Talk:0.999...

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)

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. — Preceding unsigned comment 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? — Preceding unsigned comment 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)

No, there isn't a way to represent such a number because it doesn't exist (in the reals anyway). Its a property of real numbers (and all "continuous sets", its actually what makes them "continuous") that there is never any "greatest possible value less than x", x being any real number whatsoever. And you can't represent something that doesn't exist, so the answer to your question is a definite no. Brentt 00:58, 25 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)

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)

I've never been able to understand these "proofs"

moved to Talk:0.999.../Arguments#I've never been able to understand these "proofs". -- Meni Rosenfeld (talk) 10:48, 23 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)

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)

No Way!

moved to Talk:0.999.../Arguments#No Way! Supadawg (talk • contribs) 00:43, 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)

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)

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)

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

This article is an unfortunate embarrassment to Wikipedia

Whereas mathematicians are trained in logic, etymology is sadly sometimes a total mystery. Had the authors of this article bothered to note that the 'infinitesimal' is equivalent to 1 rather than asserting that it is equal as is obviously the case by definition, then this article would have received the attention that it is due: very little.

Also, it is inherently novice to try to brow-beat opposition out of hand by refering to them as "students" or "junior". Mathematics is a science, not a political or religious arena:

Although the 'infinitesimal' converges on the number 1, thus being equivalent, it is not equal so long as the theoretical subtrahend, t, exists where $1-t=0.{\bar {9}}$

-Dworkin, --76.1.39.216 02:25, 25 October 2006 (UTC)

It's true that the decimals 0.999… and 1.000…, left unevaluated, are equivalent but not equal. But this is a vacuous observation. You could just as well say that the arithmetic expressions 1 + 1 and 3 - 1 are equivalent, or that the fractions 4/2 and 6/3 are equivalent. Okay, true enough. Nonetheless, the numbers they represent are equal, exactly and identically.

And t = 0. Melchoir 02:35, 25 October 2006 (UTC)

Again, t approaches 0 asymptotically. Thus it converges on 0 and is equivalent but not equal. It's not congruent but you could use "approximately equal" if you're hung on using the word 'equal'. (Similarly, a square is a rectangle but a rectangle is not always a square.) The essence of an asymptote is that it approaches, but never touches. t never equals 0. $0.{\bar {9}}$ never equals 1.

You're close, but no cigar. Try using the word 'equivalent' and see how little debate remains.

Note: To assert that all math books support this claim is completely untrue. Of the dozens of university and secondary level books that I have, not one asserts this claim. Converge, approach and equivalent but not equal. Equal is an unwise use of notation in this case. Rounding up doesn't count; maybe the dot above the '=' sign is missing from authors relying too heavily on the works of others but not copying correctly. Furthermore, not one of the offered proofs gives a viable reason for making that tiny, yet important step to enable the use of 'equals' rather than 'approximately equals'. Numerical evaluation would yield the value of 1 for $0.{\bar {9}}$ for all practical purposes but does not retro-define mathematical notation.

-Dworkin--76.1.39.216 04:46, 25 October 2006 (UTC)

0.999... = 1. t in your example = 0. You're close, but no cigar. What part of the proof do you dispute?

Great article, by the way, and as the above illustrates evidently still necessary.

By the way, the third bullet point under "skepticism in education" appears to address your confusion, as evidenced by your use of the word "approaches." There is additional dicussion in that section about the common misconception of the sequence as a process rather than a value. Hope that helps. Argyrios 05:27, 25 October 2006 (UTC)

One last time until I see a point to your argument. As noted above, I dispute the part that is missing in every proof. - - A point is a theoretical construct. It is defined as a dimensionless geometric object with no properties except location. You can cut the distance between $0.{\bar {9}}$ and 1 as often as you like but they are asymptotic. Unless you insert a step into your 'proof' which proves that the last point diappears, then it still exists. The case study with two options under "skepticism in Education" is an assertion, not a proof. Either way, the second assertion is not the case. - - If your assertion is true, then you contend that 'approximately equal', 'converges upon' and 'approaches' are all identical to 'equals'. There is a missing step, but you'll never get there using an asymptotic 'infinitesimal'. You can ignore it; but the 'point' is still there.

There is no confusion; only lack of proof in all of the listed 'proofs'.

-Dworkin--76.1.39.216 05:46, 25 October 2006 (UTC)

I truly am sorry that you do not understand the proofs. If nothing else will convince you to stop sabotaging the efforts of others to understand this mathematical concept by making false assertions about mathematics in response to their sincere queries, perhaps WP:OR will. That 0.999... = 1 is long- and well-settled fact among mathematicians. If you are near a college campus, try finding a math professor and try to tell him that he doesn't have it right and you do. Get it published; then come back. (Of course, this will never happen.) Your own original research does not belong on wikipedia. Regards, Argyrios 05:54, 25 October 2006 (UTC)

Alternatively, find a published article in an academic journal disputing that 0.999~ = 1. You won't be able to; no journal would publish it. I can see that I am never going to convince you that you are wrong if you haven't already accepted it, so I am just trying a different tactic here to settle the debate.

I've been on the losing end of WP:OR arguments myself, sure I was right but unable to show that my idea was accepted outside of my own head. Wikipedia is a reflection of consensus knowledge. You just don't have the authorities to back up your position. We have mathematicians. You have confused high-school students. Game over. Argyrios 06:22, 25 October 2006 (UTC)

There is no asymptotic behaviour here. There is no 'behaviour' at all. We are talking about individual numbers. Points on the real line, not functions, not sequences. Saying something like "numbers x and y are asymptotically close" has no meaning in the real numbers. We are talking the number 0.999... and the number 1. They are mathematically equal. Not just "almost equal", not "equivalent but not equal". Equal. Settling for weak true statements because strong true ones cause arguments is not the sort of thing we do on Wikipedia. Maelin 06:27, 25 October 2006 (UTC)

Wow. OK, if you insist on brow-beating rather than proof then I suggest that you check the link on the main page:

http://qntm.org/pointnine

Toward the bottom, the author who agrees with you states in his Q&A:

Q: "My mate/my dad/my mathematics teacher/Professor Stephen Hawking told me that 0.9999... and 1 were different numbers."

A: They were wrong. In science, credentials are as worthless as intuition (above). Proof is everything.

So, now that we've dropped names (like Stephen Hawking on my side) and brow-beat each other sufficiently, can we get back to the missing step in your 'proof'? As your comrade says, "credentials are as worthless as intuition." But hey, at least we know that the WP:OR wasn't violated.

-Dworkin--76.1.39.216 06:48, 25 October 2006 (UTC)

There is no missing step in the proof, and it is not my own proof. I can't force you to understand it if you are unable to do so. Unless you plan on changing the article repeatedly and then getting banned for vandalism, which is fine by me, I am going to end this conversation. Argyrios 07:40, 25 October 2006 (UTC)

Another Picture

File:Proportional decimal values.jpg

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)

.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)

Yes, what Supadawg said. I am no mathmetician, if you can't tell.Cameron Nedland 02:45, 25 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 and 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. -Nick^talk 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. — Preceding unsigned comment 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)

Woah:

You torched the article?

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

another problem students have

Moved to /Arguments. Melchoir 06:11, 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)

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)

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)

This article is just wrong

1) This article is just wrong. The LIMIT of 0.999... is 1. But that does not mean that 0.999... is exactly equal to 1. The article is based on a misunderstanding of limits and equality. 0.9 is not exactly equal to 1. 0.99 is not exactly equal to 1. 0.999 is not exactly equal to 1. No matter how many times you add a 9, the number is never EXACTLY equal to 1. It's as close to 1 as you want to get, but never gets there.

2) Also, the article title is "0.999..." and that was originally the topic. But the topic is now "0.999... is exactly equal to 1".

I noticed the article was previously nominated for deletion. I would wholeheartedly suggest it be deleted. Dagoldman 06:35, 25 October 2006 (UTC)

These criticisms are addressed in the Skepticism section. The number 0.999... is not a process. It is not a function. It does not have a limit, it does not move anywhere. It is just a single point on the real line. That point is coincident with 1, i.e. they are equal. Yes, 0.9, 0.99, 0.999, any such number with a finite number of nines, no matter how many, is less than 1. But 0.999... is not any of them. It has an INFINITE number of nines. It's true that when you write 0.9, that number is not equal to 1, and no matter how many nines you append, the number is never equal to 1. But it's never equal to 0.999..., either. 0.999... is the limit of the sequence {0.9, 0.99, 0.999, ...} and that limit is, as I'm sure you agree, 1. Maelin 06:44, 25 October 2006 (UTC)

You say "it does not have a limit". Then, you say "0.999... is the limit of the sequence {0.9, 0.99, 0.999, ...} and that limit is, as I'm sure you agree, 1". You just contradicted yourself. Which is it? Does it have a limit or not? I agree with your second statement. 0.999... does have a limit. The limit is 1. But that's not what the article says. I read the Skepticism section. The students SHOULD be skeptical. The article is wrong, and based on a misunderstanding of limits. Math is an exact science. The article is non-rigorous, an example of math misused. By the way, "It is just a single point on the real line" is just more circular reasoning. Dagoldman 07:17, 25 October 2006 (UTC)

Look more closely. 0.999... is a limit. It doesn't have a limit. "To be" and "to have" are really basic verbs; I should think we all understand the difference between them? Melchoir 07:27, 25 October 2006 (UTC)

Yes, you said 0.999... "is" the limit. I misread your comment (it would never enter my head that someone could say such an outlandish statement), and you did not contradict yourself. But where do you get the idea that 0.999... IS a limit? Isn't this just more circular reasoning? If you assume that 0.999... IS a limit, then of course it is equal to 1. But 0.999... is NOT a limit. It is an infinitely repeating decimal. It HAS a limit. You can get as close as you want to 1, but you'll never get there. Dagoldman 07:45, 25 October 2006 (UTC)

If 0.999... is not equal to 1, then what consists of 1? It seems to me like the concept of .999... is as empty as 1 is. But I'm reasoning philosophically, of course. I think the mistake here is to think of 1 as something finite and 0.999... as its opposite, the infinite. But that is not true. 1 consists of infinitesimal elements making it as indefinable as 0.999... Just musing.Moonwalkerwiz 07:37, 25 October 2006 (UTC)

Why am I wrong?

Moved to /Arguments. Melchoir 07:08, 25 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 _words^deeds 07:17, 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)