If that were true, we wouldn't be having this discussion at all, because computers, computer monitors, and computer networks (i.e., the web) would never have been invented in the first place. To the trolls: be thankful there are people in the world who like to think more than you do. - dcljr (talk) 18:22, 25 October 2006 (UTC)
Don't internalize trolling here as having something to do with trolls' viewpoints on math; trolling occurs on articles in every domain. My guess is, that here, trolls defensively respond to their lack of understanding of mathematical notation, concepts, and sophistication. You're perceiving an attack or ridicule of math, when all there really is, is trolling per se. Math and theoretical physics rule, together with classical music, great literature and history. 66.108.4.183 20:29, 25 October 2006 (UTC) Allen Roth
And, of course, some of the real trolls, in the original sense of that word, will themselves be math graduates who are personally not in any doubt about the properties of the reals, having a go at poking other people into outrage at their carefully contrived won't-be-told brand of simulated ignorance. -- The Anome 14:33, 28 October 2006 (UTC)

Bombardment

Wow, this article is being bombarded by people who apparently can't accept the fact that .999... = 1 despite the excellent proofs presented in the article. I think the big problem is that people read the first sentence which states that the two are equal, and don't read any further before vandalizing. Oh well, we can't make them read. —Mets501 (talk) 02:01, 25 October 2006 (UTC)

Yeah. Ironic, huh? Melchoir 02:02, 25 October 2006 (UTC)
Should we consider having the article protected? It's obviously rather high-profile today and that is resulting in lots of vandalism. I doubt any of our editors want to spend all day refreshing the page history to check for dodgy edits. Maelin 02:30, 25 October 2006 (UTC)
I'm an adherent of User:Raul654/protection. Melchoir 02:43, 25 October 2006 (UTC)
Some protection might be in order. I almost vandalized it myself. ;) Algr
Corrections arent vandalism, the proofs are wrong and criticism is ignored
Reversing the assertions of cited sources certainly comes close to vandalism. There are a lot of mathematicians here who, for various reasons, choose to defend the mathematics in this article. But the bottom line is that this is Wikipedia, and we don't make stuff up. See WP:V and WP:NOR. There will be no "corrections" to the article until a reliable source provides one -- and don't hold your breath! Melchoir 10:42, 25 October 2006 (UTC)
There are a lot of mathematicians here for various reasons, choose to criticise the article. But the bottom line is that the article is being sustained by brute force, not by the views of the majority of mathematicians. I fear a lot of people are abusing this concept in an attempted intellectual ego trip. Repeat after me, INFINITY IS NOT A NUMBER IT IS A CONCEPT, INFINITY IS NOT A NUMBER IT IS A CONCEPT, INFINITY IS NOT A NUMBER IT IS A CONCEPT.
Don't shout. You have no sources, and sources are power. There is no brute force on Wikipedia. Melchoir 11:46, 25 October 2006 (UTC)

Another Picture

In this conceptual representation, the heights of the digits show their relative value; each succeeding digit is 1/10 the height of the one before it.
Enlarge
In this conceptual representation, the heights of the digits show their relative value; each succeeding digit is 1/10 the height of the one before it.

I have just uploaded Image:Proportional decimal values.jpg. The digits in the number 0.999... are shown with a height according to their relatitve value; each successive digit is 1/10 the size of the previous one. I wanted to get feedback on whether this might be a good image to put on this page, the decimal page, whether it should be edited, or any thoughts at all! Thanks, Dar-Ape 02:03, 25 October 2006 (UTC)

Hmm... where in these articles do you suppose it could go?
By the way, this isn't a critique of your image, but it's a shame that 10x is such a huge gulf in size. That's why my own illustrations are all in lesser bases! Melchoir 02:07, 25 October 2006 (UTC)
No, because you can only see two 9s (3 if you look at the full size image). Just not a very visually elegant representation. Its conceptually elegant, just not visually...good idea, but just doesn't work very well when rendered. Its one of those ideas that sound like a good idea, but just doesn't work out (unlike say wikipedia, which was a bad idea, that somehow worked out.) Brentt 02:09, 25 October 2006 (UTC)
"unlike say wikipedia, which was a bad idea, that somehow worked out." I'm keeping that one for my book of awesome quotes. Supadawg (talk • contribs) 02:29, 25 October 2006 (UTC)
It's a neat idea, but it's not really clear what the image is trying to illustrate, nor how it would help someone improve their understanding. Also, it might be an idea to make the picture for a smaller base (perhaps base 3), since otherwise it's just a picture of a big .9 with a little 9 next to it. Maelin 02:10, 25 October 2006 (UTC)
And a little dot where the next 9 goes, don't forget that ... (Perhaps something like this would be better if the number were just any old decimal and it went into, say, an article on decimal representations or something). Confusing Manifestation 02:28, 25 October 2006 (UTC)
Perhaps the problem is that the first 9 is actually 100 times bigger in area, and therefore in visual impact, than the second. Zaian 06:36, 25 October 2006 (UTC)

That's a pretty good point, Zaian- making it 3 times higher (9 times the area) would be a better way of demonstrating it, I think- if you really wanted 10 for the decimal purpose, 3.163 comes close enough to sqrt(10) for it to work graphically. ChocoCid 12:42, 25 October 2006 (UTC)

.999... = 1?

Wouldn't it be 1-.ooo...9?Cameron Nedland 02:36, 25 October 2006 (UTC)

What? Do you mean that 1-0.000...1 = 0.999...? Such a number does not exist. Supadawg (talk • contribs) 02:40, 25 October 2006 (UTC)
You cannot have something after infinite 0's. --Kinst 02:44, 25 October 2006 (UTC)
Then dividing any finite number by infinity gets zero. This means a Universe that is infinite would have to have an infinite amount of intelligent life or it could not have intelligent life at all. Because if it had 800 centillion intelligent life-forms the amount of intelligent life it would have per-area would have to be zero. As zero times anything is zero it can't have any intelligent life. After that I prove black is white and get run over in a zebra crossing:)--T. Anthony 07:32, 25 October 2006 (UTC)
Which would be true, if there was such a thing as an infinite universe. -- 86.128.253.74 09:50, 25 October 2006 (UTC)
Yes, what Supadawg said. I am no mathmetician, if you can't tell.Cameron Nedland 02:45, 25 October 2006 (UTC)
Well, 1-0.000...1 = 0.999...9, but that implies that the string of 9's ends, and it does not. --M C Y 1008 22:36, 26 October 2006 (UTC)
Well... I wouldn't say that these things don't exist or can't be had. It's just that they aren't decimals, and they don't represent real numbers. There isn't even a theory of what they might be or represent, and the article touches on some of the barriers to constructing such a theory. Melchoir 02:47, 25 October 2006 (UTC)

We should try to define it better

sorry im not sure of how to edit this and my english is not very good... but i have to say this I don't find any logical sense in this article .999... should be the nearest number to 1, and so should .333... be the nearest number to 1/3. The guy who wrote Intuitive counter-proof had a good point in there our theories agree with each other. Well anyway, you should at least put the .999 equaling 1 is just a theory so people won't count as truth. ok? please ^^

It's not just a theory, it's been proven. Check the multitudes of proofs in the body of the article. To address your confusion, 0.9 is almost 1. 0.999 is almost 1. 0.9999999999999999 is almost one, and so on. 0.999... (which repeats into infinity) is 1. Supadawg (talk • contribs) 02:38, 25 October 2006 (UTC)
The real numbers have a property called density, which means that there is no such thing as a "nearest number that isn't equal". For any two numbers x and y, exactly one of the following is true:
  • There exists a number a that lies between them but is not equal to either of them (i.e. there exists an a such that x < a < y OR y < a < x) - by repeated application, we can see that there are in fact infinitely many numbers that lie between any pair of nonequal numbers. OR,
  • x and y are equal.
These "nearest neighbours" are only meaningful in sets that are not dense, such as the integers. Maelin 02:50, 25 October 2006 (UTC)
In any case if 0.999... wasn't equal to 1 then (0.999...+1)/2 would be closer to one.Geni 03:20, 25 October 2006 (UTC)

This is a really good article

Just wanted to say-- this is excellent work. I never would have believed such a great article could have been constructed on this topic, or that it could be made so accessable and, honestly, fascinating. I know a thing or two about math, but there are a lot of really creative choices made in this article that never would have occurred to me-- the page image that so perfectly illustrates the concept of string shooting off to infinity, the discussion of mathematical education, the intuitive proofs, the popular culture discussion, etc. Before reading this page, I could have written an article talking about why .999=1, but I never would have thought to make it so... comprehensive and so fun. Everyone involved should be extra proud of themselves. --Alecmconroy 02:42, 25 October 2006 (UTC)

Much agreement with the above comment. mkehrt 03:02, 25 October 2006 (UTC)

This is the most interesting article I have read on wikipedia. Very confusing, yet enlightening. 71.80.171.94 05:25, 25 October 2006 (UTC)

This article has destroyed my conception of reality. Congrats on this being a featured article. -Nicktalk 06:02, 25 October 2006 (UTC)

This is the best we can come up with?

We have fascinating articles throughout Wikipedia about Animals, Nations, Individuals, Organizations, the Paranormal, etc. You cannot convince me that more people who visit Wikipedia would be interested in reading this than, say, an article about an Ocelot or a famous military General.

Oh, I don't know. I think maybe that only 49.9999...% of them would be interested, but 'definitely' not more than half of them. -- OingoBoingo2 03:01, 25 October 2006 (UTC)
This is a fascinating article. --Zeality 03:08, 25 October 2006 (UTC)
And the kind of article that only Wikipedia would have. I can't imagine looking up this article in Britannica. This is the sort of place where Wikipedia really shines. --Alecmconroy 03:15, 25 October 2006 (UTC)
On that note, Wikipedia:Unusual articles has a bunch of such articles. Good reading for a slow day! Melchoir 03:18, 25 October 2006 (UTC)
This is a superb article. I came to this Talk page to make the same point as Alecmconroy makes above: only Wikipedia would have the guts to make 0.999... a featured article. It is so refreshing these days to see a mathematical topic dealt with properly, compared with other media that commit horrors such giving the formula of carbon dioxide is CO2. Congratulations all who worked on the article and all who agreed to feature it. --A bit iffy 04:26, 25 October 2006 (UTC)

This is an utterly asinine article. It is inconsistent with basic logic that 0.999... equals 1. I don't care how many nines you put at the end, 0.999... can only ever *approach* the value of 1. It can never equal it. Period. Anyone who thinks otherwise is fooling themselves. 1 is 1. 0.999... is not 1. The so-called proofs presented are not convincing to anyone with a modicum of intelligence. —The preceding unsigned comment was added by 64.80.201.172 (talk • contribs) 03:53, 25 October 2006 {UTC.}

64.80.201.172, print it out, work through it and you'll be convinced. --A bit iffy 04:26, 25 October 2006 (UTC)

I should add that I believe this sort of article results from the assumption by many that mathematics equals reality. I realize that this concept is particularly difficult for many to comprehend, particularly mathematicians, but there is it...this article is nothing more than an elaborate circular proof designed to make masturbatory mathematicians feel better about themselves. —The preceding unsigned comment was added by 64.80.201.172 (talk • contribs)..

If you aren't interested in it, don't read it. In less than 24 hours it will be off the main page and the featured article will be something Entirely Unrelated. Stebbins 04:25, 25 October 2006 (UTC)
It's funny that you use the word masturbatory. It is the exact same word that popped into my mind the minute I started reading the article. While the article is technically accurate, it doesn't address what I feel are more important consequences of the premise. Overall, I feel it demonstrates yet another weakness in Decimal representation. Consider the following statement from the article on Decimal:
"Rational numbers (e.g. p/q) with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation."
Therefore, the vast majority of rational numbers have a recurring representation in decimal. In contrast, every rational number can (by definition) be represented as a ratio of two integers without ever encountering infinite sequences. Also, consider the set of irrational numbers: the closest you can come to representation in either system would be an approximation. Since every decimal approximation (both terminating and repeating) has a non-repeating representation in fractional form, there is no benefit to using a decimal representation. Thus, for practical purposes alone, it seems preferable to represent real numbers as a ratio of two integers (in reduced form). The equivalence of 0.999... and 1 is simply a side effect of the definition of decimal representation itself, and is no more significant than the idea of convergence in an infinite series. --71.72.135.142 06:53, 25 October 2006 (UTC)

This is an excellent article, and I'm glad it made it to the front page. It's too bad that some people will never really understand infinity, because the concept is beautiful and almost mind altering. I think students should be taught the concept of infinity before secondary school, along with some other "college" level subjects. Falsedef 04:48, 25 October 2006 (UTC)

Very good article on something I'd never really thought about or been taught about. First Today's Feautured Article I've read through in its entirety in a while. The last few sections could do with a bit better structuring but I can't think of any particulars and it's all well written. So yes, well done everyone who worked on it. I wonder if there's any featured article that doesn't have at least one topic-related complaint when it goes up on the main page. Jellypuzzle | Talk 05:15, 25 October 2006 (UTC)

This is an excellent article about a very relevant topic - and written in a way everybody (not only mathcracks) can understand. The ongoing discussion on this page is proof enough for the relevance and necessity of the article - if only those who nag the loudest about it care to read (and think about!) the article... --FermatSim 06:01, 25 October 2006 (UTC)

It is refreshing to read 71.72.135.142's comment; criticism of the article which is actually (in some ways) correct. I agree that there's nothing terribly special about decimal expansions, or about the fact that 0.999... = 1. About the latter, it is a fact nonetheless, and given the number of people who are wrong about it, having an article is worthwhile. About the former, I disagree with the part about "no benefit"; I do not know a representation which is better suited for calculations (especially of the approxinate type encountered in applications), or a simpler representation capable of handling every real number. -- Meni Rosenfeld (talk) 17:35, 25 October 2006 (UTC)
Oh, and to 64.80.201.172: There is no circular argument in the article. Nobody has said that 0.999... = 1 has anything to do with physical reality - this article is strictly about the mathematical concept. The Mona Lisa has no practical applications, but would you refer to Leonardo da Vinci as a "masturbatory artist"? It is often the beauty, not the applications, that counts. -- Meni Rosenfeld (talk) 17:39, 25 October 2006 (UTC)

No article on Wikipedia has ever held my attention like this one does. Best article ever.Jboyler 07:54, 26 October 2006 (UTC)

Woah:

You torched the article?

Nah, a vandal torched the article and was quickly reverted. Melchoir 04:58, 25 October 2006 (UTC)

Skepticism in education

The reason why that section is titled "skepticism in education" and why it refers to "students" is because the literature on skepticism focuses exclusively on mathematics education. That's where all the data, all the quotes, all the theories come from. The arguments witnessed on the Internet seem to have a different nature, and they haven't been studied at all, so it would be inappropriate to generalize information on students to all people. Melchoir 06:21, 25 October 2006 (UTC)

Perhaps the article should note that, so as to distance wikipedia from less then diplomatic treatment of one side of the debate. Algr
Um, note what, exactly? Melchoir 06:54, 25 October 2006 (UTC)

Practical use of .999... issue

I ran into this issue when I was making some rules in a game where the length of certain objects put them into different classes, and I realized that it was difficult to describe a range in a way that made clear which class lengths exactly on the borderline fell. My solution was to use .999... notation: ClassBlue = 3 ft to 3.999... feet, ClassRed = 4 ft to 4.999... ft. Now the article is telling me that this notation is wrong, and I haven't unambiguously defined if a pole exactly 4 ft is ClassBlue or ClassRed.

How do I do this? How do I describe the maximum value of X where X<1? If .999... isn't the answer, then what is?

This also comes up with legalistic issues, like if it is EXACTLY your 21st birthday, can you buy alcohol? After all, you aren't "over 21". The article should address this, since it is how this issue effects people in real life. Algr 06:33, 25 October 2006 (UTC)

There is no such "maximum value" among the real numbers. However, it is certainly possible to describe a range of numbers that includes every number between two endpoints but not the endpoints themselves. Such a range is called an open interval. See the article interval for more information. Alternatively, you could just write like this: Blue: 3 ≤ x < 4; Red: 4 ≤ x < 5, etc. Or perhaps better yet, spell out what you mean in plain English. Fredrik Johansson 06:45, 25 October 2006 (UTC)
Real numbers don't exist in the world we live in: there is an inherent 'graininess' to the universe. If you measure a distance, no matter how accurate you are, you can't measure to more accuracy than Planck length. If you measure time, you can't be more accurate than Planck time. Irrational numbers are purely theoretical, in the sense that no physical real world measurable quantity is irrational. Maelin 06:50, 25 October 2006 (UTC)
I would contend that the entire concept of "infinity" which is a pre-requisite for the proof that 0.999... = 1, does not exist in our universe. Ansell 06:54, 25 October 2006 (UTC)
I'm confused by that. If infinity is real than why would all numbers divide by infinity be 0? Because .999... strikes me as being the same as saying that 1-(1/infinity)=1. This would mean 0=-1/infinity. So does that mean that if you multiplied zero by infinity you get +/-1? I'm admittedly a layman on this, but I do read math books for pleasure. I'm confused.--T. Anthony 07:50, 25 October 2006 (UTC)
Anyway, I'm not sure if the article can address this particular topic. Unlike, for example, -0 (number) and its meteorologists, we don't have any evidence that 0.999… is useful as a recording convention. I encourage you to look for such evidence; it would indeed be a valuable addition to the article. Melchoir 07:13, 25 October 2006 (UTC)
Well, I think I just gave you some examples of .999... being useful. Blue: 3 ≤ x < 4; Red: 4 ≤ x < 5, is just ugly as sin compared to 3.999..., and I'm guessing would be understood by few people besides math fans. (I though of it, but rejected it for that reason.) Algr 07:51, 25 October 2006 (UTC)
Blue: 3 ≤ x < 4; Red: 4 ≤ x < 5, is perfectly meaningful and standard terminology. I don't see why it should be "ugly". --Kprateek88(Talk | Contribs) 18:01, 28 October 2006 (UTC)
No offense, but are you personally important enough that Wikipedia should be reporting on your use of mathematical notation? Melchoir 09:22, 25 October 2006 (UTC)
Well, if you honestly believe that I'm the first person ever to define a range as "Under four" then I must be incredibly important. This is basic stuff, that peiple run into every day. Another example, is 12 pm, noon or midnight? Algr 17:26, 25 October 2006 (UTC)
I'm saying that 3.999… is not commonly used for this purpose; at the very least, we cannot assume that it is without some kind of evidence. Melchoir 00:33, 26 October 2006 (UTC)

I don't think so. I think it is Planck length and Planck time which are completely theoretical. The theory in them is the fact that they presume that what they measure is the truth of what they're measuring. If you base something on a calculation of the speed of light, then you're basing it on something temporary and will certainly not hold for future calculations when more precise instruments are invented. Also, to speak of a "world" and a "universe" is completely positivistic. There is no one "world" and one "universe" and certainly not one "graininess." Planck length and Planck time are theories just as much as irrational numbers are.Moonwalkerwiz 07:46, 25 October 2006 (UTC)

I would say that to speak of there "not being one world" is itself radically constructivist, and the rest of the world believes there is only one world, to some extent. Radical constructivists have a hard time coping with the entire concept of an objective science IMO. Ansell 08:09, 25 October 2006 (UTC)

Better be a radical social constructivist believing that 1 is not always equal to 0.999... than a dogmatic, intolerant mathematician.Moonwalkerwiz 00:27, 26 October 2006 (UTC)

Wow. Thank you.

I hate to clutter up the talk page with completely nonproductive and unabashed praise, but in this rare case I really must. I'm no student of math, but I found this article completely fascinating, and after reading some of the sources I can see how accurate and well-researched it is. Cheers! SlapAyoda 06:56, 25 October 2006 (UTC)

Awesome. This is certainly a more welcome "clutter" than the opposite reaction! Melchoir 07:02, 25 October 2006 (UTC)


Just tried to flick this through this article before the working day begins. Have not a clue what it all means, but I can tell just how well researched and put together this article is. Well done to all - Wiki at its best. Now, back to the Weetabix.....doktorb wordsdeeds 07:17, 25 October 2006 (UTC)

I'd also like to commend this article. It's a fantastic treatment of the subject and one of the most comprehensive and comprehendable pages on Wikipedia. Many thanks to all involved. Vincentvivi 08:50, 25 October 2006 (UTC)

what this means for other numbers

I can accept that 0.999... equals 1, but i have a question (which i think follows very logically) that doesn't seem to be addressed in the article: What implications does all this have for other numbers? In other words, is 1.999... equal to 2? Is 49.999... equal to 50? If so, why is '0.999... = 1' singled out as the archetype of this principle? If not, why? What's the difference?

I'm asking mostly out of personal curiosity, but for what it's worth i think this subject should be broached by the article somewhere. (Unless i'm just totally missing it, that is.) ~ lav-chan @ 07:27, 25 October 2006 (UTC)

Well, there's the first paragraph of 0.999...#Generalizations. Melchoir 07:36, 25 October 2006 (UTC)
Probably 0.999... is used here because (I would guess) it's the one most often taught at school/university, and also is the simplest-looking example.--A bit iffy 07:38, 25 October 2006 (UTC)
There are a couple other minor reasons. The paragraph of "Generalizations" beginning "Alternate representations of 1..." describes research which has historicaly focused on the case of 1. The first paragraph of "Applications" describes an application for 0.999… alone. Melchoir 07:44, 25 October 2006 (UTC)
It's also worth noting that by first proving the result .999...=1 we can easily get any other result. For example, I claim that 17.999... = 18. Well, we know 17.999... = 17+.999... = 17+1 (by the .999...=1 result) = 18. qed
-bobby 13:56, 25 October 2006 (UTC)

Oustanding article

I'm a confirmed non mathematician and I wanted to congratulate all concerned on an outstanding article.

My instant reaction was that this was tosh, but having read, understood and laughed out loud at the "fraction proof" and the "algebra proof", I have to hand it to you for producing an understandable article. OK, OK, you lost me completely with the "infinite series" section, but I'd understood enough by then to have the hang of the issue. Outstanding work. Deserved FA. --Dweller 08:28, 25 October 2006 (UTC)

  • Excellent! This is a problem when the One and the Infinite come together and challenge our most basic assumptions.162.116.29.39 17:07, 25 October 2006 (UTC)

Can someone tell me what "tosh" is? 129.15.127.254 19:24, 25 October 2006 (UTC)

"Tosh" = stuff and nonsense 198.241.217.15 21:35, 25 October 2006 (UTC)

Fraction proof

While I bow to the knowledge of others on this subject and don't have the background to understand the other proofs, I im sure that the frational proof is not a real proof.

Stating that 1/3 = to 0.3333.. is not acceptable. 1/3 cannot be represented as a decimal and if you try you cannot perform arithmitic on.

The step claims that 0.3333...x3 = 0.999... Well it doesn't IF 0.333=1/3 Then 0.333...x3 = 1

Let me just state again that im am not disputing that 0.999... is equivalant to 1. Graemec2 09:18, 25 October 2006 (UTC)

  • All rational numbers can be expressed as decimals, and since 1/3 is rational, it can most definitely be expressed as 0.333... — Dark Shikari talk/contribs 10:11, 25 October 2006 (UTC)
  • Yes, the fraction proof is a perfect example of circular reasoning. It's just as hard to prove that 0.333... is exactly equal to 1/3 as it is to prove that 0.999... is exactly equal to 1. If you don't accept the fraction proof, why do you still think that 0.999... is equivalent to 1? The other proofs are just as flawed, as I've pointed out in other notes. Dagoldman 10:16, 25 October 2006 (UTC)
    • Dagoldman, get over it. You obviously don't understand how the decimal number system works, which is why —to quote Wolfgang Pauli— you are "not even wrong". -- mstroeck 10:56, 25 October 2006 (UTC)
      • You've just made an ad hominem attack. Always easy to attack the messenger. But it's just as logically flawed as the circular reasoning of the "fraction proof". Dagoldman 11:38, 25 October 2006 (UTC)
        • I did not "attack" you, nor your "message". I stated that you don't know what you are talking about, which is self-evident. As to your message, I'm not aware of any coherent argument you have made that is reconcilable with the facts. I think your problem stems from not understanding how the real number system and decimal notation work. Please read this, for example, and disprove any of the statements made there, then people will perhaps consider your "message". mstroeck 12:24, 25 October 2006 (UTC)
          • Saying "you don't know what you're talking about" is an ad hominem attack. I read the link and agree with it. But the link supports what I've been saying, not what the wikipedia article says. The link says the LIMIT of the geometric sequence represented by 0.999... is 1. look for the first mention of "limit" in the article. It's way down. Dagoldman 19:05, 25 October 2006 (UTC)

Just to partially defend Graemec here, I think what he's saying is that the informal algebraic proof he's talking about that starts with 0.333...=1/3 has, as an implied axiom, that there does not exist an infinitesimal number that lies between 0.333... and 1/3. However, if that axiom is not assumed, then this particular proof isn't rigorous enough to prove that 0.999...=1. Fortunately, for those people who don't accept that 0.333...=1/3, the article does have more formal rigorous proofs using limits further down that can also be adapted to prove that, in fact, 0.333...=1/3 in the real numbers because of the definition of what repeating decimal notation means. Dugwiki 16:28, 25 October 2006 (UTC)

  • I don't see why you can't say 1/3 = 0.333...; certainly you can prove this using long division, whereas proving 1 = 0.999... isn't possible in the same way.
    • Well, where's the proof that 0.333... is exactly equal to 1/3? Do the division yourself and see. It clearly shows just the opposite. As you move to the right, there is always a remainder. That's why the division continues infinitely. At each step, it's always 10 / 3. At each step it never gets to 4. 0.333... is infinitesimally smaller than 1/3, and that's what keeps the long division repeating endlessly. Dagoldman 19:09, 25 October 2006 (UTC)
      • You haven't grasped the concept of infinity quite yet, dago. Infinity never stops, which is what the "..." represents. There will never be a remainder, because that assumes there is a point where the division stops (which it doesn't). Quantifying an "infinitesimally small" number isn't possible, anyhow. Also, using convergence theorem, you can prove 1/3=0.333... The convergence theorem is in this article. Falsedef 23:25, 25 October 2006 (UTC)
    • You can say it, but it assumes you understand limits. If you already understand limits, you will believe that 0.999... = 1 and hence won't need to have it proven to you. Therefore to assume 0.333... = 1/3 to prove that 0.999... = 1 is begging the question and isn't a proof at all. Xrikcus 09:19, 26 October 2006 (UTC)
      • Exactly, it is a LIMIT. I assume you know limits as well, and if you remember, limits do not always actually exist on a graph. Most limits are holes in the graph, and when you calculate a limit at point x, the output f(x) is where the hole in the graph is. x is approaching the hole f(x), but sometimes cannot EQUAL f(x). It is easier to say "f(x) is the limit at x" than it is to say "x gets infinitely closer to f(x), but cannot actually equal f(x)". So yes, the LIMIT of x=.999999... equals 1, but that does not necessarily mean that .999999... definitely in all cases equals 1. The "concept of infinity" (as someone else mentioned) is that it never stops. So .9999... will never stop getting closer to 1, but it will never equal 1. Anyways, this argument is all semantics, and there's not really much reason to fight about it, because it doesn't really matter. Keith.
0.999... is a number. It doesn't "have" a limit, it doesn't move anywhere, it doesn't get closer to anything. The sequence (0.9, 0.99, 0.999, ...), which is related to it, "gets closer to 1" in the sense that if you take an element in a later position, it will be closer to 1 than an earlier one. But 0.999... is not the sequence itself, it is the limit of the sequence. That limit is 1. Therefore 0.999... = 1.
I agree that the fact that 0.999... = 1 is an unimportant piece of obvious mathematical triviality. But it's not about semantics, it's about mathematics - and it's worth fighting about, because failing to understand this means failing to understand anything about mathematics. -- Meni Rosenfeld (talk) 10:29, 28 October 2006 (UTC)
If 0.9999=1, then also does 1.000...001 also equal 1? It is as infinitely close to 1 as 0.999... is.
How many numbers are we going to say equals 1? If 1 has more than one equality, then does it really exist?
Also, where does 0.999... stop equalling 1? At infinity-1? What the hell number is that? That'd be like saying 0.999999999999999999999=1, but 0.99999999999999999999 (one 9 less) does not.
Taking limits into account, and backing up my previous argument, say you have the equation f(x)=(x^2 - 2x + 3)/(x-1). If you graph this, the (x-1)s cancel out, so the graph looks like f(x)=x+3, however there is a hole in the graph at x=1. The limit of x=1 is 4, however x cannot equal 1, because the function is undefined. As x APPROACHES 1, the y value also APPROACHES 4, but it never quite gets there. At x=0.999..., y=3.999..., and likewise at x=1.000...001, y=4.000...001. x cannot equal 1, y cannot equal 4, because the function is not defined at that point (1,4). f(x) will get infinitely closer to 4, but never technically reaches 4. Therefore, 0.999... nor 1.000...001 cannot equal 1 in this sense because the function is undefined. In this case, 0.999... CANNOT equal 1. Therefore, while this theory may be true sometimes, it is not true all of the time, therefore this theory is partially wrong.
As for proofs, anything can be proven. There's probably a proof to show 2=3. Most of the time, proofs are the absolute mathematical truth, but they can easily be stretched to prove just about anything true. I don't agree with either the algebraic nor the fractional proof. It's a stretch to do arithmetic on infinite numbers.
Anyways, I said this was semantics. What I really meant was, this is stupid. I don't even know why I even typed out all of that. I'm pretty much done. Think what you want to think, but just about anyone will say that 0.999... does not equal 1. It is what it is. Keith
Not anything can be proven. There is no proof to show 2=3. Mathematics would collapse if such nonsense were true. You can either prove 0.999... is equal to 1, or prove that it is not equal. There's isn't any debate about it. There is no such number as 1.000...001, because if there are an infinite number of digits, there can be no last digit. -- Schapel 02:52, 29 October 2006 (UTC)

I would like to recommend keeping this section, even if it does not qualify formally as a mathematical proof. As someone who is not particularly knowledgeable in mathematics, I found this one instantly persuasive. —ptk✰fgs 19:44, 25 October 2006 (UTC)

Good one

I previously nominated this article for deletion in the unencyclopedic form Proof that 0.999... equals 1. I must say that the article has turned out great. The number of people claiming the article to be wrong only reinforces my belief that few people understand limits. As for me, I still do not understand the limits of masturbation.

Still, can something be not done to the title? It's a remarkably inelegant one. The choice of three 9s followed by three dots is somewhat arbitrary, and it's not right to use dots in two different senses in the same title. Also, this is not the type of name you expect to see in an encyclopedia. Loom91 09:32, 25 October 2006 (UTC)

Well, the title is arbitrary in a mathematical sense, but it does seem to be the most common form out there, and the three 9s is almost a de facto standard. It's not a perfect title, but I don't know what would be better. Melchoir 09:37, 25 October 2006 (UTC)
The article title is another example of a really creative choice that wouldn't have occurred to me, but works out quite well. When I first saw this article, I had the same thought as Loom-- "this article isn't really just about .999... this is about the idea that that .999...=1" . And really the article isn't about that-- it's about the idea that any infinite series can sum to a finite number. This article could be about .333 or .666 or anything. -- we just picked .999
But now that I look at it, I actually really like the title. It gives the article a "pop science" feel, that it wouldn't have if it were called "Infinite series with finite sums". The title feels "fun"-- sorta like that group of popular books entitled "i", "e", "pi" and "0". And then the image of the .999... shooting off to infinity just clinches it. I never would have gone with a title like this, but I like it.
And when you think about it, for some odd reason, the article really is about .999.., and not .333... . For some odd reason, I find it very easy to accept that .333...=1/3 . If you asked me if that was true, I'd say "yes", without hesitation. But when asked "Is .999...=1?", my mind replies "Basically, yes". Of course, it's the exact same issue-- but for some reason, my infernal human brain treats them different.
--Alecmconroy 10:21, 25 October 2006 (UTC)

Non-Zero Terminating Decimal

When I first read this phrase in the introduction, I mistakely read it as 'non-zero-terminating decimal', that is, a decimal number that doesn't end with 0. It took a few moments of wondering why 10, 20, 100, and so on should behave differently from other values before I realised my mistake. I suggest that 'terminating, non-zero decimal' may be less ambiguous, or maybe just an extra comma - 'non-zero, terminating decimal' - would be sufficient to satisfy pedants like me.

I don't want to edit the page directly (a) because it's currently on the main page and (b) because my version sounds less elegant, so unless it's deemed worthwhile I'm quite happy to leave things as they are. Still, in the spirit of Wikipedia I offer my suggestion up for comments. Gholson 11:45, 25 October 2006 (UTC)

Melchoir has made an edit similar to your suggestion. -- Meni Rosenfeld (talk) 13:56, 25 October 2006 (UTC)
Thanks, Melchior and Meni. And well done for keeping track of the sandstorm of comments on this page, which must be exhausting. Please accept a virtual pat on the back. Gholson 12:16, 27 October 2006 (UTC)
Just a pat on the back? I think our good friends deserve some barnstars. :) - KingRaptor 12:38, 27 October 2006 (UTC)
No problem, it's all in a day's work :) -- Meni Rosenfeld (talk) 14:51, 27 October 2006 (UTC)

Problems with 0.999....

I vividly remember at the age of 10, being quite distressed trying to understand this concept, visiting it again, I'm still not quite sure about the c=0.999... proof, although I am happier with the 0.3333... x 3 proof for some reason. Glad this has made the front page, although I'm sure you're going to lose lots of people even before the series explanation, let alone the much further proofs, which people seem to be saying are much more robust, I certainly didn't make it that far. Have fun with the trolls and vandals, I'm sure there will be many today. TerriG149.155.96.5 11:48, 25 October 2006 (UTC)

Well, I've had my fair share of dealing with trolls in the context of this article - and I can say that the vast majority of the people criticizing the article right now are not trolls. They're here for a real disucssion, and we're willing to provide it. -- Meni Rosenfeld (talk) 14:10, 25 October 2006 (UTC)

well it isnt equal but frightening near

Moved to Talk:0.999.../Arguments. --Brad Beattie (talk) 13:44, 25 October 2006 (UTC)

Show your work

Moved to /Arguments. Melchoir 20:13, 25 October 2006 (UTC)

0.999 ≠ 1, but whole fractions work

Moved to Talk:0.999.../Arguments Argyrios 15:20, 25 October 2006 (UTC)

Random question...

Moved to /Arguments →AzaToth 16:09, 25 October 2006 (UTC)

Congratulations

On having the featured article. Yes, I'll join in keeping it devandalized. Septentrionalis 16:14, 25 October 2006 (UTC)

I'd also like to congratulate the authors on this article. What seems like a trivial bit of numerical trickery in fact brings to light some important fundmentals in calculas and the real numbers. And the resulting discussions demonstrate just how passionately people will defend their intuitive gut feelings on something, even when those intuitions are incorrect. The number of people who have posted trying to "disprove" that 0.999...=1 in the real numbers despite the proofs presented is amazing. It illustrates that mathematics is, if anything, a good tool for teaching students to not jump to emotional conclusions in mathematics and science and to approach problems using logic and reasoning.

Great job to the authors. :) 4.999... stars!Dugwiki 16:39, 25 October 2006 (UTC)

They are not emotional conclusions. Your tone is typical of the arrogant attitude conveyed by proponents of the "proof." If students question the validity of the proof, it's not because they are emotional or lack abstract thought, it's because they think outside the mathematical box you're trying to force them into. It is not mathematics vs. intuition. It's one kind of mathematics vs. and understanding beyond that kind of mathematics. Present the article as a theory - rather than "in mathematics, .999... = 1" say "According to real number theory, .999... = 1. Other theories reject this notion" or something more balanced like that. Rather than simply insisting on your "fact," be open to other conceptions of the infinite, such as the notion that it is approachable but not reachable, so that it cannot be equalled. Then you might find that the objectors are not so stupid, but that they are unsatisfied with the limitations of your theory, because they see beyond it. Rather than labelling their intuitions "incorrect," appreciate the truths they are trying to convey, and incorporate them into the article.24.68.180.163 17:19, 25 October 2006 (UTC)
I agree. These proofs make unwarranted assumptions about what happens at the end of an infinite sequence, and ultimately are based on circular logic. The arrogant tone is not really the article's fault, but comes from sources like this one: [2] Apparently all mathematicians agree because you aren't allowed to graduate if you don't. This seems more like religious fundamentalism then logic. Algr 17:38, 25 October 2006 (UTC)
"what happens at the end of an infinite sequence" -- Once you figure out that there is no end, it'll all make much more sense.Falsedef 23:43, 25 October 2006 (UTC)
24.68.180.163's comments are the most insightful on this talk-page. mdf 18:00, 25 October 2006 (UTC)
24.68.180.163's comments are wya off the mark. The assumption (an arithmetic axiom going back to Archimedes) is that there is NO end to an infinite sequence. Jerry picker 18:56, 25 October 2006 (UTC)
That's right. Now, you can bring an infinitely large pile of paper to someone and say "here, have a number indistinguishable from one". They are going to say "that's all fucked up", and in turn hand you a note that has a single "1" written in the corner (or any other finite representation). To most outside observers of this exchange, there is a difference, no matter how long the line of famous mathematicians you assemble. mdf 12:27, 26 October 2006 (UTC)
24.68.180.163's statment: "It's one kind of mathematics vs. and understanding beyond that kind of mathematics" makes no sense. Is your "Enlightened-super-buddha math" somehow different from normal math? If so, can you teach me? Jboyler 07:49, 26 October 2006 (UTC)
Since 24.68.180.163 mentions no "enlightened-super-buddha math", one wonders just what you were reading. Ask anyone, even a child, and they can tell the difference between the finite and the infinite. That there are some weird ways of converting the infinite into the finite (by fiat, if not proof) will appear to them as almost capricious magic. If high-end hard-core math is needed to do this, it will seem all the more bizarre and, frankly, irrelevant to them and their normal experiences in the world. That the result in discussion has had essentially no use beyond a party trick, a novelty item, will just entrench this view even more. Blathering on about how these people "lack intuition" or are in some way incapable of abstracting is, as 24.68.180.163 notes, simply insulting. If I handed you a piece of a material, hooked it to a battery and suddenly something happened, do I criticize your reaction as an inability to intuit, or just plain ignorance? If I fail to explain the phenemona to your satisfaction, is it my communication problem or your lack of understanding? mdf 12:27, 26 October 2006 (UTC)
Can you show where any propponent of the article referred to opponents as "lacking intuition"? The only claim I've seen is that opponents lack understanding of what mathematical rigor means, and that can usually be attribtued to not having studied mathematics at a sufficiently high level (I guess you could call that "ignorance"). As for the battery example, it can be either - I cannot decide without knowing who the person was and what the explanation was. -- Meni Rosenfeld (talk) 13:11, 26 October 2006 (UTC)
I know this is a lot to ask, but I encourage you to read some of my and others' posts on the arguments page. Some of them directly address the subtle issues raised here.
I'll try to respond briefly: The article does explicitly state that 0.999... = 1 only in the context of real numbers. Even if it didn't, that's implicit since real numbers are almost always the object of interest when talking about numbers. It does mention other structures where that's not true. People may not be completely wrong when they say that 0.999... is not 1, but they are wrong when they say that 0.999... is not 1 and expect other features of real numbers to hold. They are also wrong when they say that 0.999... = 1 does not follow rigorously from the definition of real numbers. It's okay to want 0.999... to be different from 1 and to build a consistent structure around this concept; it's not okay to explain with vague notions why "0.999... is not 1, and all your proofs are wrong."
And no, Algr, not even once during my studies was I asked about 0.999... . Implying that mathematicians can't think beyond the real numbers is wrong (otherwise complex numbers, modular arithmetic, hyperreal numbers, surreal numbers, the real projective line, the extended real numbers, ordinal numbers, cardinal numbers, etc., would not exist). It's just that they know when doing that is appropriate, and when it is not. You should have some familiarity with the concept of mathematical proof before posting here. -- Meni Rosenfeld (talk) 18:09, 25 October 2006 (UTC)
The article begins with "In mathematics..." when it means "in real number theory" or something to that extent. No, I disagree that you have to have a background in mathematical proof to be part of the discussion, though I have done a little. What about a background in communication? The article doesn't communicate well. Most of the disagreements would disappear if the connection to real numbers were emphasized. It seems as though mathemeticians are so aggravated by students who won't accept their "truth" that instead of saying "you have a valid conception, but it's outside real number theory" they just insist that the students aren't understanding. What I am trying to say is that you can't assume a lack of abstract reasoning because students wish to challenge things, and the article appears to suggest this. Even when you say a person shouldn't post here unless they have familiarity with mathematical proof, you are taking the "I'm right because I'm a mathemetician" stand rather than "I might not be appreciating that you think beyond this."
I like the idea that .333 approaches but does not attain 1/3. This idea is not merely intuitive; it's abstract. It requires a willingness to abandon the concrete notion that a number must have an absolute value. Some talk in the article about the philosophical differences between the two ideas would be helpful. If this fits with other mathematical theories, it should be made more explict in the article, not just hidden way down in a later section.
The article has an overly-emotional tone that relays the need of some mathemeticians to prove themselves "right" rather than reconciling different ways of thinking; and it projects assumptions of stupidity onto people who have worthwhile questions. That is why you have a conflict -- not because people don't understand.24.68.180.163 18:40, 25 October 2006 (UTC)
I can see how my response to Algr can sound smug; I apologize for the tone. However, my claim stands: Algr has made a lot of wild (and obviously false) accusations without anything to support them. That's quite unacceptable.
As for the other things... Throughout the history of mathematical study, it has been observed time and again that dealing with vague notions lead to problems. Philosophers might find these "problems" to be what's interesting, and be happy to discuss everything vaguely. Mathematicians can't afford that. They want rigor, exactness and certainty; so that when any statement is proven, it becomes something to rely upon without fear. As such, intuition alone can't be everything in mathematical study, since intuition is often wrong. What's ultimately important is the axioms and definitions we have chosen, and the theorems we have rigorously proven to follow from them. Intuition plays an important part in suggesting which definitions to make and how to prove theorems, but that's really just a means to an end (of course, intuition is also a part of the fun). When attempting to actually make meaningful statements, intuition becomes deceptive and harmful. One can use his intuition to guide him, but must not trust it implicitly - or he will risk making nonsensical claims.
So, I repeat my original statement. It's okay to ponder the possibility of 0.999... being different from 1; But the next step must be choosing certain axioms and definitions, and proving that it follows from them that 0.999... < 1 (and, of course, every other feature we want this structure to have). On the other hand, statements like "0.999... < 1 because this idea is intuitively appealing to me, with infinite processes and numbers approaching others etc." have no place in a mathematical discussion. Anyone who makes such a statement is guilty of not understanding what mathematical rigour is all about.
And no, nowhere in the article or in the mind of any editor, exists the assumption that someone is stupid. It is asserted that most of the opponents of 0.999... = 1 are mistaken (the exception being those who actually have an alternative structure in mind and understand what's involved), but smart people also make mistakes. -- Meni Rosenfeld (talk) 20:11, 25 October 2006 (UTC)
Meni, my comment about not being allowed to graduate was not meant as an accusation, but as something that the article itself seems to imply when it relegates all opposition as coming from "students." Maybe it is a weakness in the article, or perhaps it is an accurate picture of a dark undercurrent in how mathematicians relate to the public. Just because YOU don't see this as a problem doesn't mean it isn't one. This article isn't for you, it is for people who don't already know this stuff and want to understand more. We ask questions about these proofs, and what we get back sounds more like an admonition from authority then an appeal to logic. This tells me that mathematics is not the perfect bastion of logic that some people seem to think it is. Algr 22:38, 25 October 2006 (UTC)

Mathematics is indeed the perfect bastion of logic. You should not expect the same from Mathematicians. Many (especially some of the best) are batfuck insane. I am, however, disturbed by the undercurrent of implications that the "Mathematicians" aren't understanding what it is that everyone else is trying to do, and that "Mathematicians" should make effort to appreciate the arguments made against the claim that 0.999.... = 1 and should be more open to alternative formulations. It's like insisting to a biologist that people have two eyes for purposes of redundancy rather than binocular vision, then chastising the bioligist for disagreeing, and finally complaining the that biologist doesn't understand their own field. Or accusing Shakespeare of having a convoluted writing style and irrelevant themes. Understanding 0.999... = 1 is like learning to read. It makes sense once you appreciate various fundamentals, hence all the criticisms come from "students". After you "get" it, all the conflict dissapears. No accusations of conspiracy or elitism are leveled against the community of english teachers for insisting on standard usage of plurals; why the rebellion against mathematicians so? People might expect understanding infinity to be easier than learning to read. They may be right, but they should still check out Hilbert's paradox of the Grand Hotel - it illustrates how infinity and the untrained intuition are not often compatible.

I see what you're saying, but Shakespeare did have a convoluted writing style...
I probably couldn't say it better than the above comment. I'll emphasize that people unfamiliar with any subject, let alone abstract ones like mathematics, will naturally have difficulties grasping the more subtle issues. They can then either trust the words of experts (and experts are exactly that - why would you not want to trust them?), or make a sincere effort to surmount the difficulties. Many of the questioners here do not make such a sincere effort, and it is impossible to answer in a satisfactory manner.
Anyone which is really up to the challenge of understanding why 0.999... must, in the real numbers, be equal to 1, is invited to this thread, where I have provided a rather straightforward proof and expect people to point out the exact step they disagree with. I have omitted some details, in light of the fact that a completely rigorous proof would probably be many pages long, and the understanding that I'm willing to provide any requested detail, right down to ZFC in need be. I will also sin against humility by referring you to my comments (the long monologs) near the end of this and this threads. It is my subjective opinion that a lot can be learned from them. -- Meni Rosenfeld (talk) 10:05, 26 October 2006 (UTC)
Do you shout out corrections to an actors' grammar during a Shakespeare performance? Or do you work harder to figure out what they really mean?

Cogent objection?

Moved to /Arguments. Melchoir 20:14, 25 October 2006 (UTC)

Definition of uniqueness?

I was always taught that two numbers are unique if and only if there is another unique number that lies between them, and since there is no possible number between .9~ and 1, they are not unique, but are equal. I can't find any sources on this though - everything I find on the internet uses one of the examples already provided. Could this possibly merit inclusion in the article? -- FireCrotchRIO 16:49, 25 October 2006 (UTC)

  • This is implicit in the Archimedian understanding of real numbers. Informally, any number proposed that is "between" 1 and 0.999...9 (finite) can be thrown outside that interval by simply tacking another "9" at the next decimal place. Since the number of decimal places in 0.999...(infinite) goes out to infinity, the must ALWAYS be a "next 9" to "cut in" between. Thus, there is no unique number that lies between, and so they are one and the same.162.116.29.39 17:13, 25 October 2006 (UTC)

Subtraction proof

Looking through old diffs, I see the "Subtraction proof" section hasn't been in the article very long. Someone needs to flesh it out a bit more, if it's indeed worth keeping. I don't understand how the subtraction is being done, since the usual "borrowing" technique of subtracting decimals on paper proceeds right to left and there's no rightmost decimal place to start with. Am I not seeing something obvious? - dcljr (talk) 18:10, 25 October 2006 (UTC)

I actually take issue with the subtraction proof from a mathematical standpoint. It requires a stretch of the imagination to say that 1-.999.... = 0.0000.... (even though it is technically true. I think the other two proofs convey the idea more clearly and with less ambiguity. I'd like some more input, but if nobody has any objections I'd like to remove the section detailing this proof. Alternatively, if someone would like to formalize the proof in order to better convince me I'd be open to leaving that. Thoughts? -bobby 18:44, 25 October 2006 (UTC)
A quick edit on the above comment. When I say it takes a stretch to see the result, I pretty much mean that we are accepting .999.... = 1 at that point. Otherwise there is no way to perform the proper subtraction an infinite number of times to get that result. -bobby 18:46, 25 October 2006 (UTC)
I'm going to go ahead and remove it per above. At the very least, it has no license to be in the article by virtue of being uncited Simões (talk/contribs) 19:25, 25 October 2006 (UTC)
I was the one who added this proof. I think I was being a bit overly bold and should have taken the time to mention that you should do the subtraction in the "long division" way. I.e. when subtracting 987 from 1234 you start by subtracting 9 from 12, which leaves 3. You then take this and the next digit (3) to get 33 from which you subtract 8 to get 25. As the last digit isn't 0, take it and the 4 to get 54. Subtract 7 and you get 47. Put it all together by taking the bottom number from each column and you get the answer: 247.
Apply this on 1 - 0.999... to arrive at 0.000... in the same way that you arrive at 0.333... by doing a long division of 1/3.
This method was treated back when I was in high school, so it seemed obvious to me. I gather I was being a bit over-enthousiastic :-) Great Cthulhu 14:35, 26 October 2006 (UTC)

Call me crazy

But if .999... is exactly equal to 1 (mathematically), then why bother having an article on it at all, or even bother writing about .999... when you can just say "1"? (Except for reasons of curiosity.) Almost seems like a waste of time. Also, it might be worth pointing out that .999... is only exactly equal to 1 if the number of places the .999... goes is infinite, as any lesser mantissa will not satisfy the mathematical expressions used to prove .999... is 1. 206.156.242.36 19:12, 25 October 2006 (UTC)

Copied and pasted from my above explanation: This topic is more relavant to math education than what working mathematicians actually do. Its a topic that highlights the clash between initial intuitions about numbers and sound theory. The reason why this topic is ever discussed, usually by math educators, has little to do with what .9999... equals (its indisputably 1), its about the reluctance of students to let go of intuitive concepts which don't work in theory.
And the ellipsis, in this context, means that there is no last decimal place, as does the top-bar and top-dot--i.e. "it goes on to infinity". Its not meant to be interpreted any other way. Brentt 19:37, 25 October 2006 (UTC)
If an intuitive concept doesn't work in theory, then there's either something wrong with the intuitive concept or with the theory. Also, one must keep in mind that in mathematics often it is theory that is valued, but in the "real world" it is application that becomes paramount. Not picking on you, just sayin'. 206.156.242.36 19:49, 25 October 2006 (UTC)
Yea, believe me, if it wasn't accepted that .999... equals 1, then you wouldn't be on the internet, as much of modern physics, which lead to the gadgets that allow the internet to be useful, relies on the more general concept which encompasses it, and which the .999...=1 is meant to illustrate. In this case, there is definitely something wrong with the intuition. When it becomes clear, whats intuitive changes. Its just that the common initial intuition in this case turns out to be wrong, because of a misundertanding of what the ellipsis, or top-bar/dot means. Brentt 20:31, 25 October 2006 (UTC)

A couple of suggestions for the end of the day

Even though this is a featured article, there are other things could be added (once its feature day ends), e.g.:

  • "Infinite series and sequences" could be stated in English prior to being stated in math, e.g.: "Two numbers are identical if and only if their (absolute) difference is not positive. Given any positive value, the difference between 1 and 0.999... is less than this value. Thus the difference is 0 and the numbers are identical. This also explains why 0.333 = 1/3, etc. In other words..."
  • The problem here can be considered one of notational misunderstanding as much as mathematical misunderstanding. In language, "..." means that some finite portion is left unstated or otherwise omitted (as in the above bullet point), whereas in math "..." means that some infinite portion is left unstated. Some just want to "fill in the gap" between the first and last "9" that would be implied by a linguistic interpretation.
  • It could also be mentioned that, just as human beings have difficulty understanding repeating decimals and irrational numbers, computers cannot "understand" non-dyadic rationals, because none of these "end." (Obviously, the connection would have to be put more clearly than this.)

Calbaer 20:52, 25 October 2006 (UTC)


Simple proofs

Cute article. I don't understand why the very simplest proof is not presented, namely that 1.0-0.9 = 0.1, 1.00-0.99 = 0.01, etc. so the difference must be smaller than 0.(0)1 for any number of (0)'s. If you choose any number other than 0.00..., it will have to have a non-zero digit somewhere, so it must be 0. This argument is correct in terms of the elementary understanding of arithmetic on decimal numbers (which assumes that every number has a decimal representation), and can be made rigorous (but that would be in appropriate in an introduction). --Macrakis 23:11, 25 October 2006 (UTC)

A proof similar to the above should perhaps be added. Unfortunately, the skeptics won't be satisfied, since they believe some magical "0.000...1" is the result of the subtraction. -- Meni Rosenfeld (talk) 10:40, 26 October 2006 (UTC)

Not true?

I just reverted the following addition by User:Yuraty:

'“'This is not true. .999... does not equal 1. For example, stand 10 feet away from your wall, step half way between you and the wall, you are now 5 feet away. Now step again, you are 2.5 feet away. At this rate you will never reach the wall, this rate is what we call infinity. 0.999... is infinite, saying that 0.999... = 1 is saying that you are skipping infinity, which is not possible.”

—User:Yuraty

--ZeroOne (talk | @) 21:30, 25 October 2006 (UTC)

Nice use of Zeno's paradox. Shame it isn't really applicable. GeeJo (t)⁄(c) • 21:59, 25 October 2006 (UTC)
Pfft. Zeno never took into account that while you're covering less and less space, you're covering it at a greater and greater speed. Zeno needs a lesson in calculus and the converging values of infinite geometric series. Too bad he's dead. :P --Brad Beattie (talk) 02:20, 26 October 2006 (UTC)
I never really thought of Zeno's paradox that way, so thanks for the new insight. However, I think you meant that the time period is smaller. The speed should remain constant. As an object approaches the goal, each sub section is completed quicker and each length is proportionally smaller, therefore the speed doesn't change ( s = d/t ). Since it takes less and less time to complete each subsection, the object will eventually reach the goal. Falsedef 04:57, 26 October 2006 (UTC)

Zeno's paradoxes are highly relevant. I've read this article and the talk pages up to this point, and for a good while I've been more and more thinking about adding some references to his paradoxes, and vice versa.

The real trouble with Zeno's paradox is that too many thinkers have spent too much time with 'solving' or 'explaining' them instead of pondering them. The paradoxes may have been posed originally in order to criticise the atomic theory; but they did put the light on some relevant problems with the concept of infinity. They should provoke anyone, who encounters the concept for the first time, and cannot just dismiss it with one of the common quasi explanations. In other words, the protests of school children are a positive thing; especially if they happen to have a teacher who succeeds in entering a dialogue. The first true modern solution was achieved by Dedekind; who was inspired by Eudoxos; who probably was much stimulated by the troubles Zeno and others had noted.

Both Eudoxos' theories of proportions and of exhaustion and the modern sequence limit concept are essentially based on the idea of allowing only finite amounts of (logical or mathematical) operations, but without any bounds on the amounts; and with knowledge that certain features will be preserved independently of the number of operations. Ultimately, the definition of real numbers in either the Dedekind manner or by means of Chauchy sequences are motivated by avoiding logical paradoxes, such as those by Zeno and by Berkeley; and on the other hand, without strict definitions, the 'manipulation proofs' of the equality 0.999... = 1 are meaningless.

Of course, the fundamental techniques of proof by (complete) mathematical induction and of recursive definitions depend on the same kind of understanding of 'infinite' as ' finite but unbounded amounts' of operations.

The problem I see is adding just the right amount of references to Zeno not to confuse the readers, but rather to explain the connection and stimulate them to follow the links. JoergenB 21:09, 27 October 2006 (UTC)

The reason for the confusion

The reason for the confusion seems to be that mathematicians and non-mathematicians work with a different idea of what real numbers are. More specifically what the relaton between decimal numbers and real numbers is. There seem to be two common misconceptions what the real numbers are: 1) all terminating decimals, 2) all decimals (terminating or not), i.e. all sequences {a_i}_{i\in\N} with a_i\in {0,...,9}. Of course the real numbers aren't either, set 1) is too small, set 2) is too large; it requires a fairly good grasp of mathematics to see why. I tend to think that without understanding either the Cauchy or Dedekind construction of reals any proof of 0.999...=1 *must* seem flawed, they merely replace something-not-completely-understood by something-else-not-completely-understood. Question: should any of this be mentioned in the article?195.128.251.21 21:44, 25 October 2006 (UTC)

Perhaps the "elementary proofs" should be removed. They have their merits, but it seems they just make people focus their energies on criticizing those proofs, rather than struggling with the advanced proofs. -- Meni Rosenfeld (talk) 10:43, 26 October 2006 (UTC)
Or perhaps at least introduced with a short and thus visible warning that the manipulations in themselves are pretty meaningless, if they do not presuppose what is to be proven; like in KSmrq's older version. If it's not too provoking, I'd like to restore that part of the KSmirqian text. JoergenB 21:16, 27 October 2006 (UTC)
Something like that might be a very good idea. Why don't you propose your changes here. Paul August ☎ 21:47, 27 October 2006 (UTC)
I'm not sure what you were referring to by "what is to be proven" - do we agree that those proofs are reasonably rigorous, but they presuppose quite a few non-trivial results regarding decimal expansions (which, in a proper development, would probably be before the 0.999... = 1 result)? KSmrqian or not, a note to that effect can be added, emphasizing that skeptics of such results should focus on other proofs. -- Meni Rosenfeld (talk) 06:54, 28 October 2006 (UTC)
Presently, the section opens 0.999… is a number written in decimal numeral system. Now, apart from the question whether or not it is 'written in the decimal system', the statement that it is 'a number' indeed presupposes not just results on decimal expansions, but on real numbers. This opening, effectively states that the infinite sum \sum_{n\geq1} 9\cdot10^{-n}\, exists, as it is 'a number'. The following manipulations are not completely circular, since they try to demonstrate that if calculating with this set of 'numbers' obey some laws many readers hopefully find well-known, and we accept a few other limits on faith, then this number cannot be any other number than 1; so perhaps my choice of words was a nuance to strong. Probably, there are a number of kids accepting that 0.333... = 1/3, although they don't accept 0.999... = 1; and in this case, you provide something. However, the reference to an infinite number of divisions does not prove this; and many students are (helthfully) sceptical to such arguments. Worse, I much too often meet statements such as '1/3 isn't an exact number'. Among my very worst experiences was the time when we'd given teachers-to-become the task to present the concept of infinity in schools, and it turned out that a bunch of them had spent the lesson in convincing the poor kids that 1/3 were an infinite number.
The section now also is written in a manner, which will not help a reader to separate the concept number from the concept number representation. (Is a finite decimal a number or a representation? Is an integer a representation?) IMO, this is easier to fix. The important point is not to let the skepticist get stuck with incomplete ideas.
There are some confusion about this also in later sections. It seems like if one of the editors finds the application of the formula
(1)\ \sum\limits_{n=o}^\infty ak^n = \frac{a}{1-k}\,
(such as by Euler in 1770) as a proof in itself, and the stricter and logical mathematics of the 19'th century as the results of an anti-liberal trend in mathematics. Well, formula (1) could also be used in order to 'prove' e. g. that 1-1+1-1+1-1+... = 1/2; and if my memory is correct, it was used in such manners by Euler himself. The 1770 treatment is not a proof of better standing than the digit manipulation proofs; what it says is merely that if you accept (1) on faith and absolutely, then 0.999... = 1 (and 1-1+1-... = 0.5). The mathematical historian E. T. Bell characterises Euler as the last great 'formalist', by which he means a mathematician who uncritically applies formulae like (1), without carefully controlling whether or not they are valid in the context. (He also notes that Euler started to recognise the problem, but was inconsistent.)
The article presently states: A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. This 'reactionary' view is very clearly the consenus of present-day mathematicians; for good reasons we consider the 'uncritical formalism' as nonsense. JoergenB

'Why don't you propose your changes here. Paul August' Well, starting the article something like In the mathematical theory of real numbers, would be a good start; and I hope the consensus is moving in that direction now. Another good thing would be putting back the KSmrqian section title Elementary proofs, and section introduction

Elementary proofs assume that manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition.

I'll see if I'm able to formulate a suggestion for short but adequate reference to Zeno. JoergenB 17:41, 28 October 2006 (UTC)

"Elementary proofs" is POV and undescriptive. We need to identify that 0.999… has something to do with the decimal system before asserting the decimal-system proofs are appropriate, and this goal is achieved by the present first sentence of the section. Melchoir 18:11, 28 October 2006 (UTC)
OK, what about replacing "Elementary proofs" by "The following arguments"? (I'm sorry, Melchoir, but I find it more doubtful to call them 'proofs' than 'elementary'; since they are appealed to outside a valid context.) I agree that this is not descriptive, but the descriptions would come in the sub-sections. In other words, I'm proposing the older structure, with a section which we could call Heuristic arguments or Simple explanations instead of Elementary proofs, if you agree; the one sentence introduction, and then directly proceed with a subsection Digit manipulation, retaining or improving the important discussion on what 0,999... actually should mean. This would have the effect of putting Fractional proof and Algebraic proof at the same level as Digit manipulation; but I don't think that is a real trouble. (By the way, I would prefer to replace proof by argument in these headers.) Perhaps we might move part of the analysis arguments into the same main section.
If I'm making a try at a Zeno reference, where would the best place be? On this (overloaded) talk page, in the article itself, on a new subpage? JoergenB 16:39, 29 October 2006 (UTC)

Step 2???

Moved to Talk:0.999.../Arguments#Step 2??? Supadawg (talk • contribs) 03:19, 26 October 2006 (UTC)

your loophole

Moved to Talk:0.999.../Arguments. --Brad Beattie (talk) 02:22, 26 October 2006 (UTC)

The most irrelevant FA ever

I should vote this article for deletion from Wikipedia at all. There should be just an article for infinite decimals, as is in all school and university textbooks I've read. But now, when we have this featured, should we create articles like 1.9999(9), 2.9999(9), etc. and get them featured? Huh? Cmapm 23:19, 25 October 2006 (UTC)

If someone can cook up enough unique material to justify separate articles, yes. I invite you to read through this article and consider how much of it you never would have found in your textbooks. Melchoir 23:39, 25 October 2006 (UTC)
No problem, just replace all 0.(9) entries with, say 2006.(9) and all or almost all (if I missed a couple of words in the article) info will stay true. The whole article just deals with an example of the infinite decimal representation of the real number, instead of having just the article on infinite decimals in general with the footnote: 0.(9) is an example of such a number. Cmapm 23:51, 25 October 2006 (UTC)
Unique. Melchoir 00:02, 26 October 2006 (UTC)
No it's this FA, that is unique. Find an example of such a stupidity in the mathematical literature or websites, like MathWorld, please, where there is the article entitled 0.(9) A book with pictures for pre-school kids does not count. Cmapm
By repeating the word "unique" I meant to imply that your new articles would be so non-unique, so similar to this article, that they would be quickly and decisively deleted. As for finding parallels of our articles else, I direct you to Wikipedia:Unusual articles. Would you delete them all? Melchoir 00:16, 26 October 2006 (UTC)
Yes, but the point is that this article is never comprehensive, because it focuses on one number, although the same results may be simply rewritten for a large group of other numbers. In contrast, if we rewrite the article in abstract form using in place of 0.999... this one:
b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\textstyle\frac{1}{10}}) + b_2({\textstyle\frac{1}{10}})^2 + b_3({\textstyle\frac{1}{10}})^3 + b_4({\textstyle\frac{1}{10}})^4 + \cdots .
i.e. a general infinite decimal, then we can cover all possible cases and the article would become comprehensive. Isn't this the basic method in serious mathematics-related articles? Cmapm 00:31, 26 October 2006 (UTC)
We already have Decimal and Decimal representation. If you think they're inadequate, go expand them. This article is comprehensive, because its title is 0.999... and that's exactly what it describes. Melchoir 00:39, 26 October 2006 (UTC)
It doesn't do it well in the real world. Keep in mind that Wikipedia should relate to the real world, and not to a constructed universe. In this case the constructed universe is the totality of mathematics. I do not think this article provides the necessary relationships back to real concepts that a featured article really should provide. Ansell 00:44, 26 October 2006 (UTC)
This topic has no relationship to the real world. None. The best we can do to adopt an out-of-universe perspective is to attribute results and proofs to their names and places in human history, and to comment on the reception of those ideas. The article already does this. I'm not sure what else you expect. Melchoir 00:55, 26 October 2006 (UTC)
Then it may not be needed as an article according the the fancruft definition as being something that non-mathematicians cannot relate to, and have no real interest in knowing the exact details about. Of course, I would never endorce such an action, but if what you say is true and the article describes realitic influences of 0.999... to the full extent possible then it may be true.
Personally, I think the article can do a lot more to figure out how it relates to the world, even if it is just in the illogicality that is typically perceived due to the "mathematical" nature of the subject. Ansell 01:01, 26 October 2006 (UTC)
No, that 0.999... has no relation to the real world proves only that a hypothetical article solely on the mathematics of 0.999... would be fancruft. This article contains much more; I've just described how. If you think the article can do even more, great! What sources should we be looking at, that we haven't already? Melchoir 01:11, 26 October 2006 (UTC)
(edit conflict) It's not only mathematicians that are interested in maths. Infinite decimal expansions come up in high school, so every high school kid could be interested in this article. I find the statement "Wikipedia should relate to the real world, and not to a constructed universe" very odd; does this imply that we should remove all articles about novels? If you think that "the article can do a lot more to figure out how it relates to the world" while staying within the no-original-research policy, go ahead. -- Jitse Niesen (talk) 01:17, 26 October 2006 (UTC)
(edit conflict) Melchoir: The article so far contains mostly a closed box of mathematical ideas about the subject. The skepticism in education section, which would be where reality comes into it, goes through mostly anecdotal evidence from mathematicians, as opposed to a critical review of the concept by an outside observer. Even the popular proofs section is just two anecdotes about internet communities which have so far had the issue flare up and in the Blizzard case die down. They do not describe the actual event, just the finality of the proof, which according the article is indisputable in real terms as well as mathematical terms. Ansell 01:24, 26 October 2006 (UTC)
Is it really a big surprise that researchers of mathematics education tend to be mathematicians themselves? There are no outside observers. If you find a reliable source that says anything outside of what this article already says, I'll be very surprised. Melchoir 01:37, 26 October 2006 (UTC)
This statement of how deep into maths one has to be to understand this article shows to me what you said before. How is someone supposed to verify this concept using outside references? Of course, reliable can be defined in different ways. Reliable in a movie/anime/TV setting would be exactly the opposite, to fit what people see an encyclopedia to be. In those settings reliable would only be what is said about the topic outside of the area. In this case we seem to have swapped the definition. Ansell 01:41, 26 October 2006 (UTC)
Some of the references are primary sources. Most of them are secondary sources. Some of those are even written for a popular audience. And they're all reliable sources. Do you have any better ideas? Melchoir 01:54, 26 October 2006 (UTC)
I am not sure. What I do know is that while we do recognise published material as reliable, it is an awkward subject to leave all of what students say out of it because they are not published by journals. Ansell 02:16, 26 October 2006 (UTC)
Jitse Niesen: Wikipedia does infact removal articles that are simply plot summaries and do not have relationships to real world commentaries. As for the high school students. Why are their personal opinions not represented as real world points of view, instead of simply pointing out the mathematical error in their ways. To put more weight on the mathematicians point of view as opposed to the hundreds of thousands of students who regularly fight with the concept is putting undue weight on the final mathematical stand on the issue. Ansell 01:27, 26 October 2006 (UTC)
The struggling students don't publish. All that we can verifiably say about their views is secondhand, via educational studies. The literature is unanimous that they are wrong about the real numbers. Some of it claims that they are wrong, period. It's already a tiny minority opinion that students' views should be taken seriously as describing alternative number systems, yet we've bent over backwards to accomodate that opinion in the name of neutrality. The section on skepticism was promoted as far up the article as it could be. These efforts are enough. Melchoir 01:49, 26 October 2006 (UTC)
Of course, we should take a scientific point of view of truth. I disagree that the article bends over backwards to portray the difficulties with the concept, as opposed to describing in depth the acceptance that it has by major mathematicians, who are familiar with the intricacies of what they define as "real numbers". Ansell 02:16, 26 October 2006 (UTC)
Then what difficulties does the article miss? Melchoir 02:28, 26 October 2006 (UTC)
It is possible that the article could more precisely define in laymans terms what the difference between what "real" in laymans terms means compared to the mathematical term. The real in mathematical terms contains infinity right, which is not considered realistic by laypeople. It also contains numbers which never end, ie, the subject of this page. A discussion about infinity in terms of its meaning here, and why having an infinite number of digits "actually" (in maths terms) closes the gap. Also, laying off proofs which layman see to be contradictions, ie, 0.333... = 1/3, may remove some negative inhibitions about the applicabily of the "proofs". The 1/3 proof seems way too blatantly circular reasoning for me and others. Is it posible to put a more rigourous proof down that doesn't involve the concept of an infinite number of digits in the representation of a number? I would have to do a more thorough review of the whole subject to give you many more ideas. Ansell 04:15, 26 October 2006 (UTC)
A lot of this seems to go in the OR direction that we mustn't: overreacting to the discussion on this talk page while ignoring our sources. No, infinity is not a real number, and there is no such thing as a number "ending". The 1/3 proof is not circular, and we will not suppress information just because it is unpopular among a certain fraction of our readership; many readers find it helpful, and there's documentation for that. Of course we already have plenty of proofs with weaker starting assumptions. I hesitate to call them "less rigorous" because a proof is a proof; there is no continuum of more rigourous proofs and less rigourous proofs. The proofs are explicit with their assumptions, too. Melchoir 04:48, 26 October 2006 (UTC)

Other numbers

The section titled Generalizations shows that any real number can be written in two ways with one way having an infinite length of 9s, like 0.24999... and .25. Would this apply to the 3s in 33.333... (to make 33.4) or any other number? Tim Q. Wells 23:30, 25 October 2006 (UTC)

It's not any real number, just the ones that have terminating decimal representations to begin with. 33.333... doesn't have an alternate decimal representation, since it doesn't end either with repeating 0s or repeating 9s. Melchoir 23:33, 25 October 2006 (UTC)

By the author's reasoning, 33.4 would be equal to 33.3999... (a rounding of 1/inf) Meanwhile 33.333... would equal.. gasp=> 33.3333...4. Which it doesn't, obviously; great question & it shows why this logic was wrong to start with. —The preceding unsigned comment was added by 68.211.195.82 (talk • contribs) 23:37, 25 October 2006 (UTC)

Yes, 33.3 = 33.3999…; otherwise this is just wrong. Melchoir 23:41, 25 October 2006 (UTC)
Thank you for answering (it was a bad mistake). Tim Q. Wells 01:43, 26 October 2006 (UTC)

It was a good question. It's sad that people think .999... and 1.000...1 can round to 1 but no other infinite fraction is allowed to round. People are selective about what they want to believe. The entire concept of "density" is a laugh.

Who the hell fucked up the featured article

They didn't even spell everything correctly.


Anyone who deletes this...

That edit summary is hilarious. It reminds me of Wayne's World (________ says what?). Ufwuct 00:16, 26 October 2006 (UTC)

I got this on my talk page. Actually, I was reverting unrelated vandalism at the time, and that edit got caught in the revert, but it's all good. —Ashley Y 00:58, 26 October 2006 (UTC)

Failed Logic

Moved to Talk:0.999.../Arguments#Failed Logic Supadawg (talk • contribs) 03:19, 26 October 2006 (UTC)

No!!

Moved to Talk:0.999.../Arguments#No!! Supadawg (talk • contribs) 03:19, 26 October 2006 (UTC)

Talk Page Maintenance

I've been helping BradBeattie move topics arguing against the article to Talk:0.999.../Arguments. After people reply there, proving that they know where their topic went, the leftover notice on this page is deleted. Melchoir, I don't know how you can reply to nearly every single new topic, but I commend you for it. Sometime, I'll go through and add {{unsigned}}s to the new topics, but that will be incredibly tedious. After about a month, I'll do another round of archives, but if the talk page keeps growing so fast, it might have to be sooner. Hopefully this will all die down now that we're not a Featured Article anymore, so we can get on with improving the article. Supadawg (talk • contribs) 03:19, 26 October 2006 (UTC)

Wait, if the formula is true...

Moved to Talk:0.999.../Arguments. --Brad Beattie (talk) 05:18, 26 October 2006 (UTC)

OR problem?

Does this edit acknowledge an Original research problem. "reader services" sounds horribly like an edit here put the information in. Why exactly are the "not usual" sources being used in preference to others? Ansell 04:32, 26 October 2006 (UTC)

I'm ambivalent about keeping the sentence, but there's nothing wrong with it. And you'll understand why those sources are used in that section if you read the preceding paragraph. Melchoir 04:37, 26 October 2006 (UTC)
Oh, and please don't demand citations where they've already been given. If you have a question about the scope of a footnote, ask it here. Melchoir 04:38, 26 October 2006 (UTC)
The citations in the footnote were enough to enable the use of weasel words? If the statements at the end of the last sentence actually refer to the other two sentences are really for the first two, why are they not there. The citations are not that clear, which is why I asked. And there is no law against asking in either place.
Still, there goes my motivation for editing this article. I guess "Featured Articles" will be featured articles.
I am partially annoyed by the claim that this makes the page more NPOV. Anyways, have fun reverting. Ansell 04:47, 26 October 2006 (UTC)
Well, 0.000…1 is not any kind of number taught in school, so it must be invented. And there is no last 9 in the decimal 0.999…, so that statement must be conditional. Melchoir 04:55, 26 October 2006 (UTC)

A Simpler Proof

1/9 = 0.111...

Multiplying both sides by 9,

1 = 0.999...


By the way, to say that 0.999... does not equal 1, but only approaches 1

is to say that 0.111... does not equal 1/9, but only approaches 1/9,

or that 0.333... does not equal 1/3, but only approaches 1/3.

And that, I think, is saying much more than anyone really means to say,

isn't it?

-yzgy



Unfortunately, from reading this talk page and the Arguments page, it looks like a lot of people mean to say exactly that. 153.104.208.67 04:55, 26 October 2006 (UTC)


I don't really agree. I think that the argument that 1/9 x 9 = .1111... x 9 = 9/9 is misleading. It is an infinite number, and the only way to isolate it is to pretend that it is not an infinite number. Adding the .111... together doesn't work because since it never ends, it can never be successfully added together unless in fraction form. If not, the addition continues forever, and "1" is never reached. Its why calculators always round it off. I think arguments and problems like this simply point out the flaw in decimal notation.



While I like the "simpler proof" at the beginning of this section, I can see at least one reason why some will not be convinced by it. We know how to multiply 0.4324 (or any other terminating decimal) by 9. Using paper and pencil, it's most convenient to start from the right, so that carries can be dealt with as we go. (By the way, using an abacus, starting from the left will work just fine!) So how do we multiply 0.111... by 9 using paper and pencil - how do you start from the left?
I know this objection can be answered; I just want to point out why this simple proof in this simple form may not satisfy everyone.--Niels Ø 07:26, 26 October 2006 (UTC)



So if 1/9 x 9 = .111.... x 9 is misleading, what do you get when you take 2/9 or 3/9? .222... and .333... And that said, once you get to .333... you're looking at a forumla that was already presented on the main page. As you pointed out, this seems to be more a flaw with the decimal notation system than anything else. So why is it any great leap to say, ".999... equals 1 because our written numerals cannot quantify the infinite?"Jboyler 07:34, 26 October 2006 (UTC)



How in the world is 1/9 an infinite number? --King Bee 14:10, 26 October 2006 (UTC)



This equation (0.999... = 1) disturbed me too, when I first encountered it. I saw it as a kind of hole in the universe - fascinating to look into, but terrible to fall through.

But eventually I saw that it is exactly the same as saying that 1/3 = 0.333... which is something I never had any problem grasping.

Odd thing, isn't it?

Does 0.333... really equal 1/3? Or does it just approach ever closer to 1/3 without ever quite reaching it?

In other words, is 0.333... a number, or a process?

It's the same question when we deal with the statement that 0.999... = 1

---yzgy

Yes, this is explained in the article. People who think of 0.333... as a process or sequence think of it as less than 1/3. However, 0.333... is a definite number that is exactly equal to 1/3. -- Schapel 16:49, 26 October 2006 (UTC)

while 0.333... is exactly equal to 1/3, and 3*0.333... is exactly equal to 1, 0.333...+0.333...+0.333... is not equal to 0.999... therein lies the problem. so easy to overlook, yet it tricks so many people.

No, because 0.333...+0.333...+0.333... = 3*0.333... = 1 -- Schapel 19:04, 26 October 2006 (UTC)

A Solution we all can agree on?

It seems to me that the answer to the problem we are having is already sitting unnoticed in the lead paragraph. Why are we all assuming that the whole discussion and all these proofs must be conducted in real numbers? It seems to me that the very act of understanding .999... is the first step of moving beyond real numbers and into Infinitesimals and Hyperreal numbers. What we have now is an article dedicated to proving that London does not exist. (in America) It's true, but sophomoric. The article on hyperreal numbers says "Zero is an infinitesimal, but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...)". That is .000...1. There is your reference. So lets dump all these stories about confused students and use .999... as an introduction to the hyperreal. Algr 06:20, 26 October 2006 (UTC)

Actually... (http://en.wikipedia.org/wiki/London_%28disambiguation%29#Places_in_the_United_States)
Okay, Ashkabat then. Algr
The analogy still fails as anything but an insight into your mistaken assumptions. See below. Also, 0.000...1 is a meaningless string in any system where "..." means an infinite sequence of digits. Your inference is original research, and not too convincing at that. Sorry. Argyrios 15:46, 30 October 2006 (UTC)
I totally agree, and a good start would be to change the first sentence to say something like "0.999... intuitively means the largest value that is less than 1, however this concept cannot exist in the real numbers and requires the use of infinitesimals and hyperreal numbers (0.999... intuitively equals 1-ε in such a system, is this valid/is there a strict version of this?). Since the real number system does not contain infinitesimals, in this system 0.999... is exactly equal to 1. Since this equality goes against the intuitive understanding of what 0.999... is, there have been a number of different proofs formulated in the system of real numbers to help explain the equality." The idea is to get away from the overly belligerant current first sentence. I don't have a strong enough background in maths to know if this is strictly accurate, could someone who does check it please?
There are some good points here. However, I've explained the reasoning behind emphasizing real numbers in my comments I linked to above. anon's suggestion is more or less correct, but I think it is way too complicated for an introductory sentence. Also, I think that 0.999..., more than it "intuitively means the largest value that is less than 1", is just a quirk of the decimal notation (which itself is just an arbitrary, albeit convenient, way to represent real numbers). What, then, is the largest value that is less than π? There is no satisfactory decimal expansion to that, and that has nothing to do with the numbers but only with a feature of a specific notation we have chosen to represent them.
Decimal expansions are also tightly bound to real numbers; I don't think it's possible to represent all hyperreal numbers with any similar notation. That's why I think that discussing 0.999... in the context of hyperreal numbers, while probably workable, is not very appropriate. In other words: 0.999... is equal to 1 in the only system in which it is appropriate to discuss it. That's reason enough to emphasize the 0.999... = 1 part. -- Meni Rosenfeld (talk) 11:00, 26 October 2006 (UTC)
Ok, that's a good argument (you're saying that since 0.999... notation doesn't generalise to other infitesimals, it makes more sense to use that notation in the normal real number way yeah?), although I'd like to see some of that discussion in the article.
I've said something similar before, and I'll say it again: The name of the article is "0.999...", not "life, the universe, and everything". Explaining that decimal expansions (and as a special case, 0.999...) are pretty much only adequate for representing real numbers, is probably interesting, but belongs in the decimal representation article (and if it's not there, we should add it). Similarly, explaining why real numbers are the "default" number system belongs in that article, and defining long division rigorously and proving that it works (even for non-terminating expansions) belongs in that article. -- Meni Rosenfeld (talk) 13:23, 26 October 2006 (UTC)

Meni, I have searched through your posts, and haven't found anything that explains to me why real numbers are appropriate for this discussion. (Perhaps a link?) They seem to me to be completely inappropriate since you have defined the number set as being incapable of supporting the concepts involved. You can "prove" that 1/2 does not exist if you insist on holding the discussion in whole numbers - but that is evading the question, not answering it, and this earns the frustration that this article has caused. As an aside, it seems to me that Pi is also unsupported in real numbers, because if you advance through an infinite number of digits of Pi, what must come next is even more digits, which real numbers do not allow. Algr 17:33, 26 October 2006 (UTC)

I was thinking about this and more importantly this. Apparently, I was wrong to assume these would answer your questions. I'll try again.
Integers are not anywhere taken to be the default numbers unless explicitly stated otherwise (or obvious from the context). Real numbers are, everywhere. Even if that wasn't the case: Decimal expansions are only adequate as a representation of real numbers. It is senseless to discuss 0.999..., a decimal expansion, in any other context than the real numbers. 0.999... can be used to represent elements of other structures, but the result is ugly (in the sense that most elements will not have a decimal representation) and is not normally done.
What exactly are those concepts invloved that the number set (I assume you were referring to the set of real numbers) is incapable of supporting? Is it non-zero infinitesimals? Real numbers, indeed, do not support them, but that is not a concept which has anything to do with 0.999... (should I say again that 0.999... is not adequate for dealing structures which include such infinitesimals?).
And what's that about π? The set {\sum_{k=1}^n\frac{8}{16k^2-16k+3}|n \in \mathbb{N}} is guaranteed to have a unique supremum, and π can be defined to be that supremum. Do not forget that decimal expansions are the derived, rather than primary, notion. π, now that it has been defined, has a decimal expansion like any other number. -- Meni Rosenfeld (talk) 18:07, 26 October 2006 (UTC)
I completely disagree. The real numbers are the context of most math and all real-world representations of quantity. Surreal and hyperreal numbers are a small and esoteric branch of mathematics. I certainly know nothing about them and neither do 99.9% of readers. The whole point of the article is that 0.999... is exactly equal to 1, and this statement is a true and inescapable consequence of almost everybody's accepted understanding of numbers. There need be no qualifying statement to water this down for people who don't accept that. A lot of people here seem to be grasping onto the existence of highly specific and extremely advanced branches of mathematics to justify their intuitive misconception of 0.999... without understanding anything about what they are actually doing or the implications of embracing that system. This proposal seems to me to be motivated by this impulse and will have the effect of encouraging it in confused readers.
Moreover, the London analogy seems to be betraying a fundamental misunderstanding of this article. A better analogy would be that "Los Angeles" and "City of Angels" refer to the same city, and people who don't agree with that seizing onto the existence of a "City of Angels" somewhere in Swaziland to dispute that fact. This is analogous in that "1" and "0.999..." refer to the same number, and people seize onto the existence and alternate meaning of "0.999..." in obscure branches of mathematics. Of course, this analogy fails in that it is trivial and uninteresting and uncontentious that the two names refer to the same city, whereas this article is interesting and counterintuitive. Argyrios 18:40, 26 October 2006 (UTC)
(edit conflict)Mathematically, there is no reason to think that 0.999… = 1 - 0.000…1, in any number system, nor is it connected with the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) in some ultrapower "construction". The fact that the real number system does not include infinitesimals helps prove that 0.999… = 1, but that does not mean that allowing infinitesimals necessarily causes 0.999… to break that equality. You'd have to redefine the entire notion of the meaning of a decimal, and for our purposes, that's original research.
In fact, in the mathematical and educational literature, nobody has ever brought up 0.999… together with non-standard analysis, the hyperreals, or the surreals. The entire "Infinitesimals" section derives its relevance solely from discussions like this one; we could delete it entirely and the article would actually say nothing less about 0.999…. Your point of view has already been indulged more than it should be in an encyclopedia; perhaps that was a mistake. Don't let it encourage you to push it further! Melchoir 18:48, 26 October 2006 (UTC)
I haven't read the whole thread, but I agree with the creator of this header. .999... should be used as a springboard to more complex number sets such as hyperreals and surreals. These may be "a small and esoteric branch of mathematics", but they are real and rigorous none the less. You can't simply treat them as if they don't exist. The entire basis of hte .999... equals 1 issue is philosophical, not mathematical. If you push hyperreals and surreals into the corner, then you should push this entire article there with it. Fresheneesz 03:39, 27 October 2006 (UTC)
They aren't treated as if they don't exist, they are treated as if they are a small and esoteric branch of mathematics and there are not a lot of sources which talk about 0.999... in the context of surreal/hyperreal numbers. In contrast, the real numbers are very important, and there are quite some sources talking about 0.999... in the context of real numbers, so the article talks mainly about the reals. It all seems very proper to me. What is your evidence for "The entire basis of the .999... equals 1 issue is philosophical, not mathematical."? -- Jitse Niesen (talk) 04:36, 27 October 2006 (UTC)
It's exhausting to say the same things again and again, but here goes: Discussing hyperreal numbers does not automatically make 0.999... different from 1. It does allow redefining decimal expansions in a way that might make it so, but this is not very appropriate, not normally done, ORish and misleading. 0.999... is 1 in the only context in which it is appropriate to discuss it (the real numbers), and this is a fact which can be proven rigorously (nothing philosophical about it). The decimal expansion is nothing more than a convenient notation for representing real numbers, and the existence of 0.999... is just a quirk of this particular notation which has nothing to do with the actual numbers involved, or the mathematics behind them.
I really don't know how I can be clearer than that.
Argyrios was right on the mark, explaining that some people just use hyperreals, etc. as an excuse to justify their faulty intuition (and it's not the intuition to blame, it's the undue trust placed in it), without really understanding what's involved. -- Meni Rosenfeld (talk) 09:24, 27 October 2006 (UTC)
I think it's crucial that this article refers to the reals right at the top of the article. The whole point of the article is that the non-necessarily-obvious property that the real number denoted by the infinite decimal 0.999... is the same as the real number denoted by 1 follows directly from the structure of the real number system, limits, suprema, lack of infinitesimals, and all. And that's all there is to it. Once you accept the properties of the reals (either working from "old-fashioned maths" or from an axiomatic definition of the reals), 0.999... = 1 follows directly from those properties.
Of course, anyone can prove that 0.999... != 1, simply by ignoring these properties, and making up their own non-standard number system (if they work hard, they can even make it self-consistent, see some of the examples lower in the article) and that's the point that needs to be made to the mathematically naive: if you think that 0.999... != 1, then you are no longer working in the standard real numbers. -- The Anome 11:10, 28 October 2006 (UTC)

Abve Meni Rosenfeld says, "The decimal expansion is nothing more than a convenient notation for representing real numbers, and the existence of 0.999... is just a quirk of this particular notation" [my emphasis]. I cannot see this any other way than as a massive point in favour of including the phrase "real numbers" at the very start of the article.Davkal 11:15, 28 October 2006 (UTC)

It supports the view that it is important to understand that 0.999... is a real number; but more strongly, it supports the view that no matter how you look at it, it is crystal clear (to those who know what real numbers are) that 0.999... is a real number - a fact which thus need not be stated explicitly. Also, while there will obviously be readers for whom this will not be clear, those will probably not be able to appreciate it if stated explicitly, either. -- Meni Rosenfeld (talk) 16:45, 28 October 2006 (UTC)
There should be no need for this article at all, if all wiki readers just had assimilated their Zeno, Eudoxos, Cauchy, and Dedekind a little better :-). Most readers finding this article will understand that it is about mathematics. To few of them will understand that it is about real numbers as they were formalised less than 200 years ago. However, this is neither here nor there; this is a matter of trying to maintain an encyclopædian standard. The start of the article should explicitly describe its area, even if that area implicitly is given by the fact that the article would be meaningless in any other context. Since both mathematics and the modern theory of real numbers are crucial in the article context, they should be mentioned at the beginning. JoergenB 18:04, 28 October 2006 (UTC)
Given that 0.999… was known to equal 1 long before the real numbers were invented, that is an anachronistic point of view. The formal structure of the real numbers is a model for the properties that we desire in our numbers. Those properties were developed first, and they are much more reasonable for the average reader. Melchoir 18:29, 28 October 2006 (UTC)
I really prefer the last argument of Melchoir's, in the sense that we are getting closer to discussing facts again. I do take it that both you and Meni agree with the principle: if the article in a fundamental manner depends on and concerns the modern theory of real numbers, then this should be reflected at the very beginning. If so, the 'only' unresolved issue is whether or not this is the case.
Given that 0.999… was known to equal 1 long before the real numbers were invented... I disagree. Would you please qualify that statement? Applying a formula for geometric sums does not in itself qualify, for as long as mathematicians actually used the same formula for deriving 'results' such as
1 + 2 + 4 + 8 + 16 + ... = -1
These kinds of arguments are what Bell referred to when he wrote: Modern mathematics is indebted to Cauchy for ... the introduction of rigor into mathematical analysis. It is difficult to find an adequate simile for the magnitude of this advance; perhaps the following will do. Suppose that for centuries an entire people has been worshipping false gods and that suddenly their error is revealed to them. Before the introduction of rigor mathematical analysis was a whole pantheon of false gods. (E. T. Bell, Men of mathematics, 1937; beginning of ch. 15, 'Mathematics and Windmills'.)
Bell's books are old; and he often wrote in a 'flowerishing' style. However, I know no more modern mathematical historians who are of an essentially different meaning. Yes, there were people who 'knew' that 0.999... = 1 before the rigorous definitions of limits and real numbers; but some of them also 'knew' that \sum_{n=0}^\infty 2^n = \frac{1}{1-2}\,. That 'knowledge' may be of historical interest, but has little to do with 'mathematics as we know it today'.
Now, I don't want to make this into a non-profitable cry for 'evidence' from both sides. If you retain your opinion, I'm quite satisfied if you explain what you mean by your statement, and what kind of logic (if any) would have been behind this knowledge, without quotes, Melchoir. (This won't stop me from criticising, if I disagree with the validity of your arguments.) JoergenB 16:05, 29 October 2006 (UTC)
If you look up Euler and Bonnycastle's pre-Cauchy treatment of repeating nines, both of their statements of the geometric series formula require the common ratio to be less than 1. Such geometric series were the first to have their sums determined, starting from the Greeks. After Cauchy's revolution, Abel lamented at one point that they were the only rigorously determined series. And really, the modern proof based on partial sums is already suggested by authors such as Euler and Bonnycastle, who precede their treatments of infinite geometric series with treatments of finite geometric series. Finite geometric series themselves had been completely understood and a popular textbook topic since the 1500s. And even though Euler didn't use the sequence of finite sums to prove the formula, he did use the sequence to argue that "1 + 2 + 4 + 8 + 16 + ... = -1" was ridiculuous.
So the fully correct statement of the geometric series formula was known before Cauchy and before the real numbers. The rigorous proof, using modern definitions, of this formula was of course not used before those definitions arose, but the mathematics behind the proof was already understood. That 0.999… = 1 followed.
Does the article fundamentally depend on the real numbers? Yes, in the completely trivial sense that we interpret 0.999… as a real number. You may as well ask whether "2π-π=π" depends on the real numbers. Of course it does, mathematically. But really, it's a statement of elementary algebra. If someone tried to define the real numbers in such a way that 2π-π were other than π, then that would have been a flaw in their number system, and no one would have used it. The fact that arithmetic works in the real numbers is less a miracle that we owe to the inventors of the real numbers, and more a design specification that we expect of them. And that is a reasonable expectation.
Does the article fundamentally concern the real numbers? No. The first proofs concern the consequences of decimal arithmetic, and the next proofs concern the consequences of calculus. Of course, decimal arithmetic and calculus are ultimately founded on the real numbers, but that connection is the focus of other articles. Melchoir 19:58, 30 October 2006 (UTC)
Since you've just consulted it, could you please give a precise reference to 'Euler and Bonnycastle's pre-Cauchy treatment', in original? (Not crying 'evidence'; it's just that we happen to have an edition of most of Eulers maths in original in our library. I've even once tried to read the one on the bridges of Königsberg; but my Latin isn't very advanced...) JoergenB 12:34, 31 October 2006 (UTC)

Well done

Just want to say well done to all those who withstood the main page exposure, over 500 edits in a single day! --Salix alba (talk) 07:53, 26 October 2006 (UTC)

where are the sources

In the introduction, a number of claims are made about researchers into maths education and their analyses of the reception given to this claim by students. Perhaps the relevant articles published on these points should be cited. This is one example but much of the article seems equally devoid of sources even when specific non-mathematical points are being made. Davkal 11:09, 26 October 2006 (UTC)

Per the Manual of Style, introductory sections should act as a summary, and not give any information that isn't available further down. As such, they aren't usually cited, but the relevant bits in the article that they summarise are. Have you checked the remainder of the article for the cites you're seeking? GeeJo (t)⁄(c) • 17:52, 26 October 2006 (UTC)

Could/has someone figured out a fix for this

Search .999 and yuou're redirected here Search .9 and you get no hits or redirects Search .99 and you get no hits or redirects Search .9999 and you get no hits or redirects Search .99999 and you get no hits or redireccts

etc etc etc

So, if you want to make it more searchable, still trying not to link everysingle number of nines, perhaps someone with some experties could tackle that? I sure don't know how. Balonkey 13:36, 26 October 2006 (UTC)

3 is the de facto standard to show a repeating sequence. 0.9 would probably not be an ideal redirect, since it doesn't properly represent 0.999..., however at the moment 0.9... is not redirected, though I think it should be. Falsedef
Agreed, I've created the redirect. Theresa Knott | Taste the Korn 22:09, 26 October 2006 (UTC)

One day's changes

Despite all the vandalism, this article was greatly improved during its time on the main page! See this before-and-after the main page exposure diff. —Mets501 (talk) 18:11, 26 October 2006 (UTC)

image

or  ? →AzaToth 19:15, 26 October 2006 (UTC)

The character spacing on the new image seems to be too tight for its font. If you loosen it up, it'll probably be easier to read. Melchoir 19:25, 26 October 2006 (UTC)
I agree: loosen it up and I also would make the shadows a drop lighter. —Mets501 (talk) 19:26, 26 October 2006 (UTC)
Is the shadow supposed to be subtle? The 1 is kinda hard to notice. Falsedef 19:28, 26 October 2006 (UTC)
Agreed -- it would be better with larger character spacing. The variegation on the 9's is also superfluous at the size we're using here. For now I think the grayscale image is easier to read. —ptk✰fgs 20:27, 26 October 2006 (UTC)
I don't like the new image. The first one is very charming because of its simplicity. This is an encyclopedia, and for the purposes of illustrating an infinitely recurring sequence of numbers, I find the first one to be much more concise. Less objective is the opinion of mine that the 3D and color effect on the new version are but simple effects that actually impede the image's visual integrity. Also, keep in mind that the original image prints much easier. —msikma <user_talk:msikma> 20:29, 26 October 2006 (UTC)
it might work better if the shadows were in front of the 9's instead of behind them. (Backlit) It is too fussy as is. Algr 20:48, 26 October 2006 (UTC)
Try a backlit image so we can compare. I like the subtlety. I agree that this image has great potential. I also agree with previous comment re: spacing. Good work. Argyrios 22:00, 26 October 2006 (UTC)
I'd go for AzaToth's new image if the shadow were in front of the digits so they would stand out more. Maybe even make the shadow a different color, so it's not confused with the 9s. — Loadmaster 23:28, 26 October 2006 (UTC)
I'd say we should not change the image because the current one is much more concise and works better when printed. This is just bad design; taking an existing design and applying "cool effects" to isn't what I consider to be a suitable improvement. —msikma <user_talk:msikma> 13:34, 27 October 2006 (UTC)
The reason I like the image is that the addition of the shadows is not merely a "cool effect," but actually adds meaning to the picture. After all, a large part of this article is devoted not just to the fact that 0.999... = 1, but to the fact that many people find this jarring and counterintuitive and seemingly impossible. The numbers casting those shadows is similarly jarring and counterintuitive and impossible-looking (yet, like the proof, right in front of your eyes and undeniable). I think the new image would do an excellent job representing both the mathematical and sociological / educational aspects of this article. Argyrios 18:10, 27 October 2006 (UTC)
It's badly done and I don't share the idea that the shadow has deeper meaning. I find it to be a rather shallow explanation of what appears to be, like I still think it is, an image with "cool effects" thrown onto it. I've been a graphic designer for years, and even though this new image is more sophisticated, I would never choose it over the old one, which fits perfectly with the typical black on white text of the article. Plus, I think that it's a big thing that this image impedes usability in that printers might choke on it (many people have either black and white printers, do not like to needlessly spend color ink, or have a bad DPI setting for color prints so the image won't be rendered correctly anyway). —msikma <user_talk:msikma> 22:31, 27 October 2006 (UTC)
Oh, and that said, I should note that I don't feel like I "own" this article, or anything (especially since I've not made one non-minor edit to it), but I just personally would not choose the image. You're free to do so if you do like it. I won't edit it out (straight away)... :) —msikma <user_talk:msikma> 22:32, 27 October 2006 (UTC)

I think that both images are humorous. I like the anakin/darth aspect of the second one, but sadly it's just not as clear as the other.

This doesn't seem very NPOV

It states in the first paragraph that it is exactly equal to one. I've heard a number of arguments from many different people as to whether it is or isn't. I've actually debunked many proofs myself (Too lazy right now, but if anyone wants to see sections added on counterarguments I could do so). I would call .999... an irrational number. Perhaps we should change the opening paragraph to simply say it may or may not be 1, and is frequently debated? --Nintendorulez talk 21:33, 26 October 2006 (UTC)

Hell no! NPOV doesn't mean that you can't state facts. All professional mathematicians agree that 0.999... is exactly equal to 1. Theresa Knott | Taste the Korn 21:37, 26 October 2006 (UTC)

I know quite a lot of people, including mathematicians, that don't agree. Given that there are coherent proofs to find that it does equal one, and coherent proofs that it does not equal one, I wouldn't call it a fact. --Nintendorulez talk 21:39, 26 October 2006 (UTC)
Which mathematicians do not agree? Where have they published their proofs? Please provide citations. Theresa Knott | Taste the Korn 21:43, 26 October 2006 (UTC)
In math, when one point of view cannot be supported with proofs, then that is not a point of view protected by NPOV. Argyrios 21:57, 26 October 2006 (UTC)
Before we even discuss acknowledging the "debate" on this topic, I would also like to see a peer-reviewed paper that does not agree with our article. You will find, however, that there are no such papers. mstroeck 22:21, 26 October 2006 (UTC)
An irrational number is informally defined as one that is non-terminating and non-repeating. 0.999... repeats. Supadawg (talk • contribs) 22:48, 26 October 2006 (UTC)
Oh, please. An irrational number is one that is not a ratio of two integers, i.e., is not a/b for some "whole numbers" a and b. Of course, that is logically equivalent to saying its decimal expansion neither terminates nor repeats, but no mathematician takes that to be the definition, even "informally". Michael Hardy 23:52, 26 October 2006 (UTC)
I am fully aware of the actual definition of an irrational number. I said "informal" because no mathematician takes it to be the definition. Am I not allowed to simplify for the purpose of discussion? In any case, every repeating non-terminating decimal is a rational number, as proven below by Argyrios. Supadawg (talk • contribs) 03:15, 27 October 2006 (UTC)
Aren't you splitting hairs? Every repeating non-terminating decimal is a rational number. That's an easy proof. Simply take the power of ten equivalent to the number of digits in the repeating pattern, multiply by that power of ten, and subtract the original. Example:
x = 0.293293293...
1000x = 293.293293293...
1000x - x = 293.293293293... - 0.293293293
999x = 293
x = 293/999
So every rational number is a terminating or repeating decimal and every terminating or repeating decimal is a rational number. It's not as if one way of representing the number is "right."
Incidently, 0.999... is subject to the same steps, showing it to be equal to 1/1.
Argyrios 01:34, 27 October 2006 (UTC)
So like 0.\underbrace{b_0b_1\dots}_{n}\dots=\frac{\underbrace{b_0b_1\dots}_{n}}{\underbrace{9\dots}_{n}}\! ? →AzaToth 02:03, 27 October 2006 (UTC)
It seems that would follow. I hadn't thought of that. Argyrios 02:57, 27 October 2006 (UTC)
We should probably change the opening paragraph to say that whereas some people think that 0.999... is not equal to one, this is most definitely incorrect. Besides, if it isn't 1, then what is it? O_o --M1ss1ontomars2k4 (T | C | @) 00:32, 27 October 2006 (UTC)
I think it's fine in the second paragraph. Melchoir 00:53, 27 October 2006 (UTC)
There's no need to add that this is hotly debated. It's already stated in the intro that many students of maths often reject the equality. —msikma <user_talk:msikma> 10:52, 27 October 2006 (UTC)
What about my other question? If it's not equal to 1, then what do people say it's equal to? Just 0.999...? O_o --M1ss1ontomars2k4 (T | C | @) 01:00, 28 October 2006 (UTC)
Usually something like "the largest real number less than 1", which only makes sense if you ignore that real numbers are dense. --jpgordon∇∆∇∆ 17:11, 28 October 2006 (UTC)

What about the Heaviside Step function?

As described here.
Isn't H(1-0.999999) = 1 and H(0.999999-1) = 0, or to put it another way doesn't 1-0.999999 = 0+, and 0.999999-1 = 0-. Of course you could argue that in the limit 0- = 0+ = 0, but such reasoning isn't valid if there's a Heavyside or Dirac Delta Function in the neighbourhood.
Sam 213.208.101.80 22:30, 26 October 2006 (UTC)

First of all, 0.999999 is not the issue; 0.999... is. I am assuming that is what you meant. I believe that if you define H(1-0.999...) = 1 and H(0.999...-1) = 0, what you have is not any sort of function at all, because H(0) would evaluate to both 1 and 0. Argyrios 22:40, 26 October 2006 (UTC)
(edit conflict) I don't understand what you are arguing. You are deliberatly putting a discontinuous function at x=0. Then you are approching 0 from either side and noting that there is a step. No surprise there! vWhy can you say that in the limit at epsilon >0 H(1-(1-epsilon)) = H((1- epsilon)-1) = 1/2 as per the definition of the heaveside step function? Theresa Knott | Taste the Korn 22:47, 26 October 2006 (UTC)

Simple Proof

X= 0.9999.....

10X= 9.9999

            subtract 1X

      9X= 9

    divide both by nine
         X=1

much simpler proof

I don't get what all the confusion is about. Let's put if this way: 0.9 < 0.99999999999999999 < 0.9999999999999999999999999999999999999 < 0.99... = 1. The more 9's there are, the larger the number gets, and the closer it gets to 1. When the 9's increase, the rate that the 9's get closer to 1 decreases. Since the 9's go on for infinity, the 9's get closer and closer to 1 until it arrives at 1, since 0.99... has an infinite number of 9's. An infinite number of 9's are required to reach 1. If a number can never be reached, it needs an infinite string of numbers to reach it, which is in the case of 0.99... As the rate of getting closer to 1 decreses, it doesn't stop until infinity is reached by the number of 9's, arriving at 1. Therefore 0.99...=1. Also, there are no infinite numbers except for infinity and negative infinity. AstroHurricane001 14:14, 27 October 2006 (UTC)

Your "proof" is mathematically sloppy and/or incomplete. For example, you could say, "0.8 < 0.8888 < 0.88888888.... The more 8's there are, the larger the number gets, and the closer it gets to 1. When the 8's increase, the rate that the 8's get closer to 1 decreases. Since the 8's go on for infinity, the 8's get closer and closer to 1 [all true up to this point] until it arrives at 1 [false]." Also, there are imaginary and complex infinite limits, so your statement about "infinite numbers" is wrong. Clearly you're convinced that 0.999... = 1, but I doubt your exposition would convince others that wouldn't be convinced by methods in the article (and it would upset many mathematicians in the process). Calbaer 17:16, 27 October 2006 (UTC)
[Edit conflict]That's good thinking, but it's not really a proof, and it's not really simpler. That's using your intuition to explain it, which is exactly what most opponents are doing and that we are trying to avoid. The difference is that your intuition is more loyal to the nature of real numbers - which is a good thing, but still not enough.
As for infinite numbers... Well, it's a little more complicated than what you describe, but that's beside the point. -- Meni Rosenfeld (talk) 17:17, 27 October 2006 (UTC)
The main troubles arise at this argument:
Since the 9's go on for infinity, the 9's get closer and closer to 1 until it arrives at 1
The critics (and they include classical Greek philosophers as well as school children) would point out that if indeed there are an infinite number of operations, then there is no 'after'. You don't 'arrive' at 1 in or after the last one of an infinite number of steps, since there is no last step. JoergenB 14:12, 28 October 2006 (UTC)

real numbers

I've reinserted the qualification in the first line to saythat "in the mathematics of the real numbers" since this is where 0.999... = 1 and, as the article currently says, there are number systems where 0.999... does not equal 1.Davkal 10:09, 28 October 2006 (UTC)

Absolutely agree. Mathematics is a big place, and whilst the standard reals are where all the action is in any mathematics with relevance to the real world, there are entirely valid non-standard formal systems where 0.999... != 1, even if they are not of much practical use. -- The Anome 10:41, 28 October 2006 (UTC)

Yes, but it has been removed a number of times without discussion here. The point being, here is what the article currently says: "there are number systems in which an object[hmmn - how much does it weigh?] called "0.999…" is strictly less than 1", and here is what Meni Rosenfeld says "there is no such thing as "the mathematics of real numbers" [yes there is, there had better be], in mathematics, 0.999... represents a certain real number, and that number is 1.". Davkal 10:50, 28 October 2006 (UTC)

Yeah, we can't have it both ways. Meni Rosenfeld earlier says that, "0.999... is 1 in the only context in which it is appropriate to discuss it (the real numbers), and this is a fact which can be proven rigorously (nothing philosophical about it)". However, the article then goes on to discuss several alternative contexts for 0.999. Accordingly, we can either remove those, or else we really do have to specify which context we are talking about when we make assertions. — Matt Crypto 11:06, 28 October 2006 (UTC)
The reason why I don't think it's necessary is that it's one of those things that is assumed implicitly. For instance, 10 (number) starts "10 (ten) is an even natural number following 9 and preceding 11", assuming implicitly that the decimal system is used. Lisbon starts "Lisbon (Portuguese: Lisboa, IPA: [liʒ'boɐ]) is the capital and largest city of Portugal", ignoring that there is also Lisbon, Iowa, that Lisbon is not the capital of 13th centuary Portugal, and that there are (I assume) novels in which Lisbon is not the capital. For the same reason, it's not necessary to specify "real numbers" because everybody will assume that we're using the real numbers. -- Jitse Niesen (talk) 11:26, 28 October 2006 (UTC)
That's the point: unlike the use of decimal numbers or defaulting to the more famous Lisbon, working within the reals is not assumed implicitly by the average reader. The properties of the reals are not "obvious". Most people just know -- or think they know -- about things called "numbers", which are the objects of their own personal folk mathematics, in which, for example, if two numbers have distinct decimal representations, they are different numbers -- and, for all practical real-world purposes, they're right! However, they do not have any similar intuitive grasp of the properties of limits or sums of infinite series. Put the two together and, since their intuition is not longer valid for infinite sums, perfectly reasonable confusion results. Hence the need for this article. -- The Anome 11:37, 28 October 2006 (UTC)
I agree with Jitse Niesen. The apparent contradiction with the statement "However, there are number systems in which an object called "0.999…" is strictly less than 1." is only a problem if it is taken out of context. As it stands, it is clear that that statement is about "non-standard" number system.
I still think it might be helpful to clarify that statement somewhat, for example by adding ", but such number systems will not obey the usual mathematical rules." Neither my english nor my mathematics are good enough for me to feel comfortable making such an edit, though, so I will leave that to someone else.
The "numbers" that The Anome describes are not consistent, and are thus not really part of mathematics at all. The point made by Meni Rosenfeld and others above is (I belive) that real numbers is the system in mathematics which most closely resembles peoples own "personal mathematics", and that the number systems in which an object which could reasonably be called 0.999... is smaller than 1 has other even more strange properties. Thus, unessecarily stressing that we work in the real numbers only support the mistaken idea that there could be number systems which have all the properites that we usually expect, while still allowing 0.999...<1.
Tengfred 11:55, 28 October 2006 (UTC)
We should not talk down to our readers by assuming they are stupid. Many of them -- indeed, almost all of them -- might be mathematically unsophisticated, but we should not assume that that means they are stupid.
As you correctly say, "The 'numbers' that The Anome describes are not consistent, and are thus not really part of mathematics at all." That's exactly my point: the average reader is not mathematically sophisticated (by which I mean here something like "has taken a real analysis course") and has only a "common sense" grasp of numbers (typically "numbers" = "decimal numbers" with a one-to-once correspondence with digit strings, since that's what they take home from high school), which will have served them well in real life, but will mislead them when it comes to this famous pons asinorum. They will not assume that we are talking about the reals, because they won't know what the reals are.
The reals are a logically consistent formalization of the "everyday numbers" that exist in folk mathematics, and are not the same as those useful, practical, but alas logically inconsistent "everyday numbers": and the property 0.999... = 1 is interesting precisely because it is an example of the differences between the two (see my comment above). The fact that most people rely on folk mathematics is the whole reason for the very common misapprehension that makes this article important: as a mathematical fact in the field of real numbers, 0.999... = 1 is banal, but for very many non-mathematical readers, it is not, and this article will be their first introduction to a wide range of more sophisticated mathematical concepts, and to a more formal and rigorous way of thinking about numbers. And that's a good thing.
To assume that the reals are synonymous with "numbers" in the common sense, because unlike the logically inconsistent numbers of folk mathematics, they have "all the properties that we usually expect" is itself an appeal to intuition. There are only two reasonable ways of defining "numbers": they are either a way of describing the world around us (two apples and two apples make four apples, two halves make a whole...), or they are objects in an axiomatic system. Remarkably, the reals and "commonsense numbers" seem to agree with one another to a remarkable extent, to the limits of current experimental precision, and over many, many orders of magnitude. The reals are a magnificent intellectual achievement, and they are for everyday purposes, the most reasonable definition of "numbers". However, there are lots of other number systems which are supersets of the reals (like the complex numbers) and still have "all the properties that we usually expect", and there is no evidence at all that the "actually real numbers" derived from observing the universe are, like the reals, actually continuous, or lacking in infinitesimals. (If you disagree with me, please feel free to prove either of these last two; if you can, I confidently predict that you will be the first ever joint winner of both the Nobel Prize in Physics and the Fields Medal -- and arguably, also the Peace Prize for settling the arguments on this talk page once and for all.)
To recap, the naive reader is not necessarily as stupid as they may seem; some of the issues as to whether numbers = reals are deep issues within the philosophy of mathematics, and should, of themselves, draw some more-sophisticated readers on to taking a course in non-standard analysis. But this article is not about the philosophy of mathematics, it's about the reals. And saying real numbers in the intro sentence makes that clear.
-- The Anome 12:21, 28 October 2006 (UTC)

IMO it's an excellent idea to note that we talk about the mathematics of the real numbers, already in the first line. Within the modern theory of real numbers the result is trivial, and the troubles in understanding this indeed has to do with the fact that this complicated theory is rather complicated. Actually, at our university we don't try to teach it in the first analysis courses, at least not i a consistent manner, since most of us believe that our students would be more confused than helped by exposing the real numbers as either Dedekind cuts or equivalence classes of Chauchy sequences. (We get a little closer to the 'purely axiomatic' approach, at least introducing the 'axiom of upper limits'; but I know no simpler way to reduce the consistency problem to that of the Peano, than actually to provide constructions of 'reals' fulfilling the axions.) Also, please recall that a logical foundation for the limit concept was developed in close relation to the theory of real numbers. JoergenB 14:57, 28 October 2006 (UTC)

First, my comment about "there is no mathematics of real numbers" is a naming issue - There is mathematics, and mathematics deals with all kinds of structures (real numbers, hyperreal numbers, etc.), but I don't recall having encountered the term "the mathematics of the real numbers", and do not think it is appropriate.
Second, I think the only place where I know of 0.999... being discussed other than the real numbers, is Richman's so called "decimal numbers" - a structure which is discussed only to show why it shouldn't be discussed (the reason being that it is utterly useless). This, of course, is not a criticism against Richman - it's great that he has settled the matter, to prevent some crank from coming up with the structure and claiming it is important.
I think that you will find it difficult to provide a reference where 0.999... is discussed in any other context. The reason is simple - 0.999... is a decimal expansion. And decimal expansions are good for exactly one thing - representing real numbers. Any attempt to deal with anything else in terms of decimal expansions is rather silly (though variations of it might be useful, as in 10-adics).
Of course, silly does not mean inconsistent, and the symbol 0.999... could be used for other purposes. But, as it is silly, I doubt it has been done in any serious work, and thus such a treatment cannot be considered a part of mathematics.
To conclude, in mathematics, 0.999... does not represent a Richman's decimal (because nobody talks about Richman's decimals). It does not represent a hyperreal number (because nobody talks about hyperreal numbers in terms of decimal expansions). It represents a real number. And that number, of course, is 1.
If any part of the article suggests that this is not the case, it is misleading and should be rewritten. -- Meni Rosenfeld (talk) 15:56, 28 October 2006 (UTC)
Oh, and we're forgetting another thing: While the first sentence does not explicitly mention real numbers (which is fine, per many of my and others' posts here and in Arguments), the second sentence (which clarifies the first) does, for those who have any doubt about the intention. -- Meni Rosenfeld (talk) 16:06, 28 October 2006 (UTC)
Yes. Also, the lead section did not use to say "However, there are number systems in which an object called "0.999…" is strictly less than 1.", and that is a dangerously misleading statement. It implies, for example, that there is a number system in which an object called "0.999…" is strictly less than 1. There isn't. So let's undo this recent edit along with its even more recent consequences. Melchoir 16:16, 28 October 2006 (UTC)


Spoken Article

The spoken article for this page is nuts. It's particularily loud, quality is similar to that of a telephone, and not to mention it's almost an hour long. I kind of feel sorry for whoever made it, the poor guy.

Just thought I'd point that out.

Oh, and 0.999... does not equal 1. 1=1, end of story. Any other number is different and should be regarded as such. LOL NIRVANA2764 20:45, 28 October 2006 (UTC)

You're right, any other number is different and should be regarded as such. 0.999... is not an other number, it's the same number. -- Meni Rosenfeld (talk) 21:05, 28 October 2006 (UTC)

If 0.999... is the same number, why is it written differently, spoken differently, referred to differently, requiring of infinity to understand it, have a 0 in its ones spot, not expressable as a fraction, and before 1 on the number line?--JohnLattier 03:25, 29 October 2006 (UTC)

Leave it on the arguments page. Melchoir 03:30, 29 October 2006 (UTC)
0.5 and 1/2 are the same number, but are written differently, spoken differently, and referred to differently. Any decimal with an infinite number of places will require infinity to understand it by definition. 0.999... does not come before 1 on the number line, as 0.999... and 1 have exactly the same value. -- Schapel 13:44, 29 October 2006 (UTC)

Error in logic in calculus and analysis section

Moved to Talk:0.999.../Arguments#Error in logic in calculus and analysis section by Calbaer 02:39, 2 November 2006 (UTC)

Article should give definition before proofs

The article should first define what is meant by the string of symbols "0.999..." before proving that it equals 1. It could mention that "0.999..." is defined to mean \sum_{n=1}^\infty \frac{9}{10^n} \equiv \lim_{N \to \infty} \sum_{n=1}^N \frac{9}{10^n}, along with links to Limit of a sequence and Infinite series. --Kprateek88(Talk | Contribs) 10:03, 29 October 2006 (UTC)

Most people who know enough math to understand summation notation will not dispute the premise of the article, so they are not the intended audience - which leaves verbal description and "..." notation, the latter of which is part of the "problem" as far as I'm concerned. "..." has no formal mathematical meaning (although it has a commonly agreed-on informal meaning), so it needs to be specified exactly what is meant without getting overly technical. Calbaer 18:39, 29 October 2006 (UTC)

No, it should be:

\sum_{n=1}^\infty \frac{9}{10^n} < \lim_{N \to \infty} \sum_{n=1}^N \frac{9}{10^n}

in that the first expression is 0.999... and the second expression is a limit and by definition it represents the whole number that is being approached. This may be the root of why people, including the author of this article, don't understand 0.999... isn't 1. They don't understand the concept of limits. --JohnLattier 21:07, 29 October 2006 (UTC)

Except that \sum_{n=1}^\infty f(n) = \lim_{N \to \infty} \sum_{n=1}^N f(n) is the *definition* of the infinite sum \sum_{n=1}^\infty f(n), when it exists.
The entire problem is in the notation. Once you give a rigorous logical definition of 0.999..., that it is equal to 1 is self evident and even when it's not it can be proved beyond question using real analysis. All the confusion results from interpreting recurring decimals 'intuitively', the approach used to teach them at the primary level. The truth is that recurring decimals are very complicated and entirely unsuitable for being taught to students without knowledge of limits.

Any person who had math in high-school knows limits, so I agree with Calbaer that we should present the definition of 0.999... in the introduction and then rigorously devolop its equality with 1. Then anyone who disagrees with the equality must disagree with either the definition or the foundations of set theory and logic (from which real analysis is devoloped), both reasonable disagreements that can not be contested. Loom91 08:16, 30 October 2006 (UTC)

Thanks; I added a definition that is hopefully is rigorous enough to be valid but not so rigorous that it requires much more than a tiny bit of calculus (i.e., an understanding of limits). It is quite funny that some people are so sure of their view of mathematics that they believe that there is a "controversy" or that the article is "wrong." The article is as controversial as the flatness of the Earth. It would be near-impossible to find a professor of mathematics who doubted the accuracy of the premise (although some might object to ellipsis notation being too "fast and loose" — 0.\bar{9} is better, though it's less URL- and wiki-convenient). Calbaer 19:58, 30 October 2006 (UTC)
The definition you created wasn't quite the right definition. You skipped a step by arithmetically calculating that, in fact, each term of the sequence is a number of the form 1-10^-k. While that equation is correct, it's not actually a definition of what a repeating decimal number would be.
Rather, a repeating decimal 0.AAA... would be defined as the limit of the sum sequence of all terms of the form A x 10^-k for integers k from 1 to infinity. That is, the sum of all terms {0.A, 0.0A, 0.00A, ...} (Sorry, I'm not up on wiki math formatting or I'd insert this using better looking formatting.). So 0.999... would be defined as the limit of the sum of all terms of the form 9 x 10-k for k from 1 to infinity.
Of course, from that definition, it's not a large leap to derive the sum is also the limit of terms of the form 1-10-k. But that's not the definition. And this could be a necessary distinction if you are trying talk about other repeating decimals that aren't as easily calculated as 1 minus a simple number. Dugwiki 23:32, 30 October 2006 (UTC)
P.S. Really, even that definition would only apply if A is one digit. In order to handle repeating decimals where the repeated sequence is multiple-digits, such as 0.125125125125... , the definition needs to be tweaked to be more general. But for the purposes of this article, it's probably ok to just focus on single digit repeating decimals. Dugwiki 23:42, 30 October 2006 (UTC)

This sounds weird

In the opening: "A similar phenomenon occurs in balanced ternary, where 1/2, instead of 1, has two possible expansions." My issue with this is that 1/2 has two expansions in decimal as well. What's with this sentence? --King Bee 14:27, 31 October 2006 (UTC)

Since this point gets only a single, short sentence in the body, I don't think it deserves a place in the opening anyway. Melchoir 17:44, 31 October 2006 (UTC)
Agreed. --King Bee 18:17, 31 October 2006 (UTC)
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