Talk:0.999...

Template:Mainpage date

Question about the FAQ

Hi. The answer to the second question in the FAQ says that "...an infinite string of zeroes cannot be followed by a 1." While this is, of course, true of decimal expansions, one can have a well-ordered set of order-type $\omega +1$ consisting of countably many zeros followed by a 1. At the risk of feeding the trolls, it may be worth rephrasing the answer to that question (though I cannot immediately think of the right way to do so). Molinari 19:53, 31 October 2006 (UTC)

It might be counterproductive to try to explain things concerning ordinals and cardinals to those who refuse to accept that 1=1, but you're welcome to try. --King Bee 21:27, 31 October 2006 (UTC)

Point taken. Molinari 23:34, 31 October 2006 (UTC)

The answer also says that "0.000...1 is not a meaningful string of symbols", but then goes on to discuss exactly what this string of symbols means, i.e., an infinite string of zeros followed by a one. That notion may be self-contradictory but it is not meaningless as the discussion of its meaning clearly shows.Davkal 21:37, 31 October 2006 (UTC)

The Riemann Zeta Function might not be a meaningless string of symbols either, but it's out of reach for the people disagreeing with the fact that 1=1, and probably has no place here; just as a lengthy discussion of ordinals should not be here either. --King Bee 21:47, 31 October 2006 (UTC)

That may well be so, but I don't see what it has to do with the error referred to above, i.e., not meaningful versus self-contradictory.Davkal 22:12, 31 October 2006 (UTC)

I have no idea what you're talking about. The response in the FAQ is not contradictory. It's a response to those who do not want to grasp a concept of infinity. It succeeds by asking them questions about what they just wrote down, making them find their own flaw. --King Bee 22:23, 31 October 2006 (UTC)

It's contradictory inasmuch as it says "x is not meaningful" and then proceeds to explain exactly what x means. Davkal 22:27, 31 October 2006 (UTC)

I rewrote the second answer; the above discussion seems to indicate that it was not convincing as it should have been. I believe my contribution is less dogmatic, less confusing, and appeals more to what is consistent with definitions. I also got rid of the Archimedean property link, since that has to do with abstract algebra, and gives a lot of abstract algebra before actually saying that the reals have the property. A novice would not understand it, and, worse, might say that the real numbers don't necessarily lack the property. Calbaer 23:44, 31 October 2006 (UTC)

Hi. I wrote the original answer. The current version is fine too. What I meant by "meaningless" is that the string of symbols does not represent a real number; it is not a well-formed string in the language of the system. I was reading a book by Douglas Hofstadter called Godel, Escher, Bach and picked it up from there. Maybe I am using the terminology incorrectly. Check chapter two if you have the book, entitled "Meaning and Form in Mathematics." I like what's up now just as well. Argyrios 01:02, 1 November 2006 (UTC)

Glad to hear it. I didn't like the old version (which I must admit I thought of editing prior to Molinari's comments) because saying something is meaningless without saying why seems like a dismissal of opponents without appeal to reason. If I were convinced that 0.000...1 meant something, someone calling it "meaningless" would not change my opinion one iota. However, pointing out that the "1" must have a well-defined, finite place explains why there's no such thing as 0.000...1. A doubter might say, "Well, there should be," but he or she must in the end admit that decimal notation — as it exists — doesn't allow for 0.000...1. Since that seems to be a big stumbling block — rather than claims that 0.999... isn't a real number or that not all real numbers can be expressed in decimal notation — that should be precisely addressed. Calbaer 01:17, 1 November 2006 (UTC)

If you tell us at what value N for :

$\sum _{n=1}^{N}{\frac {9}{10^{n}}}$ is 1, then I will tell you the position of the 1 in 0.000...1. --68.211.195.82 13:45, 1 November 2006 (UTC)

There does not exist a value of N for which that expression above is equal to 1. That's what you don't understand. --King Bee 14:47, 1 November 2006 (UTC)

There is no value of N for which that expression is equal to 1, this is true. But it is also true that there is no value of N for which that expression is equal to 0.999..., either. There is not a finite number of nines in 0.999..., you cannot represent that number of nines by an integer. Maelin (Talk | Contribs) 16:08, 1 November 2006 (UTC)

I explain that to students (if they ask) in the following way: You are right, 0.999... is indeed different from 1. The first represents the infinite sequence 0.9, 0.99 etc. and the second the real number 1. It just happens that the limit of the first sequence is 1. Seen like this, all this crazy discussion vanishes in dust. As the next step, I get them to agree that the symbol "0.999..." from now on denotes the limit of the sequence. Since they just agreed that it is 1, there is no more argument left.

"All this crazy discussion vanishes in dust" may not be formal enough for some students. One problem with your discussion is that 0.999... does not represent the infinite sequence, but rather its limit. Neglecting such fine distinctions allows enough wiggle room for doubters to (rightly) note a lack of rigor on your part and (wrongly) conclude that you must be wrong in your conclusion. Of course, you can fix the above by saying that 0.999... represents (as do all non-terminating decimals) an infinite sum, or, equivalently, the limit of the infinite sequence. Problem is, even though one is defined to be equal to the other, some people don't accept such definitions. And those who don't accept axioms and definitions cannot be persuaded by mathematical logic alone. Calbaer 18:15, 28 November 2006 (UTC)

You missed my point. The infinite sequence is not at all defined to be equal to its limit, at least I would not want to be understood like this. It just happens, that we denote the sequence and its limit by the same symbol. To be more precise, this is a fault in mathematical notation, not in student thinking. You may be right however, when you say that there are some folks that cannot be convinced by any means. (Classical: "Gegen Dummheit känpfen selbst Götter vergebens").

Time for a comment?

I've noticed that many pages that are subject to vandalism/controversy have an HTML comment at the beginning saying something like "The content of this article is well-established. If you plan to make a significant change, please consider discussing it on the talk page." Is it time for 0.999... to have one? Confusing Manifestation 01:49, 1 November 2006 (UTC)

That's not a bad idea. However, keep in mind that that won't deter some users with a seemingly religiously held conviction that the entire article is wrong, so we should do our best to divert them to the Arguments page. Supadawg (talk • contribs) 02:04, 1 November 2006 (UTC)

That's what I thought. What about a wording like "WARNING: This article contains several proofs that 0.999... = 1. If it is your intention to try and disprove this, please see the Arguments page at http://en.wikipedia.org/wiki/Talk:0.999.../Arguments first, as most of the common objections are dealt with repeatedly there. Also, please understand that the earlier, more naive proofs are not as rigorous as the later ones as they intend to appeal to intuition, and as such may require further justification to be complete. If there is a significant flaw with a proof, however, please discuss it on the Talk page." Confusing Manifestation 02:12, 1 November 2006 (UTC)

Done. I hope this is a good wording. } (talk • contribs) 02:23, 1 November 2006 (UTC)

Proofs need to be reexamined

Moved to Talk:0.999.../Arguments#Proofs need to be reexamined by Calbaer 01:30, 2 November 2006 (UTC)

Interwiki

Is anyone else surprised that there exists a mathematics article in 14 languages such that none of them is German? Melchoir 19:15, 1 November 2006 (UTC)

I am. I suggest we get someone to translate this to the language of our Vaterland so that the Germans can start arguing over the correctness of rigorous proofs. =) --King Bee 20:32, 1 November 2006 (UTC)

Maybe Germans just get it and so are busy laughing at these silly English-speaking people and no one has bothered to translate/write and article on such a silly topic which is understood by everyone Nil Einne 10:12, 5 November 2006 (UTC)

How can it be a real number and a hyperreal?

A representation of a number can't be two different numbers in two different number sets. If you want to express a hyperreal number that is infinitely close to 1, but not one, what would you use? Just like 999/1000 doesn't = 1 in the set of integers, $0.{\bar {9}}$ shouldn't = 1 in the reals. Fresheneesz 23:00, 1 November 2006 (UTC)

Since the real numbers are vastly more common than the hyperreals and since $0.{\bar {9}}$ is a well-defined real number, in order to avoid confusion between the real 0.9999... and the hyperreal one you would have to rename the hyperreal version. But I have yet to see a case where it was in doubt whether the real number or the hyperreal was meant. --Huon 23:07, 1 November 2006 (UTC)

There is no hyperreal named 0.999…, and the existence of an article here is not an invitation to start making stuff up. Melchoir 23:31, 1 November 2006 (UTC)

I'm no expert, but I believe there is indeed a hyppereal 0.999…, since the hyperreals are a superset of the reals. Inverse hyperreals are infinitesimals, so you could say that 1-e is "infinitely close" to 1 but not equal to 1, where e is some infinitesimal. But 0.999… is 1, whether it's a real or a hyperreal. But more importantly, any discussion of hyperreals is irrelevant to this article, which is a discussion of (and limited to) the decimal representation of real numbers. — Loadmaster 23:53, 1 November 2006 (UTC)

If '0.999...' is interpreted as the sum of an infinite series, then there is a hyperreal number infinitely close to 1, such that, the addition of that number to the rest of the series is equal to 1.

999/1000 isn't in the rationals; 0.999... is in the reals. No mathemetician uses 0.999... to mean 1-e (with e an infintesimal in the hyperreals), because that would conflict with the usual meaning of decimal expansions of real numbers. -- SCZenz 06:50, 2 November 2006 (UTC)

999/1000 isn't an element of the rationals? --King Bee 15:24, 2 November 2006 (UTC)

SCZenz probably meant "integers". The hyperreals have infinitely many infinitesimals. One of them is ε. Accordingly, there are infinitely many hyperreals infinitesimally close to 1. One of them is 1 - ε. And you still didn't explain where you got the idea that 0.999... is a hyperreal. One would rarely (if ever) use 0.999... to represent a hyperreal number (as it is a decimal expansion, and decimal expansions are reserved for reals), and one definitely does not need such a notation to represent numbers infinitely close to 1. -- Meni Rosenfeld (talk) 21:38, 2 November 2006 (UTC)

I am under the impression that 0.999… is both a real (obviously) and a hyperreal (not so obviously) because of what Wiki says about hyperreals:

The hyperreals, or nonstandard reals, (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle.

Perhaps I'm reading it wrong, but I take that to mean every real is also a hyperreal. But whatever the case, it's irrelevant to this article, which is about the real value 0.999…. — Loadmaster 22:47, 2 November 2006 (UTC)

That's a somewhat subtle issue. You can define things in a way that the reals would be a subset of the hyperreals, but the other approach (which is clearer) is to define them as two disjoint sets. My opposition to 0.999... being a hyperreal is clear with the second approach; as for the first, Fresheneesz implied that 0.999... is a non-real hyperreal, which is flawed. Your interpretation, that 0.999... is a hyperreal by virtue of being the real number 1, is more sensible, but is not what Fresheneesz was getting at. -- Meni Rosenfeld (talk) 07:17, 3 November 2006 (UTC)

Surely the difference between 0.9.... and 1 would be 1x10-(infinity), which would be the smallest possible number but still would mean that 0.9... does not equal 1.

• Jump over to the /Arguments page, where this and other misconceptions about the real number system are discussed. --jpgordon^∇∆∇∆ 14:56, 25 November 2006 (UTC)

Proposal to globally change inline mathematics into sans-serif

There is currently a proposal suggesting this addition to the global CSS of Wikipedia:

span.texhtml {
 font-family: sans-serif;
}

This would mean that inline mathematics would be displayed in sans-serif rather than serif. This proposal was shot down twice before, but it seems that the strange nature of Wikipediamocratics allow for it to be suggested a third time. Maybe you would like to take a look there and partake in the discussion, since you're all active editors in the field of mathematics. —msikma <user_talk:msikma> 06:54, 2 November 2006 (UTC)

See also

I hate "See also" sections. They assert a connection between topics without explanation, either because the connection has already been explained, above, where the link belongs; or because the connection is so tenuous and speculative that we have no words to describe it.

• Real analysis is a prominent link in the section whose title includes the phrase "real analysis". That makes it unnecessary to point out again; any second use of the link is redundant.
• Naive mathematics, Folk mathematics, Informal mathematics -- whatever you call that article, its inclusion here is original research. I've electronically searched the article's sources for these phrases and found nothing.
• Non-standard analysis is both redundant and original research.

Melchoir 16:49, 2 November 2006 (UTC)

I like See Also sections, for the simple reason that I often use them myself (which is a good reason to believe others do, too). That said, I find links such as Recurring decimal, Decimal representation and Real number to be appropriate here. Informal mathematics is a candidate, but that article currently has no real content (to do?). The others, no. -- Meni Rosenfeld (talk) 21:33, 2 November 2006 (UTC)

Okay, See Also sections are very useful when trying to navigate articles that either aren't comprehensive or don't have decent lead sections. But here? Real number and Recurring decimal are linked in the very first sentence. Decimal expansion/representation is linked in the first sentence of the third paragraph. We put the context of the article on the top, not the bottom. Melchoir 22:08, 2 November 2006 (UTC)

Wikipedia:Guide to layout#See also says: "Mostly, topics related to an article should be included within the text of the article as free links. The "See also" section provides an additional list of internal links as a navigational aid, and it should ideally not repeat links already present in the article." -- Jitse Niesen (talk) 04:46, 3 November 2006 (UTC)

The word "ideally" suggests that this depends on the circumstances, and in my view important links can be duplicated. Suppose a user has finished reading the article, and would like to read some more about decimal expansions (a plausible scenario). Should we demand that he searches the article for that link, instead of finding it in a tidy list at the end? -- Meni Rosenfeld (talk) 07:02, 3 November 2006 (UTC)

I agree with Meni. The See Also section of this article, I feel, should only link to articles with obvious and direct connections to 0.999.... For the less obvious links, like informal mathematics, leave the links in the article. Maelin (Talk | Contribs) 03:50, 4 November 2006 (UTC)

I, too, agree with Meni, but does that not mean we all disagree with the guide? Here's what the guide could say:

"Mostly, topics related to an article should be included within the text of the article as free links. The "See also" section provides an additional list of internal links as a navigational aid. In short articles, it should not repeat links already present in the article. In longer articles, it may repeat important links for cnvenience."

Perhaps, "longer" should be defined as "more than one or two screens", say. - Should this be brought up on the talk page for the guide?--Niels Ø 09:58, 4 November 2006 (UTC)

A proof of the more general concept

One interesting thing to note is that there are other ways of lexicographically representing real numbers in a fixed range (say, [0,2]) by infinite strings of digits (or bits or other alphabets), and they all have the multiple representation "problem." By lexicographically, I mean that x >= y if and only if x is after y in alphabetical (dictionary) order. In any such system, there will be no string between "0hhh..." and "1000..." where 'h' is the highest digit of the alphabet, 9 in decimal. If these two strings represent two different real numbers, their average will fail to have a represenation. Thus they must represent the same real number. An interesting example of an alternative representation is LCF ([1]), in which the representation is based on continued fractions, and the alternative representations, rather than being 0.999... and 1.000, are $0+{\frac {1}{1+0}}$ and $1+{\frac {1}{\infty }}$. I'd add this, but I'm not sure how to do so without possibly causing additional confusion. Calbaer 20:58, 5 November 2006 (UTC)

A counterproof?

Is this a valid argument?

Tan(89.999999....) = infinity

Tan(90) = undefined

Savager 16:56, 8 November 2006 (UTC)

No. --King Bee 17:04, 8 November 2006 (UTC)

To be precise, the reason it's invalid is that it assumes that, if $\lim _{x\uparrow 90}x=90$, then $\lim _{x\uparrow 90}\tan(x)=\tan(90)$. This is untrue; if something were always equal to its limit given a parameter that is equal to its limit, there would be no need for calculus! Calbaer 17:40, 8 November 2006 (UTC)

Umm, but 89.999999.... = 90 so Tan(89.999999....) = Tan(90) = ±∞. -- kenb215 talk 03:08, 10 November 2006 (UTC)

Yes, but Calbaer was (succesfully in my opinion) trying to understand the intent of the proof, which was based on the fallacy that tan(89.999...) is the limit of the sequence { tan(89.9), tan(89.99), tan(89.999), ... } rather than the tangent of the limit of the sequence { 89.9, 89.99, 89.999, ... }. -- SCZenz 03:13, 10 November 2006 (UTC)

Infinity and undefined are the same number.

What? Where did you get that idea? "a*b is undefined" means (with the standard interpretation) "the ordered pair (a, b) is not in the domain of the operation *", and "a*b = ∞" means "(a, b) is in the domain of *, and its image under * is ∞". And what is ∞, you ask? Well, taking a projective geometry approach, we could define it as $\{(x,0)|0\neq x\in \mathbb {R} \}$. -- Meni Rosenfeld (talk) 08:49, 1 January 2007 (UTC)

But we can agree that 1/0 is infinity and if you're talking about tan with a uniform circle and at 0,1 the tan is "undefined", we also know that tan is equal to opposite divided by adjacent so it's just the words are different but the result is the same (as 1/0 is infinity)

But we can agree that 1/0 is infinity — Actually, we can't. $\lim _{x\uparrow 0}1/x=-\infty$. $\lim _{x\downarrow 0}1/x=+\infty$. Therefore, $\lim _{x\rightarrow 0}1/x$ is undefined and the informal "1/0" (which would be considered using limits) is undefined, too. Calbaer 02:13, 2 January 2007 (UTC)

Student stories

Many of the examples of students misunderstanding this concept are apocryphal and not referenced. It seems that they should be rewritten or removed. —The preceding unsigned comment was added by 209.174.60.3 (talk • contribs) 02:44, 9 November 2006 (UTC)

Could you please be more specific? Melchoir 02:45, 9 November 2006 (UTC)

Some of the papers are not well-attributed. For example, a paper which I assume is "Conceptual and Formal Infinities, Educational Studies in Mathematics, 48 (2&3), 199–238" is only attributed as "Tall 2001, p. 221." The references could use more info by whomever added them on. Calbaer 03:13, 9 November 2006 (UTC)

The full citations are in the reference section, rather than in the footnotes. -- SCZenz 03:17, 9 November 2006 (UTC)

There is no Tall 2001 in the reference section. Calbaer 03:30, 9 November 2006 (UTC)

Gosh, you're right; sorry about that. Can someone add it, please? -- SCZenz 03:38, 9 November 2006 (UTC)

Whoops! Actually, those should simply read "Tall 2000"; I messed up my own records. I'll fix them now. Melchoir 04:01, 9 November 2006 (UTC)

Other number systems

I think that the introduction to the "Other number systems" section, as well as its subsections, could use some redoing. Right now, the introduction seems to imply that 0.999...=1 doesn't hold in any of the example number systems, when, in fact, it does for some of them. It would be good to say at the outset of each section whether, in fact, 0.999... = 1 in each of them and whether the number system had much useful application. Otherwise some people are going to think, "A-ha! I knew 0.999... really didn't equal 1! Real numbers aren't the way to go; I want insert non-standard number system here." Also, the section on p-adic numbers, although interesting, seems completely irrelevant. However, my knowledge of non-standard analysis is quite limited, so perhaps someone more informed can do this? In the meantime, I might do a touch-up here or there. Calbaer 22:45, 16 November 2006 (UTC)

I agree that the overall intro to 0.999...#Other number systems needs further explanation, but I don't know what could be said in the subsections. In particular, 0.999...#Infinitesimals does not cite any reliable sources that even mention 0.999…, unless you count this website, and I wouldn't. If we had to say anything explicit about 0.999… in those alternate number systems, I think the only honest message is that no mathematician has ever found it useful or necessary to even consider what 0.999… might mean in them. The entire question is undefined and maybe even logically inappropriate.

As for the p-adics, the naturality of asking what happens if the 9s are allowed to repeat in the other direction was evident to a seventh-grader, and it's been published in two journal articles; that's already enough relevance for me. It's also a natural place to bring up the whole 0.000…1 nonsense. And it reprises some of the old proofs about 0.999…, an important goal considering that you don't really understand a proof until you can adapt it to a new situation. I think it's great to be able to show readers how the proofs work in a different context. Melchoir 06:51, 19 November 2006 (UTC)

Another Proof

This proof is intuitive, but poorly worded. I would like to see some more refined version of this in the article. Does anyone agree, does anyone have criticisms?

0.999… is a real number. If you are thinking that it is not – that’s okay; but, understand that we’re referring to two different numbers. I am specifically referring to the real number 0.999…

Since 0.999… is a real number, and since 1 is a real number, then if they are different, there are an infinite number of real numbers between the two, by the properties of real numbers.

Then we will construct a real number between 0.999… and 1 by modifying one or the other by any digit in order to construct a number between the two.

Case 1: We will modify 1.000… by changing some digit to construct the number. If we add any digit to any place after the decimal, i.e. change the kth digit from 0 to some other numeral – then we have constructed a number greater than 1. If we change 1 to 0, then the number we constructed is 0, which is < 0.999… Lastly, if we change 1 to any other numeral, then the number we constructed is > 1. Hence, we cannot modify any digit that makes up 1 to construct a number between 0.999… and 1.

Case 2: We will modify 0.999… by changing some digit. Since 0.999… is a decimal followed by an infinite string of 9s, then any digit on that string that we change will construct a number not > 0.999…, since 9 is the largest numeral in base 10. Hence, there is no way to construct a number between 0.999… and 1 by changing some particular digit of 0.999…

Conclusion: Since we cannot construct any number between 0.999… and 1, and since there are an infinite number of real numbers between any two distinct real numbers, then 0.999… is not distinct from 1; i.e. 0.999… = 1.

Flame on! Tparameter 02:44, 2 December 2006 (UTC)

I see why that "proof" might be warranted what with all the people who (implicitly if not admittedly) believe that there's something holy about the decimal representation of numbers. But it's a bit awkward and not at all formal. Also, it seems like a lot of the trouble is with people who refuse to believe that there cannot be two adjacent real numbers with nothing in between or those who believe "0.999..." could not possibly represent a real number. Better we stick to the shorter, clearer proofs. Calbaer 03:22, 2 December 2006 (UTC)

I like that proof. However, a lot of people who come here with misconceptions about the real numbers might not understand why there must be any number between the two. For some reason, people seem to think of the two as beads on a string, right next to & touching each other. So going into the whole background of WHY there are always an infinite number of real numbers between any two real numbers might complicate the proof beyond what you intended. Good thinking, though. Argyrios 16:19, 1 January 2007 (UTC)

The validity of various proofs

Fraction Proof assumes that ${\frac {1}{3}}=.333\ldots$, so therefore ${\frac {3}{3}}=.999\ldots$.

My understanding tells me that this is incorrect. ${\frac {1}{3}}\neq .333\ldots$.

My understanding tells me that $.333\ldots <{\frac {1}{3}}<.333\ldots 4$.

Algebraic Proof ${\begin{aligned}c&=0.999\ldots \\10c&=9.999\ldots \\10c-c&=9.999\ldots -0.999\ldots \\9c&=9\\c&=1\end{aligned}}$

My understanding tells me that $9>9c>8.999\ldots$, or that $9c=9\times .999\ldots$.

The only way for the above statements of mine to be true is for 'Fractional Infinitesimals' to exist. Therefore ${\frac {1}{3}}=.333\ldots {\frac {1}{3}}$ and $9c=8.999\ldots ({\frac {9}{10}}\times .999\ldots )$

My understanding tells me that the following statement is true.

$\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {9}{10^{k}}}=1.\,$

However, when limits are not used, the function ${\frac {1}{10^{n}}}$ where $n=\infty$ is no longer equivalent to 0.

${\frac {1}{\infty }}\neq 0$; ${\frac {1}{\infty }}=0.000\ldots 1$, This supports the existance of asymptotes.

Thefore we can conclude that $1-{\frac {1}{\infty }}=.999\ldots$ 202.150.120.237 12:02, 2 December 2006 (UTC)

The problem is that mathematics based on your understanding (your conception of the real numbers) would not be consistent. For instance, you could easily prove that $0.999\ldots =1-{\frac {2}{\infty }}$ instead, and after that it would not require many steps to prove that 1=2. Mathematicians have struggled with such inconsistencies, and finally resolved them with the modern conception of the real numbers, in which 0.999...=1.--Niels Ø 12:52, 2 December 2006 (UTC)

Or you can just do a little math. We know that any multiplication/division you perform on the top of a fraction you do the same to the bottom. Let's say you had the .3... You could multiply by 100, which gets you 33.3..., then subtract one onehundredth of your original number from it (.3...) the result is 33. And since on the top you subtracted 1 onehundredth of the original decimal, you subtract 1 hundreth of 100 (which is one) from 100 (the denominator) and get 99. 33/99 reduces to 1/3 if you divide the top and bottom by 33. You can extend that operation to any repeating decimal to get a fraction with a terminal number as the numerator. If you do exactly the same steps with .9... it will give you 99/99 which is 1. — Preceding unsigned comment added by 66.66.92.167 (talk • contribs)

Another rant

For the past few years, I have come to Wikipedia every week for valid, interesting, complete and unbiased information. But now I see this poor excuse for an article, pushing a personal opinion as fact, and feel greatly saddened.

No, I have no degree in advanced mathematics, but I have done much reading and thinking on this subject; many people have. And all that can be said is that this is a matter of opinion. What this often comes down to on a simpler level is whether or not the infinitisimal equals zero, and on a more complex level it is often assumed that because something can be applied in one area of mathematics, it can be applied in all areas with the same result. While I admit I cannot discuss the most complicated of reasons here, I can say that that simpler 'proofs' (fraction proof, algebraic proof) are simply restating the contention that .9... = 1, not proving it. The fractional proof is horrendously flawed in this regard. In fact, the fractional proof contradicts the introduction which states that the belief that 0.999… should have a last 9' is erroneous'.

All that can be done with these sorts of topics is a discussion - no proofs can be made in some circumstances. Yes, I can tell you specifically why I personally disagree with this articles contents, but that's not the point; the point is that this always has been and always will be a matter of opinion.

This page should be completely scrapped in favour of an article on .9 recurring which discusses both points of view. In its current state, it is nothing more than an argument for a point of view.

It absolutely disgusts me that this sort of article remains on this otherwise fine site - it is no more than a form of complex trolling, labelling anyone who disagrees with it as simply ignorant and stupid: 'an inability to understand', '[a belief]...that arithmetic may be broken'. This article discourages all forms of free thinking and discussion, and promotes the acceptance of personal opinions based on the fact that some people with that opinion can use large, complex words.

Intelligence =\= knowledge.

If people absolutely insist on keeping this point of view as an article, please at least have the parts which treat people who disagree like useless idiots removed. They are childish and useless.

Noone can prove anything about infinity, because we don't have one.

P.S. Yes, this is going to make no difference to anyone, ever, I understand that not a word of the article will be changed over what I say. But I have to at least say something - after all Wikipedia has taught me, I am sickened by this kind of thing and feel I should at least try to make a difference, especially considering that I might switch to a better encyclopaedia after this. 220.152.112.132 14:16, 13 December 2006 (UTC)

Discussion regarding the article and the article itself, therefore, are kept separate from each other for a reason. It has been established that 0.999...=1 through axiomatic, limiting, etc. ways, but ironically it is the people who do not believe who are causing the problems by trolling. It is almost impossible to reason with someone who has to use expletives and flaming to attempt to make a point. Also, regard the fact that Wikipedia is not censored. x42bn6 Talk 14:38, 13 December 2006 (UTC)

There is no room for opinions in math. Dlong 15:14, 13 December 2006 (UTC)

1. First, I must say that I find myself personally offended by your comment "Yes, this is going to make no difference to anyone, ever, I understand that not a word of the article will be changed over what I say." The whole point of Wikipedia (and anything else in life, for that matter) is to change whatever is found to require change. It is insulting that you do not trust us to acknowledge this simple concept.
2. I agree, though, that there will probably not be any substantial changes to the article, for the simple reason that the article is fine as it is.
3. Did you consider the fact that if Wikipedia does such a great job on any other topic, perhaps it did a good job on this, too, and what the article says is actually true?
4. Did you consider the fact that there is no mathematician who would argue the main point of the article, and that your disagreement might be caused by lacking sufficient mathematical background?
5. If there are inappropriate sentences in the article, they might be removed, but as I see it, they serve the important purpose of emphasizing that this is not, in fact, a matter of opinion (an impression many people wrongly have, yourself included).
6. The fraction proof, etc., are in fact good, but indeed, not enough context is provided to justify them (that is, the other theorems on which they rely). This is a flaw, and I have half a mind to do something about it (not too sure what, though).
7. There is a difference between stating (and proving) mathematical theorems, and interpreting them. There is no problem with proving theorems about infinity (in fact, virtually all of mathematics deals with infinite objects). Trying to figure out what these theorems actually mean in real life is another matter entirely, but that's not what we're trying to get to here.
8. And of course, the statement that the real number represented by the decimal expansion 0.999... is the number 1 is a provable mathematical theorem. There's no matter of opinion here, and this is the point of the article. Again, this has nothing to do with any interpretation we may have for this result, which (for better or worse) is not a focus of the article.

-- Meni Rosenfeld (talk) 15:58, 13 December 2006 (UTC)

Let me give an example. In the right context, any one of the following statements can be true : 1+1 = -1, 1+1 = 0, 1+1 = 1, 1+1 = 2, 1+1 = 10, 1+1 = 11. Does this mean, then, that any article which states that 1+1 = 2 should be deleted because this is a matter of opinion? -- Meni Rosenfeld (talk) 16:04, 13 December 2006 (UTC)

Dlong, I disagree completely. There is much room for opinion and personal descision. You can do math with or without the Axiom of Choice. You can do math in a model of set theory where all subsets of the Reals are measurable. You can define compactness in terms of convergent bounded sequences or finite open subcovers. You can define the Reals as a complete ordered field, or as the equivalence classes of some Cauchy sequences. Having said that, 0.999... = 1 is one of the proven facts that you don't have any choice about. Also, incidently, mathematics is awesome *specifically for* it's ability to prove facts about things we don't have. Endomorphic 21:04, 13 December 2006 (UTC)

But none of that is opinion, but rather are either choices of which axiomatic system to do mathematics in, or which of many equivalent definitions of Reals you want to use at any given moment. A mathematician who is working in any one system is not going to say "because I'm working in this system, I don't believe 0.999... == 1 in the reals". --jpgordon^∇∆∇∆ 21:13, 13 December 2006 (UTC)

A question about the FAQs

This is about the 1-.00000...0001 well wouldn't the 1 be at the oo-1 decimal place becuase 1-.9999= .0001 and notic that they are three 0s and four 9s. This contunes with any number of 9s I would expect that it would apply to infinity number of 9s.

Ps: if you say that oo-1=oo then wouldn't -1=0 because of subtracting oo on both sides of the equation.

There is no ∞-1 decimal place, and there is no ∞th decimal place. There is only 1st, 2nd, 3rd, 4th, 5th, and so on (for every natural number $n\in \mathbb {N}$, there is an nth decimal place).

We did not (I haven't recently, anyway) said that ∞ - 1 = ∞. However, if we do wish to add some sort of infinite element to our system (in which case, it is likely that we will say that ∞ - 1 = ∞), that would require giving up some arithmetic properties, such as the ability to substract something from both sides of an equation. -- Meni Rosenfeld (talk) 23:32, 14 December 2006 (UTC)

But in agreement with the last person, we'd also have to give up some operations for the other 'special' number, 0. If you just multipled both sides of any equation (especially ones that aren't true) by 0, you'd get 0, which is why you can't do any operation and have it be true.

Rationality of 0.999.....

Would 0.999... be irrational as $9/9$ is understood as one and there is no other way to show 0.999... as a fraction ($3*1/3$ doesn't count becauseit is $9/9$)? Definition of an irrational number: any real number that cannot be put in the form $n/m$, where n and m are integers. 60.228.126.134 02:13, 20 December 2006 (UTC)GHS Nerd

No. Since 0.999... = 1, and 1 is rational, then 0.999... is rational. In fact, all recurring decimals are rational. Maelin (Talk | Contribs) 02:16, 20 December 2006 (UTC)

This is a dispute that 0.999...=1, so saying "1 is rational" doesn't really satisfy anything. And as for "all recurring decimals are rational" can you write 0.999... in the form $n/m$, where n and m are integers, without assuming 0.999...=1? I seriously doubt it! 144.131.111.196 08:57, 22 December 2006 (UTC)Anonymous

Obviously 0.999... cannot be written as n/m unless n and m are equal. If you count that as "assuming 0.999...=1", then we can't write it as a fraction of integers without "assuming" it to be equal to 1. That's why there are proofs to show that it's equal to 1 (so it's no asuumption), then rationality is a consequence. --Huon 09:35, 22 December 2006 (UTC)

Is this a dispute that 0.999... = 1? It seems more like a query whether 0.999... is rational to me. Anyway, none of the proofs rely on an unproven assumption that 0.999... is rational, so it is a valid logical argument to first prove that 0.999... = 1, and then use the rationality of 1 to deduce the rationality of 0.999... Maelin (Talk | Contribs) 15:13, 22 December 2006 (UTC)

I think that this is symptomatic of the problem with this whole article. The believers assume their own conclusion, and then prove themselves "right" through intentional misrepresentation of the questions. It is absurd to assume that someone asking if .999... is rational would simply accept .999...=1 as an answer. Algr 11:33, 29 December 2006 (UTC)

Tell me, is $\sum _{n=1}^{\infty }{\frac {1}{(n\pi )^{2}}}$ rational or not?

What does this have to do with anything, you ask? Simple, we can show that this is equal to 1/6, and 1/6 is rational. I don't think there's any easier way to show that it is rational, and there's nothing wrong with it. Similarly, we can show that 0.999... is 1, and 1 is rational, so 0.999... is rational. -- Meni Rosenfeld (talk) 16:32, 29 December 2006 (UTC)

People actually argue about this?

Why, oh why... Nice article, by the way – Gurch 22:10, 23 December 2006 (UTC)

Yeah, check out Talk:0.999.../Arguments. It's pretty absurd. Larry V (talk | contribs) 22:57, 23 December 2006 (UTC)

Indeed. Imagine how many articles could have been written if all the time spent on those pages was directed elsewhere – Gurch 23:17, 23 December 2006 (UTC)

It seems almost every online forum I've been to has had the subject crop up, wasting everyone's time. I'm not sure how much of it is ignorance, and how much is trolling, but having this article to direct them to (and the arguments page if they still won't give up..) saves a lot of effort. It's kind of like the Monty Hall problem in that common sense and truth don't completely overlap. 69.85.162.224 01:39, 24 December 2006 (UTC)

Well, at least the Monty Hall problem (which intellectually I know is proven, but emotionally seems utterly insane to me, even though it's correct) can be demonstrated and simulated and at least that generally shuts up the doubters. This one, though, has no physicality, no way to demonstrate it in other than mathematics. --jpgordon^∇∆∇∆ 02:48, 24 December 2006 (UTC)

There should be a POV tag

This article is blatantly POV, and it's of little importance that the POV may actually be correct. Stating that opponents are naive just to make a point, any point, is not an encyclopedic tool. Worse, it's not even efficient. Obviously, the trap of this situation (as of POV in general) is that it makes it harder for readers to trust what you are saying. Why not just try to simply state the facts? Luciand 12:27, 28 December 2006 (UTC)

Please point to an excerpt of the article that you claim is POV, and I'll take a look. –King Bee ^{(talk • contribs)} 14:02, 28 December 2006 (UTC)

It's the emphasys (placement in the second paragraph) on the students that's making it POV. The whole thing is seen through the eyes of an irritated teacher. (Is it only a problem with students, really?) I think the misperception part should be in the second half of the article or even in an article of its own. Luciand 22:42, 28 December 2006 (UTC)

I disagree. I think part of the beauty of the article (and partly what made it a FA) was the controversy in the classroom surrounding the equality. To leave it out would be a bad idea. To answer your question, no, it's not only a problem with students, but since Calculus teachers all over the country have to show this to their children every semester (myself included), this is where most of the data is gathered. And believe me, that paragraph sounds exceedingly calm compared to how I feel when my students exasperate me. =) –King Bee ^{(talk • contribs)} 22:49, 28 December 2006 (UTC)

I have to say that this article has devastated my faith in the "perfection" of mathematics, and as a big science and education fan, that is saying something. It seems to me so obvious that these proofs assume their own conclusion, and yet all the discussions instantly degenerate into Pro1 namecalling, appeals to authority, and argument by intentional misinterpretation of the other side. In fact, I just checked one of this article's main references, and am amazed to find Fred Ritchman swiftly demolishing the "proofs" that this article continues to display. [2] This is beyond POV. Algr 11:21, 29 December 2006 (UTC)

Let me guess... you only got access to the first page. Why not read the rest before you assume he's "demolished" anything. Of course the quick, easy proofs aren't as rigorous as the advanced ones, and so they can be demolished (sort of). -- SCZenz 11:31, 29 December 2006 (UTC)

Appeals to authority are not only legitimate means of argumentation on wikipedia; they are the only legitimate means. Just saying... Argyrios 12:56, 29 December 2006 (UTC)

The point here is that the wiki article assumes that the reader ought to find nothing wrong with those proofs. Fred Ritchman recognizes as legitimate the very objections that I was attacked for making. Essentially, the article lies to the reader, and then demands trust from those who recognize that the early proofs are invalid but don't know calculus. (And leads me to believe that calculus simply hides the same assumptions behind deeper symbolism.) "Escape to inaccessible theology" is not a tactic that people interested in the truth ought to be making. This article has EARNED the reception it has gotten. Algr 11:45, 29 December 2006 (UTC)

Yes, it has earned the reception it has gotten; it was designated a featured article. -- SCZenz 12:00, 29 December 2006 (UTC)

I believe I mentioned appeals to authority. I first learned about this article when it was on the main page. I'm looking for the original nominating discussion. Algr

I repeat: appeals to authority are not only legitimate means of argumentation on wikipedia; they are the only legitimate means. Stop badmouthing them. Argyrios 12:56, 29 December 2006 (UTC)

Since you probably don't have access to JSTOR (as SCZenz noted), you really should wait to read the whole thing. Besides, did you really read what the skeptic must assume to debunk the arguments on the first page? That subtraction of real numbers is not always possible? The skeptic assumes nonsense here. –King Bee ^{(talk • contribs)} 13:44, 29 December 2006 (UTC)

What is wrong with that assumption? Infinitesimals, being a form of infinity, don't seem to be in the set of reals. But other non-real numbers such as infinity and the imaginary unit are nevertheless important. I'd certainly like access to the rest of what Fred Ritchman has to say. I don't think I'd get very far citing a pro .999... proof that wasn't online. Algr 14:13, 29 December 2006 (UTC)

What's wrong with that assumption is that the real numbers are a field, hence closed under addition (hence subtraction). –King Bee ^{(talk • contribs)} 14:21, 29 December 2006 (UTC)

Basically, what Richman has to say is that if, for just a moment, you put aside usefulness, elegance, beauty and common sense, and focus on the main issue of making 0.999... different from 1, then you could define a clumsy, ugly, silly, meaningless and useless structure, which is similar to the real numbers, but where there are some additional elements of no understandable nature, one of which is different from 1 and represented in this system by the symbol 0.999... . I am by no means badmouthing Richman - in my view, discussing this structure is important, but only for the purpose of demonstrating that it is so useless that there is no reason to discuss it. I could just as well have defined a consistent structure where 0.999... = 3.4725. Alas, we want to deal here only with sensible structures, and the only thing decimal expansions are good for is representing real numbers, and the real number represented by 0.999... is obviously 1.

By the way, I don't see Richman mentioning infinitesimals anywhere (he doesn't say that 1-0.999... is an infinitesimal; he just says it isn't defined). Please don't confuse whimsical structures such as Richman's decimals, with meaningful structures such as the hyperreal numbers, where nonzero infinitesimals exist but in which it is senseless to discuss decimal expansions. -- Meni Rosenfeld (talk) 16:07, 29 December 2006 (UTC)

I should also mention, as I have in the past, that the basic proofs are, in fact, perfectly legitimate, only that the preliminary results that they rely on are not explicitly stated in the article. This is okay - the name of the article is 0.999..., not A complete and thorough development of all of mathematics starting from ZFC. However, due to the amount of confusion and controversy they give rise to, I have a mind to do something about them, only that I don't know what it is, and no one has yet suggested anything constructive. -- Meni Rosenfeld (talk) 16:14, 29 December 2006 (UTC)

Someone turn that ^^^ red link blue! Just kidding. =) –King Bee ^{(talk • contribs)} 20:01, 29 December 2006 (UTC)

So if you're just kidding, the page shouldn't be created with a redirect to 0.999...? AstroHurricane001 23:03, 29 December 2006 (UTC)

I don't think Fred Richman "demolishes" anything; he merely says one can construct a system whose elements are something other than real numbers which are represented by decimals in the same way, but within which the two symbols represent different things. That does not detract from the proofs that when decimals represent real numbers in the conventional way, the identity holds. And please note: his name is Richman, not Ritchman. Michael Hardy 02:16, 30 December 2006 (UTC)

"calculus teachers all over the country?" so this is about some teachers and some country (what country?) wasn't it supposed to be about mathematics? "sounds exceedengly calm?" well, why not angry it up a little bit, then? "beauty?" A POV may be indeed beautiful, but it doesn't belong in wikipedia Luciand 09:49, 1 January 2007 (UTC)

The entire "Skepticism in education" section is extremely well-referenced. First of all, it clearly talks about education, where most people are either teachers or students. That the students get it wrong more often than the teachers is likely, although shortcomings on the teachers' part are also mentioned. I suppose "the country" is the one prof. Tall writes about, whose articles are explicitly cited and to whom most of that section's content is attributed. For all these reasons, I fail to see a POV. For clarification, what would be the opposing POV? Are there any sources for it? --Huon 11:20, 1 January 2007 (UTC)

I fail to see what you're trying to prove. I was saying that the whole teacher approach doesn't belong in the second paragraph. You're saying it's well-referenced. What does oane have to do with another? Luciand 11:33, 1 January 2007 (UTC)

Why's there a POV if the "teacher approach" is in the second paragraph? You said the article deserved a POV tag, which I took to mean that it embraced some non-neutral point of view (which I fail to see, since every non-neutral-seeming statement is attributet to somebody, while there seem to be no sources for any other POV). The problems students and others may have with of 0.999...=1 are an important part of 0.999... and definitely deserve a mention in the introduction. Sure, probably it's not just students who have these problems, but that's all we know about.

What kind of approach would you prefer? Just removing any mention of the problems people have with this notion surely is inappropriate due to the prominence of these problems. And when they are mentioned at all, I don't see why the summary shouldn't contain a short note about them, just as it now does. --Huon 11:58, 1 January 2007 (UTC)

Because it's an article about "0.999..." not "teachers and their frustration about having to explain year after year to some stubborn kids that 0.999... actually equals 1". Make a separate article on that Luciand 12:37, 1 January 2007 (UTC)

I still fail to see why speaking about the educational problems surrounding 0.999... deserves a POV tag. Even if it completely missed the article's subject (which imho it doesn't), it's still written from a neutral point of viev.

Anyway, why shouldn't the special difficulties surrounding 0.999... be part of an article on 0.999...? An additional article, say, 0.999... in mathematical education, seems rather cumbersome, and we don't have all that much on the topic. Right now, the 0.999... article covers all aspects of 0.999... - the purely mathematical, the educational, and even the popular culture. That gives, to me, a well-rounded impression.

Finally, it's not at all about "teachers and their frustration". Teachers are barely mentioned at all, and frustration only in connection with students, and then it's attributed to Tall. --Huon 15:28, 1 January 2007 (UTC)

And of course, keep in mind that this is an encyclopedia, not a mathematical textbook, which is why the article should discuss issues beyond the trivial mathematical fact. -- Meni Rosenfeld (talk) 18:05, 1 January 2007 (UTC)

"The equality has long been taught in textbooks" Come on!Luciand 22:36, 1 January 2007 (UTC)

I don't understand why you think that is POV. –King Bee ^{(talk • contribs)} 22:43, 1 January 2007 (UTC)

How about saying "The equality has long been taught in textbooks and teachers have long had a hard job in convincing students that it holds true" I mean, it would be no more POV Luciand 23:24, 1 January 2007 (UTC)

Being NPOV does not mean giving equal time and coverage to every opinion. It means "representing fairly and without bias all significant views that have been published by a reliable source." There is no support in the article for the belief that 1 does not equal 0.999... in the real numbers because there is no support for that opinion in published, reliable sources.--Trystan 02:10, 2 January 2007 (UTC)

1/9

One divided by nine equlas 0.111...
Times that by nine and you get 0.999...
Same thing as the article example right? --The Dark Side 01:55, 2 January 2007 (UTC)

Yes, the fractional proof that a third times 3 equals 1. x42bn6 Talk 01:57, 2 January 2007 (UTC)

Notes into two columns

I made the notes into two columns to try and make it smaller. If anyone objects please revert, I don't want to step on feet. Thought I'd leave a note. (haha) Cheers,--Flying Canuck 06:57, 2 January 2007 (UTC)