Talk:0.999.../Archive 11

This is an archive of past discussions about 0.999.... Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

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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)

They actually know that, regardless of all so-called "proof" or argument, logic must dictate that 1 cannot be equal to any other number. (in my opinion - and you are entitled to it) ¢Joseph C 13:52, 26 January 2007 (UTC)

HA! Well, just one thing though - 1 is in fact not equal to "any other" number. The fact is that 0.999... and 1 are THE SAME number. Tparameter 15:25, 27 January 2007 (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)

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)

I don't know where to begin. undefined is something which has not been defined, and this is very context dependent. In the real numbers, the definition of division is:

For every

a,b\in \mathbb {R}

where

b\neq 0

, a/b is the unique real number c such that cb=a.

At this point, 1/0 is undefined for the simple reason that we haven't defined it (the definition above doesn't address the case of a 0 divisor). A different question is why we have chosen not to define it (remember, we have a choice what to define and how!). The reason is that any way to define it as a real number would not make any sense, as can be easily understood. Similarly, we can define the function tan, and we will choose not to define it for π/2 etc because any such definition would not make sense.

However, we need not restrict ourselves to real numbers. We can instead consider a structure such as the real projective line (see linked article). It can be constructed as equivalence classes of pairs of real numbers, but for our purpose we will assume that it consists of the real numbers and an additional element called infin;. We can then define arithmetic operations on this structure. Note in particular that ∞=-∞ and that 1/0 = ∞. It also makes sense to define tan(π/2) = ∞. However, there are still some things which cannot be defined in any sensible way, and are therefore left undefined, such as 0/0, ∞+∞ and sin(∞).

So, it is true that what is undefined in one context can be ∞ in another context. But

{\sqrt {-1}}

is undefined in the context of real numbers, and equal to i in the context of complex numbers. That doesn't mean that "undefined" and i are the same. -- Meni Rosenfeld (talk) 12:23, 2 January 2007 (UTC)

Well I never said any context can mean any other context. Sure, I left out that when you have a vertical line if you go from negative infinity to positive infinity you have a positive infinite slope, and when you go from positive infinity to negative infinity, you get a negative infinite slope, as both are in a superposition, but I said the math made 1/0 (but not -1/0, I apologize if you're a stickler) equal undefined. But, now that I think of it, if you're really dicking on the definition, sure, I'll give in and say undefined means both a positive and negative infinite slope. Undefined is the definition of a slope where something is divided by zero. Infinity is a number going on forever, which can be defined by any number except 0 being divided by 0. Undefined and infinity are not separate "contexts", as the both apply in the same situations, to the same entities, in the same way. For the last part, i is not undefined. It is defined as the square root of -1. Also, the square root of -1 and real numbers are only separated by your definition you apply to them, as they work perfectly together in equations. They are not separate contexts. It'd be like saying fractions and decimals are separate contexts and that you can define something in one to mean anything in the other. On the other side, I could see if we were talking about maybe adding matrices and real numbers or something. (and don't start trolling if you want to say that you know how, it doesn't matter here (this applies to everyone))

I have read your response. -- Meni Rosenfeld (talk) 16:24, 3 January 2007 (UTC)

Things don't "equal" undefined. That doesn't make any sense whatsoever. –King Bee ^{(talk • contribs)} 17:27, 3 January 2007 (UTC)

How about you find a situation where I'm wrong? Any situation that has undefined is have a vertical line. It has an infinite slope, but you could see it as "going" from negative infinity "to" positive infinity, which gives a positive infinite slope. You can also see it the opposite way with positive infinity "going to" negative infinity, which would give a negative infinite slope. I await your response, King Bee.

Most of what you've written is meaningless, there's nothing to respond to. "Situations having vertical lines?" Nonsense. –King Bee ^{(talk • contribs)} 13:54, 9 January 2007 (UTC)

But anyway, the function

\log(x)

over the real numbers is undefined at the value -17. There is no "vertical tangent line" (which is probably what you mean). The function just isn't defined there. –King Bee ^{(talk • contribs)} 13:56, 9 January 2007 (UTC)

Also: For the last part, i is not undefined. It is defined as the square root of -1. Also, the square root of -1 and real numbers are only separated by your definition you apply to them, as they work perfectly together in equations. They are not separate contexts Actually, that's a good example of the misunderstanding here. The square root of -1 is quite certainly undefined in the reals. That's why the complex numbers were invented: because in the reals, there's no way to multiply a number by itself that results in a negative number. Likewise, It'd be like saying fractions and decimals are separate contexts -- well, they are, if by "fractions and decimals" is meant "rational and reals". The sets work differently; one is countable, the other is not (for example). --jpgordon^∇∆∇∆ 14:37, 9 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)

As reflected on the current arguments page, the p-adic numbers seem to confuse more than help. People who don't understand 0.999...=1 will be confused by it. People who do understand 0.999...=1 will be confused by it. It doesn't address 0.999...=1. It addresses a similar phenomenon that almost no one thinks about. If WP:RS were the only measure of whether something should or should not be in the article, this article could contain the contents of books on the history of numbers, on real analysis, on mathematical education, etc. But it shouldn't. It should be readable, and anything that hurts more than helps should be excised. Does anyone think that this section helps more than hurts? Calbaer 19:22, 25 February 2007 (UTC)

If that arguments page ever has such a large-scale effect on the content of this article, I really will push to destroy it. Its lifeblood participants are self-selected to be prone to misunderstanding, and it can only mislead us about how the article is read by our audience as a whole.
0.999...=1 isn't the full scope of the article.
The sections "Generalizations" and "Applications" contain plenty of similar phenomena that almost no one thinks about. Let's face it, this is an obscure topic to begin with. That doesn't mean that its connections can't be relevant and interesting. And when I read an FA-quality encyclopedia article, I expect to be told about all the similar phenomena that the average website or textbook left out.
…999 does not merely appear somewhere in the same book as 0.999…; it is directly relevant to whether or not some common manipulations on 0.999… can be trusted. Now, Fjelstad's journal article is 4.5 pages long, but its contents are summarized into a single paragraph. DeSua's article is 3 pages long, but here we give it 2 sentences. The so-abbreviated material is buried as the last of three subsections of the sixth section of the article; among even more esoteric and difficult material, some of which has truly tenuous connections to the article; and where few readers are likely to reach it at all. The only way we could possibly bury it any further would be to split off summary-style sub-articles and banish it from the parent article, and I'm personally supportive of even that option. So please don't make it sound like the merest presence of the 10-adics somehow signals a takeover of the article. There are shades of gray.
Yes, I think the section makes the article more interesting and therefore helps to interest the readers – not only in the main content, to which it links back repeatedly, but in other topics on Wikipedia they may never have heard of before. Prepared readers will even learn from it. Unprepared readers…? Sorry, I do not believe that one is harmed by being passively exposed to something one does not understand. The only possible harm is if they become vocal about their excitement and lack of understanding, and annoy the people around them. To which I say, again, don't take the arguments page so seriously. Melchoir 20:49, 25 February 2007 (UTC)

Points taken. I would not want to take out a useful part of the article just because someone on the arguments page had a problem with it. I just think it's more confusing than helpful, not just to skeptics, but to me. This might be solved by having the headers and introductions providing more context. For example, since the article is "0.999..." and the section is "other number systems," it might be assumed that the section addresses 0.999... in other number systems. Perhaps "analogous phenomena in other number systems" might better express what the section is going for. And perhaps each section should make it explicit that 0.999... is 1 in those number systems. e.g., "Another extension of the real number system, and thus another system in which 0.999... = 1, is the p-adic numbers. When asked about 0.999…, novices often believe there should be a 'final 9.' However, there is no 'last 9' in 0.999….[40] For an infinite string of 9s including a last 9, one must look elsewhere...." Calbaer 21:17, 25 February 2007 (UTC)

How about just "Analogues in other number systems"? It still wouldn't be a perfect fit, since the series implied by "0.999…" doesn't converge in the 10-adic numbers. As for the introductions providing more context, yeah, that's still a problem that no one has fixed since you brought it up at the start of this thread. I'm tired of editing this article; why don't you give it a shot? Melchoir 23:20, 25 February 2007 (UTC)

This whole section sounds strange to me. It talks about "behavior" and "phenomena", when I think you guys might mean "algebraic structures" and "properties". Anyway, the section sounds very experimental, and maybe it should be worked out in someone's sandbox before it's presented in an article. As a matter of fact, maybe it should be eliminated altogether. After all, probably any number could be fit into some different algebraic structure for this or that application. So what? Is this section just to make the skeptics happy? Tparameter 13:31, 26 February 2007 (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. — [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)

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.
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.
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?
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?
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).
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).
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.
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)

What has that got to do with anything!? Luciand 08:42, 2 January 2007 (UTC)

What it has to do with anything is that things on Wikipedia need to be verifiable. Since there is no support for the opinion that 1 != 0.999... in the published literature (concerning the real number system, mind you), there's no way we can put it in. This is especially true of mathematical articles, where the sources need to be published articles or books. Mathematicians rely on proofs, and if you make a claim, you better have a proof for it, or be able to point to an article (or book) that does. –King Bee ^{(talk • contribs)} 16:14, 2 January 2007 (UTC)

No one said anything about puting any opinion in. So what has it got to do with anything!?!? Luciand 04:52, 3 January 2007 (UTC)

What exactly is the POV that you think the article is slanted towards? It seems nobody else can see what your complaint is. Please describe the POV itself, then give an example of this slant, and a description of how the example is slanted towards the POV. Maelin (Talk | Contribs) 05:48, 3 January 2007 (UTC)

It's easy to see when you start looking from outside the fishtank. This should not be about how *teachers* in the *USA* have a hard time explaining *math students* that 0.999... ...you know. If reference "Tall" studied this particular phenomenon it doesn't make it the quintessential thing about 0.999... It's no more than trivia. When you make trivia into a main discution point, that is bias, hence POV. It's like stating, in the second paragraph of an encyclopedia article about the ocean, that it "is the source of water for the fishtank, 'cos thus said mr. platypus^[1]" Luciand 07:50, 3 January 2007 (UTC)

Please stop this nonsense, remove that disputed tag, and start being constructive. There's nothing interesting about 0.999... as such - after all, it's just 1. Well, there are lots of interesting things to say about 1, but see 1 (number) about that. What is interesting about 0.999... (as a representative of the infinitely many decimal numerals ending in 999...) is the confusion it causes. So we do not need an article about 0.999... as such, but we need one about this confusion. There are two sides to that: Explaining why 0.999... is in fact equal to 1 (in the only number system we all use), and explaining how people get it wrong. If anyone can think of a better title for this article about the confusion, perhaps we should rename this article (again), but removing everything from the article that makes it interesting (in a general encyclopedia, that is, not in a math textbook) is not constructive. Suggesting a better title would be.--Niels Ø (noe) 10:39, 3 January 2007 (UTC)

Well, it's not a newspaper neither :) I agree that the dispute is the most interesting thing about the subject. Still, it does by no means define it. That's why it doesn't belong in the beginning of an article named 0.999... (The renaming suggestion may be a solution). And it should be reworded so as not to sound like a teacher's resignation note. "confusion it causes" - that's nicely worded and neutral. it should belong in the article Luciand 11:36, 3 January 2007 (UTC)

Please do not re-add the POV tag. It is a violation of WP:Point; specifically, read this. Argyrios 12:19, 3 January 2007 (UTC)

I find that rather insulting. You should have read this Luciand 13:49, 3 January 2007 (UTC)

Nobody else can understand the problem. Thus it is appropriate for you to come here and see if anybody else agrees that NPOV is being violated in the article. It is not appropriate for you to put a POV tag on an article after it is clear that no other editors agree with your POV claims, that is where you violate WP:Point. Maelin (Talk | Contribs) 14:06, 3 January 2007 (UTC)

So what should the title be? "Confusion over 0.999..."? "Equality of 0.999... and 1"? Or - my favorite - "0.999..." (i.e. no change)? I'm sure lots of articles do a lot more than defining the term in their title; e.g. they also give the history of the term and/or of the thing it defines, state its significance, present significant opposing views, etc. Are the opposing views here insignificant just because they are incorrect? I think not: As we agree, they are what makes it interesting in the first place. So why not stick to the simple title? If we called it something else, "0.999..." should redirect there anyway, unless someone wrote a separate "0.999..." article - probably a rather uninteresting one, the most important and useful thing in it would be the link to the present (but renamed) article! If anything, one should move the non-trivial proofs to a new article, "Proofs that 0.999... equal 1" (leaving just a "main article"-style link behind), and focus this article on what is truly relevant in our general encyclopedia.--Niels Ø (noe) 13:16, 3 January 2007 (UTC)

Than why did you get into this dillema in the first place? Luciand 13:49, 3 January 2007 (UTC)

You mean why I entered the discussion here? Because I hoped I could make you remove that POV tag yourself. About a name change: I wouldn't mind a reasonable name change, but I do not actually want it. What I do mind is having POV tags where they do not belong. I generally try (occasionaly with success) to push things towards lasting solutions that nearly everyone can accept, in order to stop perpetual edit wars. The worst edit wars are not those between editors A and B, but those involving numerous editors over time. It's not particularly useful to tell an editor who disrupts an otherwise reasonable compromise to read the talk page first, when the talk page is long and confusing. Instead, I hope the article can reach a form where people can leave it alone unless they have something valuable to add.--Niels Ø (noe) 14:30, 3 January 2007 (UTC)

I fail completely to see what this has to do with POV. Luciand seems think that because:

1) the discussion on students rejecting the equality is merely "trivia" (false - it is important enough to merit attention in the article)
2) it is given too much attention in the article (despite the fact that the vast majority of the article is about numbers rather than skepticism in education)

Am I correct in stating that, because of 1 and 2, you think the article is POV? Even if the issue was trivia (something it is obviously not), WP:TRIV says to integrate trivia into the content of the article. WP:NPOV says absolutely nothing about it, except for perhaps undue weight, which the article complies fully with. The statement that 0.999... = 1 isn't even a POV but rather a fact, and since it is well substantiated, it gets the most weight (actually all, since there are no verifiable sources supporting otherwise). - KingRaptor 14:22, 3 January 2007 (UTC)

No, you are not correct. I said the (begining of the) article is POV because one can easily realise that the author is an irritated math teacher who thinks that 0.999... is only about what they do in classroom. (see "fishtank" above). However, it is the least of my intentions to start a revert war. Do however you think it's best for this article. Luciand 15:30, 3 January 2007 (UTC)

While 0.999... is a subject on which I am sure many students attempt to instruct their teachers, the students have failed to publish as of yet. As a published, significant view, Tall's study is entitled to be reflected in a NPOV article. Criticizing formal studies done by math teachers in a mathematics article as POV is like criticizing an article on dolphin anatomy for only reflecting the views of marine biologists.

As for the name, I would suggest either Proofs that 0.999... = 1 and reasons why people think it does not or simply 0.999....--Trystan 16:18, 3 January 2007 (UTC)

A few points:

An article should not be marked as POV simply due to someone not liking the title.
A lot of thought and discussion has gone into determining the title — the title under which it was featured — so any title change should be a matter of consensus.
If the only problem is one of tone, anyone can change the text to what they think it should be. The worst that can happen is that it gets reverted. And it takes a lot less time than participating in the above conversation or constantly reverting.
Once again, reliable sources are the golden standard. If someone has one that disputes or contradicts Tall, please share it. Otherwise, arguments that it's just one point of view are merely fatuous. Calbaer 18:08, 3 January 2007 (UTC)

If some marine biologist wrote a study about how students think that the dolphin is a fish not a mammal, that's science. If some frustrated teachers used that study to make the second paragraph of the article on dolphins, it would be POV. Check again on who exactly didn't like the title. Everybody seems to argue about anything else, while ignoring the only point i stated. It's true that time is an issue for me and everybody else as well (still, i won't edit the text as long as i have not managed to raise a doubt, as it would be futile), but it seems that this is a sect here, so there can be no point in hanging around. Goodbye and have funLuciand 20:28, 3 January 2007 (UTC)

Goodbye. Argyrios 14:00, 4 January 2007 (UTC)

If the only interesting thing about dolphins was that many people insisted they were fish, then it would be worthy of note in the second paragraph. While Dolphin has much fascinating information which completely overshadows any fishy misconceptions, there is really nothing interesting about 0.999... other than the common initial rejection of its equality with 1.

If we restored the old Proof that 0.999... equals 1, and added Skepticism that 0.999... equals 1, what would that leave for 0.999...? The first two are the only two interesting areas of discussion for the third. I don't see the benefit in splitting the two interwoven areas of discussion into separate articles.--Trystan 22:47, 3 January 2007 (UTC)

Nuts, looks like I'm late to the party! I have just one point to add-- it's a little backwards to argue about the content of the lead section, since it's supposed to summarize the body of the article. It would be more productive to argue about the content of the skepticism section and/or whether or not it is accurately summarized in the lead. Melchoir 23:22, 3 January 2007 (UTC)

I'm sorry, i tried to stop posting, but this is , in a bizzare way, too amusingly ridiculous. 1. Indeed, i think the lead is POV 2. After making a poor argument, now you're showing that the whole example is inappropriate. Than why did you bring it up in the first place!?!? Is that a hobby for you people? 3. Do you have any reference who sais that the only interesting thing about 0.999... is the common initial rejection or is that your point of view? Luciand 23:47, 4 January 2007 (UTC)

P.S. If you want to keep bringing me back, just throw some nonsense. I can't resist to that, otherwise, I'd really like to go :)

The notability of 0.999... = 1 being rejected by some is easily proven by the debates all over the Internet about it (this page included).
In any case, splitting the article in any way will only condemn one of the articles to perpetual stubdom, which would force us to merge it back in, and we're back to square one. Since only one section of the article is about the skepticism, the rest of the article being about proofs, axioms, and the like, you'd have to put all the proofs/axioms/etc. in one article (since it makes absolutely no sense to spread it out over both articles). This will result in one article being almost as long as the current one is, and the other being absurdly short. The shorter article, since it is related to the bigger article, and due to its inability to be expanded in any way, is required by Wikipedia policy to be merged or deleted.
Moreover, nothing about the second paragraph in the opening statement jumps out as being written by "annoyed teachers". The research is documented, the issue is notable, the statement is neutral (sure, it only talks about students, but students are the ones who are commonly given the equation, and the ones whose reactions have been studied); thus, it is NPOV.

Or, in short, everything Trystan said.- KingRaptor 07:15, 5 January 2007 (UTC)

Just adding the PoV tag would vastly reduce the amount of arguing that goes on here. If there was an absolute truth to this matter that automatically disproved all alternatives, then there wouldn't be any argument at all. An unfortunate truth in this world is that Wikipedia is used by all kinds of perfectly ordinary people; not just mathematicians... and exactly the same is true of numbers as we know it. The number of people who fiddle about with recurring decimals isn't limited to mathematicians either (which means this axiom of the set of "real numbers" is not automatically assumed true by everyone).... but yet the entire article as it stands is represented as rather irate mathematical propaganda. It -could- be represented as the conclusions of the mathematical community without trying to push those conclusions down everyone's throat... but apparently nobody wants it to. Rather than basically stating: "this is the way we found it", it comes across more as "this is the way it is and you're an idiot if you disagree". The article is deliberately antagonistic. Something needs to be done about that. ~ SotiCoto 195.33.121.133 15:14, 30 January 2007 (UTC)

But decimal expansions are only meaningful as a convenient way to represent real numbers. Anyone who thinks otherwise is indeed an "idiot", not in the sense of actually being an idiot, but in the sense of not knowing what he is talking about (due to lack of mathematical knowledge). As an analogy, there are many people who make decisions regarding their health every day, but just because they think something is good for them doesn't usually mean that it actually is. Physicians, on the other hand, do usually know what they are talking about as far as medical issues are concerned.

And for the nth time, the article does mention explicitly that it discusses real numbers and not anything else. -- Meni Rosenfeld (talk) 15:12, 31 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)

That's correct, except I'd say "multiply that by 9" instead of "times that by 9". Using "times" as a verb in that way is what teachers do when talking to small children that the teachers think cannot pronounce "big" words like "multiply". Michael Hardy 00:48, 4 January 2007 (UTC)

Only if for whatever reason you believe that such fractions can actually be expressed in decimal terms, rather than more securely assuming that 1/9 approximates as near as damn it to 0.1111... (recurring). Of course I naturally believe that all these attempts to "prove" the matter are based on circular reasoning. ~ SotiCoto (29/01/2007)

1/9 can be proven to be equal to 0.111... via the infinite sum of the series 0.1 + 0.01 + 0.001 + ... Dlong 16:26, 29 January 2007 (UTC)

If you believe that the attempts to prove the matter are based on circular reasoning, then you do not know what the reasoning is. The concept of decimal expansions can be well-defined and their properties can be proven rigorously. I could explain it to you if you want.

For the time being, I'll describe it intuitively: A decimal representation for a real number is a recipe to obtain finite decimal expansions arbitrarily close to it. When we say that 1/9 is represented by 0.111..., this can be understood as saying that not matter how close you want some finite expansion to be close to 1/9, you will find one of the form 0.111...1 with some finite number of 1's. You should agree with this observation. As this is the meaning of "1/9 is represented by 0.111...", you should agree with this latter statement.

Of course, you can prove many things about this - for example, every nonnegative real number has such a "recipe", and two different numbers cannot have the same recipe (but one number can have two recipes). Now, just as when I say "3" I actually mean "the real number represented by the numeral 3", when I say "0.111..." I actually mean "the real number represented by the decimal expansion 0.111...". As this number happens to be 1/9, I can safely say that "0.111... = 1/9". -- Meni Rosenfeld (talk) 14:51, 31 January 2007 (UTC)

-- Meni Rosenfeld (talk) 14:51, 31 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)

It doesn't show up as two columns in Safari. Anyone else experiencing this issue? –King Bee ^{(talk • contribs)} 16:15, 2 January 2007 (UTC)

Works in Firefox 1.5, but not Internet Explorer 7. Supadawg (talk • contribs) 19:37, 2 January 2007 (UTC)

It also works in Firefox 3.0a2pre (Minefield) but not in Opera 9.10 (though that doesn't even display the images in the article - maybe I need to alter some settings). Raoul 20:18, 2 January 2007 (UTC)

I think a simpler proof

2 - .999..... = 1.000...... because the .999... goes on forever you can never get 1.000......1 idk if this continues a rational proof I am just saying. Sorry for grammar. Or lack of math vocabulary. 72.178.44.126 21:06, 3 January 2007 (UTC)Barry White 21:08, 3 January 2007 (UTC)

The problem I would see is that the familiar algorithm we were all taught in grade school for subtraction begins with the rightmost digits, which obviously is not possible with 0.999....--Trystan 22:36, 3 January 2007 (UTC)

but it continues forever so you cant stop at a certain place to subtract so it would equal 1.000... which = 1

You are simply assuming your conclusion, just like all the other proofs here. 2 - .999..... = 1+ an infinitesimal. The irony is that while none of the proofs here prove anything to my satisfaction, I know of one approach that optionally does seem to come out with .999=1. However it does so in a way that I think the others here will not like. Algr 07:04, 4 January 2007 (UTC)

None of the proofs in the article assume their conclusion. You may have come to the conclusion that they do because you do not recognise the results on which they are built, but the proofs are all built on solid mathematical results that are derived inevitably from the axiomatic construction of the reals. If you really feel that a proof is assuming its own conclusion, please describe how you think it is doing so on the Arguments page. Maelin (Talk | Contribs) 07:49, 4 January 2007 (UTC)

 I corrected some spelling

Is 0.999... even a number?

In my analysis book there's an explanation of a mathematical algorithm which produces a decimal representation of any real number, starting from the axioms. And in the end it has a note to the effect of "it is impossible to produce a decimal representation ending in an infinite number of 9s". Which means that 0.999..., according to the algorithm, doesn't correspond to any real number. Which is, of course, because the real number which would have corresponded to 0.999... will actually be converted to 1.0000..

Does this have a point? I dunno, but I'd like to see it at least briefly mentioned in this article, I just don't know where and how. So I'm posting this on the talk page instead. Cheers, Arag0rn 02:22, 4 January 2007 (UTC)

Firstly, it helps to tell us what book. Secondly, the book is not saying that the number 0.999... doesn't correspond to a real number. It is saying that the output 0.999... will never be produced by the algorithm, because it will instead produce the more common equivalent representation, 1.000.... As many have noted, alternative representations include 1/1 or i⁴, and the algorithm will not produce these as outputs either. Put another way, algorithms do not determine number properties; math does. This article does not discuss algorithms, but rather mathematics. And it is made clear that "1" is more common than "0.999..." for day-to-day use. Calbaer 03:05, 4 January 2007 (UTC)

Wikipedia does not report original research, and since I have no source for the following, it does not belong in the article. Anyway, I think the World would have been a slightly simple place if mathematicians had created a one-to-one correspondence between reals and decimals (ignoring the possibility of leading or trailing zeros, inclusion/omission of decimal point in integers, and inclusion/omission of plus sign before positive numbers, and the use of dots to represent infinite digit strings) by banning decimals ending in an infinite string of nines. However, that's not how they've done it. Of course, it is merely a matter of habit, convention, tradition, definition. Calbaer is probably rigth: Your book points out that that particular algorithm does not produce infinite string of nines; it would be unconventional if it states numbers cannot be written that way.--Niels Ø (noe) 06:07, 4 January 2007 (UTC)

The book is in Serbian and it's simply called Analysis I. I doubt it's going to help, but since you asked for it... :-p

I think the World would have been a slightly simple place if mathematicians had created a one-to-one correspondence between reals and decimals ... But this is exactly what the algorithm does! It creates a bijection between the set of real numbers and the set of decimal representations not ending in an infinite number of nines. And the other-way-around proof, that every decimal string has its real number, hangs on the assumption "for any S, we can find T>S such that the Tth number in the string is smaller than 9". So I naturally asked myself "where do all these people get the idea that 0.999... represents anything?

...of course, not long after posting that here, I found a way of modifying the proof so that strings which end in all 9s are possible, making my point moot. However, that makes the "usual" all-zero-ending strings impossible, and I still haven't found a way to make them both work at the same time :( Nevertheless, it's been fun. Thanks. --Arag0rn 19:36, 4 January 2007 (UTC)

Even a Serbian may need more to go on than Analysis I (i.e., author, ISBN, etc.). Anyway, that's probably not necessary at this point. And you're contradicting yourself. You claim that the correspondence is one-to-one between reals and decimal representations, then go on to say that it's one-to-one between reals and decimal representations not ending in an infinite number of nines. Clearly these are two different things.

The way repeated zeros and repeated nines "work at the same time" is similar to the way "1/2" and "5/10" work at the same time; they're just two representations representing the same thing. In the case of the algorithm you mention, as well as by convention, one does not generally use the 0.999... form. That does not mean it's meaningless, however, just that it's not, in some sense, "canonical." If I asked you to represent fractions in irreducible form and then said, "What about 0.5?" would you say, "That's meaningless," or "That's 1/2"? Hopefully you'd say the latter.

Perhaps it would be useful to insert a sentence to the effect, "In practice, 1 is preferred to 0.999... in common representations." However, this might be a tad too obvious to merit inclusion in the article. Thoughts? Calbaer 20:26, 4 January 2007 (UTC)

Well, if you can think of a natural place to insert it, you may as well give it a try. Melchoir 04:26, 5 January 2007 (UTC)

(By the way, most of the world might have been made simpler if reals and decimals were one-to-one, but the ternary characterization of the Cantor set would be less simple.) Melchoir 04:26, 5 January 2007 (UTC)

well...part of me wants to insist that "no, 0.9999... is not a number. Neither is 1. Both are representations of particular real value that we assign the symbol '1' to by convention." --jpgordon^∇∆∇∆ 00:00, 5 January 2007 (UTC)

That's actually a pretty important point. Perhaps we should mention that distinction in the article, and stress that neither 1 nor 0.999... is an actual number, they are merely different representations of a number. This might ease people's confusion. Maelin (Talk | Contribs) 02:26, 5 January 2007 (UTC)

Maybe. I've found it hard to get people to understand the difference between numerals and numbers in my day. If someone can coherently add it to the article, I'd be all about it. –King Bee ^{(talk • contribs)} 03:27, 5 January 2007 (UTC)

While I agree that "1" isn't a number, 1 is a number. Melchoir 04:23, 5 January 2007 (UTC)

You think? Or is just a name for a number? --jpgordon^∇∆∇∆ 07:11, 5 January 2007 (UTC)

Is France just a name for a country? Melchoir 08:31, 5 January 2007 (UTC)

Is the present King of France just a name for a person? ^_^ Argyrios 14:50, 5 January 2007 (UTC)

"The map is not the territory." --jpgordon^∇∆∇∆ 15:02, 5 January 2007 (UTC)

I Have a New Proof for why 0.999...=1

Can we add the proof in the article that 0.999...^2=0.999... and in our number system there are only two values that when you square them they equal themselves so 0.999... must be equal to 1.

Sorry, but I doubt this proof is a useful expansion of the article. There are a few problems:

You would have to know how to multiply decimal representations, and even if you know the principle, it's quite a task to show that the result really is 0.999...
Those who believe that there's a tiny amount missing to 1 won't be convinced anyway, since to them for 0.999...^2, there's a larger, but still infinitesimal amount missing to 1, but 0.999...^2 isn't 0.999...

The "simple" proofs we give rely on long division and multiplication by 10, both of which seem easier than multiplication of decimal representations. --Huon 20:43, 9 January 2007 (UTC)

1.999...

Does 1.999... equal 2? I would think that it would but I'm just making sure.-- Jacroe | Talk 06:41, 15 January 2007 (UTC)

Yep. Given that

0.999...=1

, we can then see that

1.999...=1+0.999...=1+1=2

. Maelin (Talk | Contribs) 07:10, 15 January 2007 (UTC)

This article could be better

I made a few edits. I'll see the reception before making more. Here are my thoughts: It is not stupid to be uncomfortable with the equation. Many justifications students give are misguided. They do need to get past them. If they stop worrying, great! If they do retain some discomfort they might make good mathematicians. At some level the equation is true because that is the way that we do real numbers. There are very good reasons for doing it the way we do. I'll expand on this but maybe elsewhere if there is too much furor here. Here is a quote from The Mathematical Gazette, Vol. 64, No. 429. (Oct., 1980), pp. 149-158 (an address by the president of the British Mathematical Association: Zeno, Aristotle, Weyl and Shuard: Two-and-a-Half Millenia of Worries over Number):

...if the construction of the real numbers troubled such an acute intellect as Weyl's as recently as 1917 and still worried Godel in 1940, it is not to be wondered that some of our first year undergraduates find it hard to stomach. Perhaps they are wiser than we are.
I think there is a moral in this for all our teaching, and not only in analysis for first-year undergraduates. It is this: if, year after year they seem to find particular difficulty with something, it may be because the difficulty is reallythere. Our modern axiomatic method is very powerful; but it has a power which can sometimes be used to sweep difficulties under the carpet. And a study of history can help by indicating when the danger of this is present.

Gentlemath 11:43, 19 January 2007 (UTC)

I don't think your mathematical attitude is at all controversial, but I have to question the factuality of one of your points. Everything that was previously said in the lead section was a reflection of points made in the body of the article, which are in turn cited to published sources. So, what evidence do you have for the new claim that students make misguided arguments for the equality? Melchoir 18:56, 19 January 2007 (UTC)

I did say in the one line description that it was in line with Tall (1978). That is the reference D.O. Tall and R.L.E. Schwarzenberger (1978). "Conflicts in the Learning of Real Numbers and Limits". It starts with responses by first year math students (were they math majors I wonder? I wouldn't be suprised.) when asked about the equality. Of the 5 given, three are from students who say that yes they are equal:

"The same, because the difference between them is infinitely small."

" The same, for at infinity it comes so close to one it can be considered the same."

"Just less than one, but it is the nearest you can get to one without actually saying it is one."

"Just less than one, but the difference between it and one is infinitely small."

"I think that 0.999...=1 because we could say ' 0.999... reaches 1 at infinity – although infinity doesn’t actually exist, we use this way of thinking in calculus, limits, etc."

Look at student one and student four. They could be the same person. To student four we say "NO! You don't get it! Look at this arithmetic again. etc. etc. eventually he turns into student one (which is a relief) and then we say "well, not exactly but carry on." I wanted to quote a passage later on in the paper.

Perhaps the students were a little confused when they first saw the proof, but the passage of time, and the knowledge that the mathematical world accepted the proof, allayed their fears. To misquote a certain proverb, “Familiarity breeds content.”

I didn't beacause it is about the irrationality of sqrt(2) (which he says troubles some 3rd year honors math majors). I figured I'd get called on it but I am sure it applies to our equation. Gentlemath 02:53, 20 January 2007 (UTC)

Oh! Hmm. I guess the first time I read all that — it seems like ages ago — I must not have thought that it was necessary to make the point in the article, and eventually I forgot it was there! Well, please do edit the body of the article to include this information.

As for the familiarity quote, I for one wouldn't have a problem with it if it were included in a way that avoids misrepresenting its context: one could write something like "On the comparable subject of contradiction proofs and sqrt 2 in particular, Tall writes, "…"". But then you have to wonder just how comparable it really is… Perhaps it would be better to include his analysis in Square root of 2 and/or Reductio ad absurdum and link to those articles from somewhere here? Melchoir 04:24, 20 January 2007 (UTC)

Algebraic proof

 	        Here is an algebraic proof involving decimals. If you take any
  	        two equal numbers and do the same equations to each of them, then
 	 	you will come out with the same number as your answer.
 		For instance:

               X = Y
 	 	
 	 	( X - .9 ) * 10 = Z
 	 	
 	 	( Y - .9 ) * 10 = Z
 	 	
 	 	IF
 	 	
 	 	X = .9999...
 	
               Y = 1
 		
	         Then
 		
 		(.9999... - .9) * 10 = .9999...
 	
	        (1 - .9 ) * 10 = 1
 		Therefore, 1 is equal to .9999...!

Can someone explain to me how this proof does not work? I tried adding it to wikipedia but the editors keep taking it off. — [Unsigned comment added by 71.113.137.69 (talk • contribs).]

It's fundametally identical to the proof already in the article, just with more problems. What you're really doing with (.9999... - .9) * 10 is 9.999...-9, which is exactly like the proof in the article. —Mets501 (talk) 04:12, 23 January 2007 (UTC)

He subtracts .999.... but I olny subtract .9 Also after this he still continues to go on I don't.

He subtracts .999... from 9.999... to get 9, and you subtract 9 from 9.999... to get .999... It's the same thing. —Mets501 (talk) 04:22, 23 January 2007 (UTC)

I see what you are saying. I guess they are pretty similar. But maybe having both (not as different proofs, but as two ideas to help the reader) can help more people understand them.

Do you agree? — [Unsigned comment added by 71.113.137.69 (talk • contribs).]

I don't see how this "proof" proves anything. Defining a function f(x)=(x-0.9)*10, you first say (in function notation) that if X=Y, then f(X)=f(Y). Then you observe that f(0.999...)=0.999... and that f(1)=1. Where's the proof?

Let me prove to you that 1=2: Define g(x)=x^2-2x+2. Now we can compute g(1)=1-2+2=1 and g(2)=4-4+2=2. Wouldn't your logic now allow me to conclude 1=2?

Really, the only things we have proved are that 0.999... and 1 are both so-called fixed points for your function f, and that 1 and 2 are fixed points for my g.--Niels Ø (noe) 07:37, 23 January 2007 (UTC)

That is not how I had to put it. I could say if X=Y then with the same equation you will come put with Z. So if .999... = 1 then the answer would b the same. I tested it and it worked. also, your proof of 1=2 you have changed the equation. You need to use the number in the equation. If I start every equation with 1-a+a then my answer will always be 1. Do you see what I'm saying? — [Unsigned comment added by 71.113.137.69 (talk • contribs).]

It's not a valid proof. Your argument does not prove its conclusion. You have only shown that 0.999... and 1 are fixed points of the function

\textstyle f(x)=10(x-0.9)

. You haven't shown that they are equal. Maelin (Talk | Contribs) 13:03, 23 January 2007 (UTC)

My reasoning is, however, that the two numbers have the same change when multiplied and subtracted from, which only happens in like numbers. — [Unsigned comment added by 71.113.137.69 (talk • contribs).]

No it doesn't. I take two numbers, 3 and 5. I multiply them both by 10, subtract 20, divide them by 2, multiply them by 1/5, add 2 and I'm left with what I started with. They are fixed points of myfunction, but this doesn't make them the same number. Be more precise in your explanations, then we will be able to help you understand why your proof is invalid more easily. Maelin (Talk | Contribs) 01:16, 24 January 2007 (UTC)

But you multiplied by 10 then divided by ten. I 'm not dividing. it's different. — [Unsigned comment added by 71.113.137.69 (talk • contribs).]

Please sign your comments using four tildes. The reason my example is "different" to yours is that you did not explain precisely what you meant. Regardless, your proof is not valid because you haven't proven your assumption that "all fixed points of

f(x)=10(x-0.9)

are equal." Maelin (Talk | Contribs) 01:46, 24 January 2007 (UTC)

If I were to change it so the answers were not the same would it work? 71.113.137.69 04:05, 24 January 2007 (UTC)

Not quite. What you really need to do is prove that there is only one fixed point of your function - that is, prove there is only one solution to

10(x-0.9)=x

. It shouldn't be too difficult. After that, your proof should be pretty much valid, although it will be a little long and messy. Maelin (Talk | Contribs) 12:13, 24 January 2007 (UTC)

Your proof involving a unique fixed point can be made completey rigorous, but it won't sway anyone who's already inclined towards

0.999...~=1

. Skeptics must first swallow the usual real analysis hoopla: either "

x

and

y

satisfy

|x-y|<\epsilon

for any

\epsilon >0

therefore

x=y

", or perhaps "

x<1

and

x>1

both being false forces

x=1

", or even worse, explicitly mentioning the Archimedean property. How would you prove uniqueness to someone who believes that two particular distinct real numbers can be an "infinitely small" distance apart? Endomorphic 01:30, 25 January 2007 (UTC)

Here's my comment to this: (x*2+50)/2-25 = 25, no matter what x is. This equation does the same thing. 87.227.40.242 15:38, 9 March 2007 (UTC)

Counter-example: Let x=1. Then (1*2+50)/2-25 = 1. Tparameter 23:30, 9 March 2007 (UTC)

Suggested Removal as a Featured Article

The article itself is unashamedly biased to the point of being insulting in favour of the views of conventional "real" mathematics. It clearly has no place being considered one of the "best" articles of Wikipedia as basically it spends most of its duration being indirectly insulting about anyone who might ever doubt the mathematical status quo on this matter. Now I'm definitely not one to start claiming that I'm right and the mathematicians are wrong, but neither do I think its appropriate for them to repeatedly (ad nauseum) claim absolute dominion over truth in this case. Afterall, I have rationally concluded that absolute truth is incapable of existing in reality, that all "knowledge" is based initially on unfounded assumptions, and consequentially that all claims to absolute truth without exception are undeniably false. In other words... an article that is basically a one-sided p!ssing contest has no business having a gold star. ~ SotiCoto 195.33.121.133 16:39, 29 January 2007 (UTC)

The article is not much changed from when it was featured. A lot of people don't like the article, but the criteria for being featured is not likability but quality. If you've "rationally concluded that absolute truth is incapable of existing in reality," then Wikipedia — and indeed any encyclopedia — is not for you. An encyclopedia is a collection of facts, not opinions. The three cornerstones of Wikipedia are Reliable sources, Verifiability, and No original research. If I'm interpreting your post correctly, you don't believe in reliability or verifiability, and you're using original research to make your argument. Given all this, if you want your view to be considered, I'm afraid you'll have to find another website to do so. Calbaer 17:26, 29 January 2007 (UTC)

That which we call fact is basically just general concensus. Its not necessarily any more true than any alternative, but apparently it makes more sense to the vast majority of people. The problem with this is not only that its a rather specific concensus from a very particular kind of people (and indeed argued against by vast numbers more), but that it is actually borderline insulting to those who do not conform to its views. Whether its true or not isn't what I'm striking at here. Surely there has to be a better way to express all possible views while clarifying to whom they belong. ~ SotiCoto 195.33.121.133 10:07, 30 January 2007 (UTC)

You can deny the theorems of math, or you can advocate that every entry in every encyclopedia/textbook/reference first state "other than the 'nothing is absolutely true' idea..." However, math is simply logic constructed from some basic axioms and definitions and rules, and that every theorem is TRUE **GIVEN THOSE AXIOMS**. I think you should be in some philosophy article arguing about what is "true" in logic rather than in this somewhat obscure detail of a particular area of math. Otherwise, you may find yourself arguing with a cashier at WalMart about what **IS** fifty cents, and is it **TRUE** that the price is... Tparameter 16:43, 30 January 2007 (UTC)

...all claims to absolute truth without exception are undeniably false. -- So is that an absolute truth, or...? =) –King Bee ^{(T • C)} 18:17, 29 January 2007 (UTC)

I get that response a lot, and my typical answer is as follows: Probably not. I believe that it is possible for an exception to exist, thereby stopping the statement from being absolute, but experience leads me to believe every time that the probability of an exception actually existing (irrespective of whether it is capable of doing so) is negligable. ~ SotiCoto 195.33.121.133 10:08, 30 January 2007 (UTC)

[Edit conflict]I actually agree completely with your statement 'all "knowledge" is based initially on unfounded assumptions'. Mathematicians call these "unfounded assumptions" axioms. Every mathematical statement is only true within the context of some axiom system. However, if it is clear which axiom system we mean, we do not mention it explicitly. The statement "0.999... = 1" is not true on its own, but neither is "1 + 1 = 2". We allow ourselves to say "1 + 1 = 2" without reservation because it is clear that we mean the integers, and likewise, we allow ourselves to say "0.999... = 1" without reservation because it is clear that we mean the real numbers. In this sense, the statement "0.999... = 1" is much more true than a statement like "George W. Bush is the president of the United States of America" - I am able to prove the former, but can only rely on what I see on TV for the latter. You will not proceed to claim that articles which mention the latter statement are of poor quality, will you?

Building up on this analogy, would you say that the article United States is biased towards the US? After all, it says a lot of things about the US and there is hardly any mention of France. I guess that you would not, as that article is about the US. Similarly, this article is about the number 0.999..., and there is (effectively) no object called 0.999... in any structure other than the real numbers. So this article is about a real number, and again, in the context of real numbers it is true that 0.999... = 1.

So, you should focus instead on articles which state things like "1 + 1 ≠ 1". This is actually wrong in some meaningful structures, so if any statement can be called "biased", it's ones like this. -- Meni Rosenfeld (talk) 18:42, 29 January 2007 (UTC)

Excellent! Yes, mathematical theorems, given the axioms from which they are constructed, are as close to "truth" as we have in human knowledge. Of course this is gold star article. Tparameter 05:53, 30 January 2007 (UTC)

This "axiom" system is all well and good when absolutely everything reasonable and capable of being experienced suggests that it should be so, but even if it is then used it is folly to brand it as absolute truth. Rather it is a practical conclusion supported by consistancy. If the article wasn't spending so much time belittling anyone who disagreed with the statements therein, and was actually willing to admit to not being infallible (irrespective of it being concensus truth or not), then there wouldn't be a problem with it being "featured". As it happens though, my own disagreement with the whole process I have thus far led back to the conclusion that 1/3 of 1 cannot be expressed in decimal figures at all (and is merely proxied by 0.3333...), any more than can 1/9, 1/7 or many such others. Thats just my opinion though, and I'm not trying to shove it down anyone's throat... but I would appreciate not being belittled on what is supposed to be an objectively written article for holding that opinion. ~ SotiCoto 195.33.121.133 10:07, 30 January 2007 (UTC)

Even if this article's content were false, you'd still have a serious problem getting it removed. For almost every statement, including those parts you consider belittling, is attributed to reliable sources. Feel free to find some reliable sources claiming the opposite. Up to now, all opponents of 0.999...=1 have found that a formidable challenge. By the way, there are even proofs that don't rely on a definition of 0.999..., but only on some "common-sense" properties (that a number denoted by 0.9999... should satisfy) and a property of the real numbers. For an example, see the archive. --Huon 10:53, 30 January 2007 (UTC)

The place where you are going wrong is where you say, "Thats just my opinion though." There is no space for opinions in discussion of mathematical truths under a given axiom set. None whatsoever. This is where maths differs from the more hands-on sciences. In maths, it doesn't matter a whit what you think. What matters is what you can prove. It has been proven rigourously that under the real numbers, 0.999... = 1. If you don't like this and refuse to accept it, you have three options:

Reject the axiom set of the real numbers, and propose an axiom set you think is better. Note that nobody has found any actually useful axiom sets in which 0.999... and 1 represent things analogous to what they represent in the reals, but where they are unequal. Note also you will not have rendered this article untrue (since this is concerned only with the real numbers), you will merely have found a different axiom system in which the statement 0.999... = 1 is false.
Find a hole in the proof. Note that there are quite a few proofs, most are quite simple, and they have all been carefully checked by many mathematicians. Nobody has yet found a genuine problem in any of the proofs, though some skeptics claim they have when in fact they are have found a statement that is true but is not proven in the same place.
Avoid thinking about it.

I'm not trying to say, "if you don't like it then go away," but you have to get past this idea that the legitimacy of opinions, which is totally valid in other fields, extends to mathematics. Maelin (Talk | Contribs) 12:36, 30 January 2007 (UTC)

No space for opinions in discussion of mathematical truths? HAH! All "knowledge" is opinion. All truth is subjective. If you discount opinion from mathematics, then the entire field is left empty (much like anything else). The fact that I admit my fallibility is simply accounting for as many circumstances as possible so as not to be blatantly violating logically sound principles. The article's insistance that it is correct (rather than merely stating what it believes to be correct and what leads to it without exclusive self-confirmation as such) is a clear failure to account for the fallibility of human "knowledge" and as such is not reliable as "truth". (And I don't take issue with other non-mathematical articles based on "fact" because they don't make statements implying that opposition to the statements made therein is incorrect)... I cannot (and would not try to) prove the contents of the article to be false... not because they are true, but because "proof" as I see it is an impossible concept (due to its reliance on "axioms" full stop). Rather than demonstrate distinct fallacy here, I merely seek to point out fallability and thus the inaccuracy of commenting on particular things being "correct". And just because something is more "useful" or "practical" than anything else does not make it absolute truth by any stretch of the imagination. The "real numbers" axiom is something which is useful for everyday existence, but to declare it law at the expense of all else is unjustifiable. Its just a convenient fallback invented to avoid leaving an unknowable hole... much like the christian god: A personally useful thing to believe in on occasion, but hardly essential to universal existence. ~ SotiCoto 195.33.121.133 14:57, 30 January 2007 (UTC)

Mathematical proofs are impossible because they rely on axioms? I think you might be a little confused as to what mathematics is. Also, this article is not (by any stretch of the imagination) furthering some "view" or "opinion." It contains statements that are true in the real number system, and no one here claims that they are "absolute truth" except for you.

This isn't the weirdest thing out there, by the way. Why don't you check out Banach–Tarski paradox? –King Bee ^{(T • C)} 15:18, 30 January 2007 (UTC)

Not just mathematical proofs, but all proof. Nothing is beyond doubt (including that statement, but getting into that is a whole different ball-game)... and hence statements claiming to be beyond doubt are most likely incorrect in that statement. Having examined several more related articles, I have perhaps better identified the problem: that mathematicians will take certain axioms for granted and not even make mention of them, despite said axioms themselves being beyond proof. While it might be meant (and understood by mathematicians to mean) that all "proofs" are under the assumption that said certain conditions are true... this is not something automatically assumed by the average person reading the article, who would rightly perceive the article to be failing to fully justify itself or demonstrate itself as being conditional. And of course, being a Nihilist as I am I tend to reject and further question rather than blindly accept, given the choice (that applying to everything). And incidentally, if having "(correct)" as referring to the mathematical concensus toward the beginning of the article isn't a suggested declaration of absolute truth, would you mind telling me what it is? ~ SotiCoto 195.33.121.133 15:42, 30 January 2007 (UTC)

Mathematical proof is different from "scientific proof", or "proof that Jones killed his wife", etc. To say that "no type of proof anywhere ever" is possible, then you are equating things that are very plainly different from each other. While it is true that the axioms of ZFC are not stated here, it would be extremely cumbersome to have to list all of the axioms all the time at the beginning of every proof. Certainly many readers are not familiar with ZFC, but to have every mathematics article on Wikipedia start out by listing them is a horrible idea.

I would say that if an article said "this is absolute truth," then that would be a claim as to it being absolute truth. I removed that nonsense about "the view," because it's not a view at all. It's a proven fact about the real numbers, like it or not. You do not get to weigh in with an opinion on its "truthhood" unless you have counterproofs. This is how mathematics works and has worked for many years.

You do indeed like to question things instead of blindly accepting them; this is a good quality to have. You would probably make a good mathematician, actually. However, understanding the nature of mathematical proof is vital to success in understanding how people do mathematics. –King Bee ^{(T • C)} 16:03, 30 January 2007 (UTC)

If the different kind of proof all have one thing in common that itself is inherantly flawed... then all kinds of proof reliant on that commonality are thus flawed as well. If these flaws are actively ignored or discounted, the only possible implication is that they don't exist... which is false. When making an argument on serious terms (rather than frivolous or minor matters), one should state one's assumptions. While I accept that going through ZFC in every mathematical article would be in many cases superfluous (often as nobody cares that much to doubt it), in particularly controversial articles like this one it would be strongly advised to at least make mention of the key axioms assumed and on which the entire argument functions. That seriously would slow or stop the argument (albeit at a kind of ceasefire or stalemate). Remember that this numerical matter crosses over into normal, less-mathematical life... where things aren't conveniently black and white, and people won't be thinking in terms of real numbers. Its like people from Holland arguing with people from Zimbabwe about the colour of grass when each is assuming (but not saying) that the grass in their own country is the subject. Of course I would agree that if one is assuming the axiom thingie of Real Numbers that 0.999... would equate to 1, but at no point did I ever personally accept that axiom as totally true.... and I'd bet that most of the people trying to "disprove" the mathematical conclusion are thinking the same way (whether they know it or not).

So to return to the point, if this article is to be worthy of being "featured", it should at the very least be changed so as not to be so antagonistic, and should better justify its prior assumptions. ~ SotiCoto 195.33.121.133 16:31, 30 January 2007 (UTC)

The article begins: In mathematics, 0.999… is a real number... Thereby, it as stated clearly in which context this is true. A few clicks, and you will find the axioms and all. Absolute truth? I guess not, but a lot closer to it than 99% of the statements made in all other Wikipedia articles. My car is blue. Is it really "my" car? Is it really "blue", or does it just look that way to me? Is Bush president of the United States? You can question anything, but if that should stop us from making statements, supported by "reliable sources", there'd be no wikipedia, no knowledge, no civilisation.--Niels Ø (noe) 16:17, 30 January 2007 (UTC)

To 195.33.121.133 - I don't think anyone accepts these axioms as "true" in the sense that you mean when you say "true." However, this is completely beside the point. You don't have to accept the axioms to prove statements within the system. Take a look at why Banach and Tarski came up with their paradox; they were trying to show that the Axiom of Choice would lead to an inconsistency. Tarski wasn't a fan of the Axiom of Choice; but that didn't matter. He could not show it led to an inconsistency (together with the rest of ZF), and that's all that matters. –King Bee ^{(T • C)} 17:49, 30 January 2007 (UTC)

The question we should ask ourselves is: Why is there so much confusion surrounding 0.999...? The main reason, as I see it, is a problematic way of teaching decimal expansions at schools. No emphasis is made on the point that decimal expansions are just a convenient way of representing real numbers, with some useful properties such as easily comparing numbers and performing arithmetic operations. Students then make the flawed assumption that decimal expansions have some deep meaning on their own. It is then natural to assume that two different decimal expansions must represent two different numbers. But there is no legitimate grounds for this view, no suggestion for the construction of an alternative number system (indeed, any structure based on this foundation would be ridiculous). All there is is a misunderstanting of what decimal expansions are all about. As such, there is nothing wrong with forthcomingly stating that the reason for the confusion is this misunderstanding. No emphasized justification for the foundations is necessary either - once the article establishes that it is concerned with real numbers, the "whole package" comes with it, and anyone is free to follow the link to find out exactly what this package actually is. What the article does need is a clear statement about the truth of the matter (which is, that in the real numbers, 0.999... = 1) to dispel any confusion, and a discussion about why this confusion existed in the first place. Of course, you could argue about how "true" a proven mathematical theorem really is, and I could agree, but this has nothing to do with this particular article. -- Meni Rosenfeld (talk) 22:02, 30 January 2007 (UTC)

Here's the thing: mathematics doesn't provide facts about the universe; merely implications, but with rigid certainty. The statement "If the sky is made of cheese, then the sky is made of cheese" is an absolute truth. Irrespective of the composition of the sky, the implication is utterly true. Similarly, in mathematics: "if we're using the real numbers, then 0.999... = 1" is an absolute truth. Demonstrating the truth is slightly more complicated, hence mathematicians exist. Your epistomological equivocation might find a home in medicine, or geology, or law; anywhere except mathematics, because mathematics concerns true implications and not facts of the universe. Endomorphic 07:17, 31 January 2007 (UTC)

But if the conditions under which the statement is true are not given then by default it is implied to be true for all conditions under which the statement can be made... which is incorrect. Furthermore, repeated emphasis on it being "correct" and "absolute" still without any reference to the conditions under which it is so can serve no purpose but to antagonise those who would disagree. Its not often I'd say this... but the article needs serious re-working to make it less offensive. While I'm typically all for the notion of being honest having greater importance than being nice, phrasing things unpleasantly when it need not be so and there is nothing significant to gain by doing so is entirely redundant. Since I seem to subconsciously enjoy using strange metaphors... its rather like stating that "pigs -can- fly and people just claim they can't out of ignorance" without stating that it requires the pig to be on an aeroplane or helicopter (or equivalent flying machine).

Oh... and just to fight back with obstinate semantics, regarding the sky statement... its not absolutely true under all conditions. It is only true if you discount the probability of unrelated intermediary change (i.e. if its not relative to time). In other words... if at the point of the comma in the stated sentence the sky ceases to be made of cheese then the statement becomes false... because it was cheese at the first instance enough to satisfy the condition, yet not cheese at the otherwise identical conclusion, rendering it false. In other words... the subject of the statement must be entirely stable throughout and not subject to change due to being named, observed or otherwise identified. FURTHERMORE, there is no absolute guarantee that you're talking about the same sky each time.

If there is one axiom I will live my life by, it is this: "Nothing is absolute"... and yes, that statement included. I won't detail what the exception to that one is. ~ SotiCoto 195.33.121.133 12:11, 31 January 2007 (UTC)

its rather like stating that "pigs -can- fly and people just claim they can't out of ignorance" without stating that it requires the pig to be on an aeroplane or helicopter. Except that our article very clearly does state the specific context in which the statement 0.999... = 1 is true. The very first sentence specifies that we are discussing the real number represented by 0.999.... There is no, "Surprise! it's on an aeroplane!" trick here. Real number is linked on the very first line.

You need to understand that facts of logic are an entirely different kind of thing to facts about the world. You can feel free to decide whether you believe in absolute truths about the real world, but there is not even any question of it in logic and maths. Nobody talks about absolute truth in mathematics because in that context there is no other kind of truth. A mathematical statement can only be considered true or false in the context of some axiomatic system. There's no other kind of truth in maths. Frequently, the context in which a statement is being considered is implicit and not actually stated, but that doesn't mean it's not necessary or not there.

Saying you reject such a notion of absolute truth in discussions about mathematics is kind of like saying you reject the notion that the world can be described by a set of fundamental governing laws in a discussion about physics. Sure, you can if you want to, but if you do then there's really no point in having the discussion. Maelin (Talk | Contribs) 12:47, 31 January 2007 (UTC)

I suppose there isn't... Afterall, yet more delving has revealed to me the humourous notion of the "Law of Excluded Middle"... which is so incredibly ironic because its absolute opposite is the primary governing axiom of my conscious existence (i.e. that no thing nor concept can be 100% "true" or "false", but rather some combination thereof... with the necessary exceptions of all existence and non-existence). I mean... I denied it because it has the gaping flaw of only being consistant within its own variable bounds, concerning itself with nothing outside... and as such can make itself correct simply by changing those bounds on a whim. But in any case, I'd agree to disagree on certain matters... You apparently insist on the article being definite and unyielding in its adherance to absolution... whereas I insist that doing so is a grievous fallacy... Big deal...

That wasn't the issue I primarily had with the article though; that it was antagonistic was the issue I had with it. It actively encourages people to bicker and squabble over it because it is effectively set out like a challenge. ~ SotiCoto 195.33.121.133 13:41, 31 January 2007 (UTC)

...if at the point of the comma in the stated sentence the sky ceases to be made of cheese then the statement becomes false. --- No. The statement "if the sky is made of cheese, then the sky is made of cheese" does not depend on whether the sky is really, honest to goodness made of cheese. It is a tautology; it is of the form "If P then P," and doesn't depend on P being true or false. –King Bee ^{(T • C)} 13:36, 31 January 2007 (UTC)

That you wish it to adhere to the rules only there stated was not initially declared and thus the exceptions are not invalidated. In other words, it was never declared that the validity of P at any given point is irrelevant, and as such to take that as an assumption would be redundant. ~ SotiCoto 195.33.121.133 13:41, 31 January 2007 (UTC)

Seems odd to me that someone would claim that there is no such thing as absolute truth, then choose some obscure particular mathematical article about which to argue that point. If you can't believe that anything is absolutely true - even structures that have well-defined rules and axioms, and step-by-step rigorous proofs - then what's the point? But, again, I wonder if you question the arithmetic used when you're at the cashier at the grocery store!? If not, maybe you should start there. Tparameter 17:05, 31 January 2007 (UTC)

Before this discussion goes any further, you need to start here. –King Bee ^{(T • C)} 18:39, 31 January 2007 (UTC)

...it has the gaping flaw of only being consistant within its own variable bounds, concerning itself with nothing outside... - but that's exactly what mathematics does; theorems are built from axioms, which have little-to-no direct relation to the universe. Hence: mathematical statements can be utterly true, as theorems are. It's not a gaping flaw, but a wonderous featue. Perhaps your problem, SotiCoto, lies not with 0.999... = 1, but more with philosophy of mathematics as a whole? Endomorphic 22:15, 31 January 2007 (UTC)

Bingo! Tparameter 02:28, 1 February 2007 (UTC)

Do you believe that the article would be better without a discussion of the reasons for the confusion? In particular, do you believe that it will have higher encyclopedic quality, and do you believe that it will reduce the likelihood of people to reject its content? If so, can you explain why, without attributing it to some inherent ambiguity in the statement itself (which for our purposes, there is none)?

By the way, I should probably say that on some points, I agree with your view more than with the other view presented in this discussion. Once again, though, I shall remain silent, as it is this article we should be discussing. -- Meni Rosenfeld (talk) 14:29, 31 January 2007 (UTC)

^ Tall 2000 p.221

[1] Tall 2000 p.221

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