Talk:0.999.../Archive 6

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This archive page covers approximately the dates between 2006-02-11 and 2006-02-23.

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Archive 5
The talk page had grown to 140k, far over the recommended limit. It has been archived. Please limit future discussions to relevant topics, as described at Talk pages. Note especially the following guidance: Also please observe the rules stated there about signing, formatting, and civility. Thank you. --KSmrqT 19:27, 11 February 2006 (UTC)
 * Wikipedians generally oppose the use of talk pages just for the purpose of partisan talk about the main subject. Wikipedia is not a soapbox; it's an encyclopedia. In other words, talk about the article, not about the subject.


 * I do appreciate that, although when the dispute is over the fundamental basis of the article (you can't accept a proof for something that is false) I think there is reason to debate it. Also, I think you need to check the advice on How to archive a talk page where it says Regardless of which method you choose, you should leave current, ongoing discussions on the existing talk page. I would point out that the last 5 or so sections on the talk page had had contributions (including several significant ones) in the past two days, which I would count as "ongoing". Unless I see a reasonable argument to the contrary, I will soon return the sections from "What is wrong with this discussion?" onwards to the main talk page. Confusing Manifestation 11:44, 12 February 2006 (UTC)


 * Ordinarily, and in the past here, I have left dangling discussions. However, the perpetual discussions on this page rarely even mention the article itself, and a review of the archives shows that such a pattern of violating the purpose of the talk page is likely to continue unchecked no matter what the starting point. There has been no significant contribution to improving the article. Frankly, this needs to stop, or move to a chat room. One can always hope. --KSmrqT 13:47, 12 February 2006 (UTC)


 * We could try setting up a free speech zone outside of the talk page. Seriously. Something like Talk:Proof that 0.999... equals 1/Off topic... Melchoir 20:38, 12 February 2006 (UTC)
 * Speaking of staying on topic: I sympathize with ConMan's points but I disagree. KSmrq may have violated the recommendations at WP:ARCHIVE, but it was the right call. In fact, there were no protected "ongoing discussions", since they didn't concern the article. And no, there is no dispute over the fundamental basis of the article here; there is only a mockery of a mathematics classroom for the obstinate, and we don't need to support those to write an encyclopedia.
 * Even if the IPs did have a point, talk pages are not for conducting original research on alternate theories of mathematics. Let them hash it out on Everything2 and Usenet; let them write their viewpoint on Uncyclopedia, where it belongs. If they're right, let them publish first. But the crap has interrupted too many actual content debates, and that hurts the encyclopedia; any method of discouraging it is fine with me. Melchoir 21:02, 12 February 2006 (UTC)

Why this article is nonsense.
Saying that 0.999... is exactly equal to 1 is like saying 3.14... is exactly equal to pi. Does the latter statement make any sense? 158.35.225.229 16:20, 13 February 2006 (UTC)


 * The difference between 3.14... and 0.999... is that it is not clear what 3.14... means, whereas 0.999... can be used to clearly identify an infinite decimal expansion. I personally don't like the notation 0.999... very much, but it is clear that if it, or 0.9(rec) or anything with that meaning, is used to represent a real number, that number must be 1. JPD (talk) 11:31, 14 February 2006 (UTC)

Nonsense. Nothing is clear about 0.999... and it is very clear what 3.14... means (some series expanion for pi). What this is saying is that it makes no sense to set 0.999... equal to 1 when you can't find a limiting value for pi. 68.238.109.124 14:06, 14 February 2006 (UTC)


 * Why is it clear that 3.14... means some expansion of pi? It could mean 3.14141414... or something like that. Pi is a real number - what do you mean by a limiting value for it? I am more and more convinced that the article should have something near the beginning about exactly what is meant by the notation 0.999... JPD (talk) 14:28, 14 February 2006 (UTC)


 * The article does say exactly what is meant by 0.9999...: A recurring decimal. Should that be expanded and repeat the definition of a recurring decimal? I don't think so. Huon 18:05, 14 February 2006 (UTC)


 * If it gives a clearer context and makes people less likely to misunderstand it, why not? It shouldn't be necessary to spell it out, but then again, the page as a whole isn't strictly necessary. Mind you, the similar section at recurring decimal should probably include a link to this page. JPD (talk) 11:50, 15 February 2006 (UTC)

One of the posters in the archive (MikeKelly) stated that the difference of two real numbers is real. What about:

pi - 0 = (log -1) / i

Here we have the difference of two real numbers equal to the ratio of two unreal (complex) numbers. So what, you say? Well, the complex numbers are built on a non-workable idea, i.e sqrt(-1) and with a little help from trigonometic series cos and sine (and Euler's formula) we now have Fourier series, complex contour integrals, complex polynomials, etc and many uses for this mathematics under the name of complex analysis. Complex numbers are built on a fundamental contradiction. The idea that 0.999... = 1 was born out of a misinterpretation of the original ideas. Concepts such as infinite sum and limit of infinite sum were assumed to be the same by some ignorant mathematicians. Melchoir quoted a result completely out of context from a reference he found pertaining to the disputed subject. Look, what I am saying is that if you retain this article, you should state clearly on what grounds you reach this conclusion. You need to clearly define all the terms you use. What good is it if someone reads this and does not understand the definitions? You need to define infinite sum, limit of infinite sum and what you mean by 0.999... It is not universally accepted by mathematicians that 0.999... = 1. Most mathematicians will not even bother with the topic because frankly it is such nonsense and has no use whatsoever except perhaps to frustrate students. 0.999... is less than 1 by definition. Contrary to the author's opinion, a radix-based number system has much to do with all these 'strange' results. Finally if infinite sum and limit of inifinite sum are considered to be the same thing by respected institutions, then these same institutions do not deserve to be respected for there is a clear difference between the two phrases. 68.238.100.221 13:46, 15 February 2006 (UTC)
 * Wikipedia is such an institution. If you have decided not to respect it, I wonder why you are still here. Melchoir 14:03, 15 February 2006 (UTC)

Very good. As long as you admit that you are not a respectable organization and no one else posts on this page, I will refrain from posting. If anyone else continues to post, I will respond and you can go and hop it! 68.238.106.152 19:25, 15 February 2006 (UTC)


 * For a start, I dislike your statement that complex numbers are based on a "fundamental contradiction", but rather on the idea that it may be possible to take the square root of -1 if you consider the possibility of numbers outside the reals. As for "infinite sum" and "limit of infinite sum" being "taken" as the same by "ignorant" mathematicians, not that in this archive I managed to prove the equality. I will admit the proof is not 100% rigorous, but it is based on the few definitions that both sides of the argument were willing to accept, and I'm fairly sure it wouldn't take long to fill in the holes. As for your other point, that the article fails to define its terms properly, I would point out that most of the key terms used in the article are Wiki-linked to the appropriate articles, which is where the definition should be kept in any case. Examples include recurring decimal, limits, and infinity. If you can point out a term that is not sufficiently defined either in this article or in the linked article, feel free to point it out. Confusing Manifestation 04:50, 16 February 2006 (UTC)

You cannot take the square root of any negative number and f(x) = a^x is undefined for a<0 and frac(x) <> 0 where frac(x) is the fractional part of x. So, it's a given: sqrt(-1) is just not possible. So what do they ('mathematicians') do? They reason that i^2 = -1. This is false since sqrt(-1) does not exist. In actual fact, an entire branch of mathematics has been developed on this falsehood. In fact sqrt(-1) might as well be anything since all we care is that the result of the squaring is -1. We could equally well have defined i as (1/sqrt(2))*sqrt(-2) or anything whose square yields -1. How can you take the square of a number whose existence is questionable? Not very logical eh? Thanks to cos(x)+isin(x), we now have 'complex analysis'. As for your proof: yes, it is not 100% rigorous but that does not matter as much as the fact that you insist on having 0.999... equal to 1. You know, I could argue as follows with the Archimedean property(corollary 1):

If a and b are real numbers with a > 0, there exists a natural n such that na > b.

Case 1: Let a = 0.999... and b = 1:     I can find an n (say n=2) such that na > b. Check:  2*0.999... > 1          (In fact any n > 1 will do)

Case 2: Let a = 1 and b = 0.999...:     Choose n = 5 Check: 5*1 > 0.999... (In fact any n will do)

Case 3: Let a = 1 and b = 0:            Choose any n.

For corollary 2: If x is a real number greater than 0, there exists a natural n such that 0 < 1/n < x.

If you set x equals to 0.999... then you can choose many such ens (e.g. n=2 => 1/2 <0.999...)

So when you do not try to see what happens at infinity (which you cannot do anyway), you have no problems. Now the third corollary is where you might get into a bit of hot water:

If a and b are real numbers with a < b, there exists a rational number c such that a < c < y. There appears to be a problem when you say 0.999... < c < 1 because you think that you know the full extent of the sum 9/10+9/100+9/1000+... As was explained in the archive, this is an infinite sum that is indeterminate (except for its limit). In the archive it was shown that you can find an infinite number of such sums between 0.999... and 1 so that you can indeed show that many such c exist to make this corollary true. Simply let A=99 and use base 100. Then, A/100+A/10000+A/1000000+... is always larger than 9/10+9/100+9/1000+... and always less than 1. Very simple to see.

Now let's consider the statement that the infinite sum is actually equal to the limit of the infinite sum. This is a stupid remark someone like Prof. Hardy would make. Let's assume that this is true. Now suppose I give you a number 3.14......6 and the dots in between this number are placeholders for 1000000000000000000000000000000000000 digits. I now ask you to tell me whether this number is greater or less than pi. How are you going to do it? Hmmmm. Big problem because the only way you can do it is by comparing the partial sums, i.e. digit by digit. But wait! This is a contradiction because an infinite sum is equal to its limit - so what is the limit of pi? Oops! You weren't thinking again, were you? As for 0.999... and 1: you are comfortable comparing the limit of 9/10+9/100+... with 1 rather than the partial sums. Is this not contradictory? You compare the limits of some numbers and for others whose limits you don't know, you compare using partial sums. This stupidity makes me extremely angry at the idiotic Phds who insist 0.999... = 1. Never mind, those who are guilty know what I think. So what is the moral of the story? The moral of the story is:

'An infinite sum is NOT the same as the limit of an inifinite sum'.

70.110.91.141 15:00, 16 February 2006 (UTC)


 *  Couldn't we just archive this page, so as per anon, As long as (...) no one else posts on this page, I will refrain from posting and just hope he goes away (or cools off)? I mean, it's rather hard to assume good faith. Jesushaces 05:32, 16 February 2006 (UTC)


 * Wow, this diff is a microcosm of the entire history of the talk page. Could we all please agree not to respond? I appreciate the irony in my posting this, but I'm serious. Melchoir 17:24, 16 February 2006 (UTC)


 * I agree with Melchoir. Nobody else besides the Wiki authors, sysops and Wiki administrators read this junk anyway. So perhaps if all Wiki staff would just stop posting, someone else might actually post something that has some value. Give the readers a chance. They can make up their own minds. 71.248.149.184 02:16, 17 February 2006 (UTC)


 * I wonder just how important this is to math majors and their professers.Timothy Clemans 03:56, 17 February 2006 (UTC)


 * I would say pretty important. Look at how many math majors (including Melchoir and his idol Prof. Hardy) who in their 'infinite wisdom' don't know the difference between an infinite sum and the limit of an infinite sum. You can look at the archive to see how Melchoir misrepresented certain information to back up his claims. 70.110.89.172 12:24, 17 February 2006 (UTC)
 * Oh yes, I idolize Hardy and I misrepresent information for fun! Fear not, brave IP address, your brilliant narrative of the mathematics war has been published at last:
 * http://uncyclopedia.org/wiki/0.999...
 * Did I get it right? Melchoir 21:57, 17 February 2006 (UTC)


 * Wow, you've summarized the arguments of the entire Wikipedia team in this link. Well done Melchoir! 70.110.89.172 23:27, 17 February 2006 (UTC)

A question
If you feel that 0.99999.... is not equal to 1, could you please state what the value of 1 - 0.9999.... is. If the two things are not equal, then their difference must be something other than zero. Tompw 13:27, 23 February 2006 (UTC)

Perhaps you might answer my question first? How do you compare pi with the number I presented above? 68.238.99.53 13:45, 23 February 2006 (UTC)


 * No thanks, I won't answer your question. The reasons is that my question relates directly to the disputed issue, so an answer to it would settle the dispute. Thus answering any other questions would be a side issue. So, I ask again: what is the value of 1 - 0.9999... ? Tompw 14:11, 23 February 2006 (UTC)


 * Neither will I answer your question because my question has everything to do with the dispute. It is directly related because the dispute is about comparison of numbers in the decimal system or any other radix system. 68.238.99.53 14:59, 23 February 2006 (UTC)


 * Whether or not your question is diectly related is not the issue here. My point is that you are argueing that 1 and 0.999... are different. If they are different, tell me what the value of the difference is. Tompw 18:36, 23 February 2006 (UTC)


 * The difference is the same as that between pi and its predecessor. Now if you can tell me the exact value of the limit of any pi series, I will tell you the exact value of the difference. Deal? 71.248.147.163 01:31, 24 February 2006 (UTC)

Use of ellipses
I'd say there's some confusion in the article in that "..." is being used both to represent "recurring", but also for "and the series continues in a non-repeating manner", eg, "1.41421356...". I don't think it's helpful, and it may confuse some to believe that "0.999..." just means that the sequence continues, but not as a recurring decimal (as the earlier poster did when he said it was nonsense). We need to explicitly be clear that we are talking about the recurring decimal here. Mdwh 15:38, 18 February 2006 (UTC)


 * The use of ellipses ("…") in both contexts is standard, and the sentence changed is explicit about lack of repetition for irrationals, while the 9s usage is explicit about recurring. --KSmrqT 17:18, 18 February 2006 (UTC)


 * The use of "..." is not standard and only you know what you are talking about. In my understanding your article is nonsense and 0.999... < 1. 68.238.102.30 22:14, 18 February 2006 (UTC)
 * It is standard for most things, though some people denote a recurring decimal, in this case, 9, as
 * $$0.\dot{9}$$
 * although the LaTeX notation for the dot is actually for derivatives.  x42bn6   Talk  08:46, 19 February 2006 (UTC)


 * Whilst the use of ellipses may be common for both, it's still misleading to use the same notation for two different concepts in the same article. Why do you object to my changing this in order to make it clearer? Even if you think it's currently "good enough", how is the article better without my change? In addition, why did you remove the wikilinks to the relevant articles?


 * This is a topic that causes a lot of people confusion (see above for someone who says "Saying that 0.999... is exactly equal to 1 is like saying 3.14... is exactly equal to pi."), and we need to be exact as possible with the definitions. Note how the articles pi and square root of 2 avoid usage of ellipses, presumably in order to be clear as possible. Mdwh 17:38, 19 February 2006 (UTC)


 * The use of notation on Wikipedia is descriptive, not prescriptive. "0.999..." is a common notation; I have no research claiming that it is the most common, but I wouldn't doubt it. Other notations, such as the overdot or overline, should be mentioned in the article as alternatives, but I don't think they should be emphasized. The different meanings of ellipses might be more explicitly examined in the article; the sooner we get the reader thinking about the meaning(s) of mathematical notation, the better.
 * Part of the confusion underlying this article's topic is that two notations might refer to the same number at all; we mustn't paper over that issue. If we rewrote the article to use the best possible notation, it would read, "Proof that 1 equals 1", and it would be very short! Melchoir 22:37, 19 February 2006 (UTC)


 * Firstly, I think you've misunderstood what I've said. I don't object to using 0.999..., as that is one of the accepted definitions, albeit not the best one in my opinion (as described at Recurring decimal). My objection is using it for 3.14... and so on. Firstly I'm not convinced it is entirely correct terminology, and secondly, it's misleading to use the same terminology for two significantly different concepts in the same article - especially in an article where clear definitions is of utmost importance. It's not clear to me why including this is better than simply a wikilink to the articles for those numbers.


 * I agree that Wikipedia is descriptive, but I don't see how that's an issue, as what I propose is within the standard notation, and not prescribing how it should be done. I'm saying we should use as precise and clear notation as possible, especially in a Mathematics article. In particular, I'm not convinced that "..." does have a mathematical usage in the sense of describing pi as 3.14... (indeed, Ellipsis says "it is not a formally defined mathematical symbol. These dots should never be used unless the pattern to be followed is clear.") It may have a common usage, but I believe Mathematical articles should use precise mathematical notation rather than ambiguous non-mathematical notation.


 * This is nothing to do with descriptive versus prescriptive, but rather mathematical terminology versus common usage. A common usage description of why "0.9 recurring" equals 1 is not going to convince the sceptics.


 * I disagree with your last statement. The difference between "3.14..." and saying "3.14 to 3 significant figures" is one of misleading notation versus correct mathematical terminology. The difference between "0.9..." and using the line is a matter of notation. But on the other hand, showing that "0.9 recurring" equals 1 is not a purely a matter of notation - it requires proving that a sequence tends to 1 as the number of terms tends to infinity. Mdwh 02:49, 20 February 2006 (UTC)


 * Okay, I absolutely misunderstood you. If you have a good replacement for "numbers like √2 = 1.41421356… and π = 3.14159265… with an endless number of digits that do not repeat", you may as well try it out on the article.
 * Nonetheless, I have to stand by my last point. There are plenty of textbooks-- I've seen them, but I don't have one on hand-- that define real numbers as decimal expansions, and they simply set 0.999... and 1 equal by fiat. For them, the fact that a decimal expansion represents a certain power series in 1/10 is a theorem, not a definition. We even have a Wikipedia article acknowledging this approach, albeit briefly: Construction of real numbers. So if that's your conception of a real number, the only difference between 0.999... and 1 is notation. It might be confusing to mention that approach in this article, but it's worth keeping in mind. Melchoir 03:11, 20 February 2006 (UTC)


 * To say you absolutely misunderstood is being rather nice to yourself isn't it? 68.238.105.214 14:21, 20 February 2006 (UTC)


 * The general meaning of ellipsis ("…") is that something has been omitted. When we write 0.999… we state explicitly that we are talking about a recurring decimal, with the 9s repeating. When we write 3.14159… for &pi; we state explicitly that we are talking about a decimal without repetition. It is true that in the first case we have available other notations that indicate which of the trailing digits recur; for example, we could write 1&frasl;81 = 0.0123456789 . The essential point here is that the ellipsis alone is not being used to carry the weight of informing the reader of what is omitted; the text (or context) does that. Nor is such a convention restricted to mathematics; it is, in fact, common in text elision as well. Ellipsis consistently represents only one concept: omission.
 * Furthermore, on theoretical grounds we can do no better; in the most general case we cannot say what is omitted, because almost all real numbers are not computable. That is, we have no effective way of writing down a description of what the omitted digits should be. In this sense, numbers like 0.999…, 1.41421356…, and 3.14159… are very special cases, because we can give an algorithm for producing the digits.
 * Another argument for the uniform notation is that we interpret the numbers with a uniform convention. That is, we do not need to know the sequence of digits in order to explain how we will associate the notation with a specific real number. This is a Good Thing. The "order" proof converts any digit sequence to a Dedekind cut; the "limit" proof, to a Cauchy sequence. We have other options as well; for the standard reals, all yield the same results. --KSmrqT 06:12, 20 February 2006 (UTC)


 * Please stop using words whose meanings you do not know. You do not provide any proof that 0.999... equals to 1 for two reasons: Your definitions are inconsistent and your proofs are incomplete/wrong. About the most convincing attempt to prove this fact was Rasmus's attempt that was demonstrated to have holes in it. Another Wiki contributor tried to accomplish the same and did not manage to do so completely because he uses four facts not entirely in context. Look, using your logic I can prove that 0 = 2,000,000:  Let's see, I start off with a number 'a' very close to zero but greater (now I am reasoning like you) and since the difference is almost zero, a is equal to 0. Let's choose another number greater than 'a' but very close to 'a' and call it 'b'. By the same logic, we can deduce that a = b. We can proceed in this manner until we reach a number whose value is 2,000,000. Hence 0 must be equal to 2,000,000. Not only do you not have the vaguest idea what 0.999... is but you don't know what '1' is and here you are trying to define 1 in terms of 0.999... 68.238.105.214 14:21, 20 February 2006 (UTC)


 * If you're talking about my proof, where I started off with four definitions that I made sure were agreed on, why not tell me how I used them "out of context"? Tell me at which step in the proof I have stated something that is completely fallacious (and don't tell me that I make that mistake at the conclusion, because it has been carefully derived from the previous steps). It is not a rigorous proof, and I admitted as much, but show me where the lack of rigor leads to logical falsehood, not just to something that requires further detail. Confusing Manifestation 05:46, 23 February 2006 (UTC)


 * OK, how about this, to appease the people who are trying to argue that ... doesn't say it properly - does anyone object to adding a sentence like this somewhere, even though the article already describes 0.999... as a recurring decimal: "In mathematics, a decimal expansion finishing with an ellipsis (...) often represents an infinite sequence of unwritten digits, whose values are usually defined by context, so that for example when writing &pi; = 3.14159... the first few missing digits are 265358979. In this article, 0.999... is taken to be $$0.\dot{9}$$, that is, the ellipsis represents an infinite sequence of 9s in the decimal expansion."? Confusing Manifestation 05:46, 23 February 2006 (UTC)

The problem is with your conclusion unfortunately. Look, can you answer my question: How do you compare any number with an irrational number? 68.238.99.53 13:48, 23 February 2006 (UTC)

Contradictions and stupidities...
Sorry JPD, nothing personal but I am going to attack what you wrote here because it reflects the same ignorant mindset of all Wiki authors.

You made a statement: 'Pi is a real number - what do you mean by a limiting value for it?'

Well, on the one hand you know that every real number is bounded from above so how can you ask such a stupid question?

Now I will take great pleasure in attacking the biggest fool of Wikipedia - Ksmrq. He writes:

"...That is, we do not need to know the sequence of digits in order to explain how we will associate the notation with a specific real number..."

This fool insists that radix systems have nothing to do with real numbers, yet he is the author of this "incredible fable". If the sequence of digits are not required, how can one using the decimal system associate any representation with a real number? Furthermore, how can one even begin to compare two numbers in the decimal system or any other radix system? If an infinite sum is equal to the limit of an infinite sum, then we should never be able to compare any number with pi, e or any other irrational number. The sequence of digits and the order is of utmost importance in comparing numbers in any radix system. Finally if you compare limits of infinite sums, then you ought to be consistent and do the same for irrationals such as pi and e. 71.248.144.48 18:16, 22 February 2006 (UTC)


 * I have warned you at User talk:71.248.144.48. Regardless of future IP addresses you use, if you continue to make personal attacks on Wikipedia, I will add more warnings to that page until there are enough to block you. If you wish to be tolerated here, let alone taken seriously, you will need to reverse your attitude. Melchoir 20:36, 22 February 2006 (UTC)

Civility aside, I'm sorry to say this, 71.248.144.48, but your claim is just wrong. I'll try to put it simply: In the structure of real numbers, there's no other way to define an infinite sum than as a limit of finite sums. Look at any first-year calculus book for the definition. For example, we say
 * $$\sum_{k=0}^{\infty} \frac{1}{k!} = e$$

When what we actually mean is
 * $$\lim_{n \to \infty}\sum_{k=0}^{n} \frac{1}{k!} = e$$

Similarly we have all other sorts of sums, such as:
 * $$\sum_{k=0}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}$$
 * $$\sum_{k=0}^{\infty} q^k = \frac{1}{1-q} \quad (|q| < 1)$$
 * $$\sum_{k=1}^{\infty} \frac{1}{10^k} = \frac{1}{9}$$
 * $$\sum_{k=1}^{\infty} \frac{3}{10^k} = \frac{1}{3}$$
 * $$\sum_{k=1}^{\infty} \frac{9}{10^k} = 1$$

Now, when we write a decimal expansion a0.a1a2a3a4a5... what we actually mean is:
 * $$\sum_{k=0}^{\infty} \frac{a_k}{10^k}$$

So 0.33333333.... is equal to
 * $$\sum_{k=1}^{\infty} \frac{3}{10^k} = \frac{1}{3}$$

and 0.9999999... is equal to
 * $$\sum_{k=1}^{\infty} \frac{9}{10^k} = 1$$

That's for the technical part. Also, if you knew a bit more about where real numbers come from, you would have known that 0.99999.... can't possibly be any other number than 1. So I suggest you study the matter a little, and the article real number would be a good place to start.

Now about Wikipedia, many users hold doctoral or other degrees in their field of writing, so having an argument with a multitude of people and saying Wikipedia is not a respectable source, is probably unwise. I'm not saying one should always follow the herd, but in a case like this it would be best if you research the matter on your own and find the correct results instead of being confident that it is the world that is wrong.

I hope this helped. -- Meni Rosenfeld (talk) 13:27, 23 February 2006 (UTC)

On second thought, I'll make life easier on you: We say that
 * $$\sum_{k=0}^{\infty} a_k= S$$

iff for every ε>0 there is $$N \in \mathbb{N}$$ such that if $$n \in \mathbb{N}, n > N$$ then
 * $$|\sum_{k=0}^{n} a_k - S| < \epsilon$$.

I'll leave it as an exercise for you to prove the summation formula for an infinite geometric series... It is much better to use explicit mathematical definitions than discussing all sorts of "ghost" quantities. -- Meni Rosenfeld (talk) 13:39, 23 February 2006 (UTC)

Meni: I have no disagreement with what you wrote above except that you claim an infinite sum is the same as the limit of an infinite sum. This (even if it were defined this way) is untrue because you then have no way of comparing numbers consistently. Read my post on the inconsistency of comparing numbers such as pi, e, etc! As for people here having Phds - this means nothing to me. They are not in the least smarter than I am. And if the world (as you falsely 'claim') insists on this, then the world is wrong. If Newton said this, I would call him an idiot. As for real analysis, it was developed by fools. It is rife with contradictions and major errors. Perhaps it is you who ought to study these things and try to reach a correct understanding. 68.238.99.53 13:43, 23 February 2006 (UTC)
 * Like I said, I am by no means implying that having a degree makes a person infallible. And I agree one should have confidence in his own intellect. But maybe your entire objection to the world, PHDs, Wikipedia and everything simply comes because you misunderstood some concept, and just understanding it will make it all vanish? That would be worth trying to find.
 * Now, about your argument, I didn't understand it, and it refers to "ghost" operations that aren't defined properly. Math isn't philosophy, math is about things that are defined rigorously. So one should start with a definition of "real number". This can be done in several ways, either by specifying an axiom system for reals, or for example constructing real numbers as sets. Either way, we have a structure that is defined by certain axioms - an example can be found at Real_number. Now the question is what is your claim against the reals. Is it that the axiom system is inconsistent? Explain why, in terms of the axioms. Do you think it does not unambiguously define the reals? Present a statement the axioms can't solve (but stay within the context of reals, since building on ZF for example there are many unresolvable statements, but those are beside the point). Is it something else? Tell what it is. What you did in your argument is referring to things that aren't defined properly, and so cannot be dicussed in the context of mathematics. -- Meni Rosenfeld (talk) 14:05, 23 February 2006 (UTC)
 * Quite the contrary my young friend - my arguments are very rigourous and sound. To say that 0.999... = 1 is "ghost" mathematics. 68.238.99.53 15:04, 23 February 2006 (UTC)

The Conclusion
If anyone defines an infinite sum to be equal to its limit then pi, e and any irrational number is undefined and it's time to go back to the drawing board. 68.238.99.53 13:58, 23 February 2006 (UTC)

Pi is the unique real number satisfying
 * $$\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = \frac {\pi}{4}$$

In the sense described above. e is the unique real number satisfying
 * $$\sum_{n=0}^{\infty} \frac{1}{n!} = e$$

That these numbers exist and are unique is easy to show. I can also present other definitions if you'd like...-- Meni Rosenfeld (talk) 14:09, 23 February 2006 (UTC)

Here's an even better definition: Assuming we treat real numbers as dedekind cuts, Pi is the set {A, B}, where A is the set of rational numbers r for which there exists a natural number N such that
 * $$\sum_{n=0}^{N} \frac{8}{(4n+1)(4n+3)} > r$$

And B is the set of all rational numbers not in A (I'm not sure I got the formula right, but that's the general idea). -- Meni Rosenfeld (talk) 14:18, 23 February 2006 (UTC)

You may want to quit cluttering up the page with well-known formulae and answer the question. Again, how do you compare any number with pi, e or an irrational number? In particular, how do you compare 3.14.....6 where the dots represents 100000000000000000000000000000000000 digits with pi? You must be very young Meni? Just out of college or still in your first year? 68.238.99.53 14:52, 23 February 2006 (UTC)
 * I am 21 and have finished a bachelor's degree 3 years ago. Thank you for asking. Now, if you'll agree to quit cluttering up the page with personal insults and return to the matter at hand, what do you mean by "compare"? -- Meni Rosenfeld (talk) 14:57, 23 February 2006 (UTC)


 * That explains it. I am more than twice your age and have thought about this a whole lot longer than you. If you don't know what 'compare' means, you ought to start by looking it up in the dictionary. What do you think one says when one claims that 0.999... = 1 ? This is your first clue. I leave the rest as an exercise for you. 68.238.99.53 15:02, 23 February 2006 (UTC)
 * I know very well what "compare" means, but it seems you mean it in a different manner entirely. And I told you already. When one says that 0.9999...=1, he means that a certain infinite sum, defined in a rigourous way, is equal to 1. The decimal expansion is just a way of representing a real number. There are many ways to represent real numbers - Decimal, binary, factorial base, continued fraction, and so forth. But they all represent the same set of real numbes, defined by certain axioms. And it just happens that the decimal representation (as do many others) has the flaw that a given real number can have 2 representations. Namely, 0.99999.... = 1, in the same way that in hexadecimal, 0.FFFFFF = 1, or in factorial, 0.1234567.... = 1. You keep thinking that a decimal expansion is a synonym for real number, which it is not. And if you wish to challenge the entire world regarding this claim, I can only wish you good luck (you'll need it). And you still haven't expressed your concerns in terms of rigorous mathematical statements, which is evidence that your argument is basically founded on nothing. -- Meni Rosenfeld (talk) 15:13, 23 February 2006 (UTC)

There is no flaw at all in any of the radix systems. I disagree there is duplicate representation. You do not realize that any radix system is based on approximation. Now where did I ever say that a decimal expansion is a synonym for any real number? Nowhere. Look, there is nothing you can tell me that I don't already know. I learned real analysis before you were born and I was a top student - this means I passed with first class passes. So look, do me a favour and don't waste my time. As for not expressing my concerns in terms of rigorous math: I have done exactly that - I pointed out how stupid you and Wiki authors are in comparing 0.999... and 1 differently to the way in which you would compare 3.14.........6 and pi. You have such a cheek making a statement like this or were you just not thinking again? And it seems you failed to use your clue regarding comparisons of numbers. Use it and maybe you will learn something regarding rigorous math. 71.248.147.163 01:44, 24 February 2006 (UTC)
 * That's your third. Melchoir 01:46, 24 February 2006 (UTC)


 * (via edit conflict) When you say compare, you don't say how, which I think is the problem. Do you mean "compare the two numbers two see which is larger"? "Compare the two numbers to see if they are equal"? "Compare the two numbers to see which one is more interesting"? And, for that matter, why does saying that the infinite sum equals its limit mean that numbers like pi and e are undefined? If you define the infinite sum to be the limit of the partial sums, then the universe does not collapse. In fact, now that you have some kind of meaning to the notation of an infinite sum, you can actually perform mathematical operations with it! (Incidentally, Meni, I believe those formulae are not actually definitions of pi and e, but rather formulae proven to equal those values. Pi, for example, is defined as the ratio of a circle's circumference to its diameter, and I can't exactly recall which of its many properties e is defined for, but I'd bet even money on either being the base of the exponential function which is its own derivative or else the limit of the "infinitely compounding money" problem.)
 * And on that note, I am going to point out a little page I have created to try and help me understand some of the arguments from the other side: User:ConMan/Proof that 0.999... does not equal 1. Please follow the rules as I have set out, and if you can provide a proof that I can follow and find no logical holes in, then I will be able to consider your arguments with a little less skepticism. Confusing Manifestation 15:19, 23 February 2006 (UTC)
 * Well, as with any other concept that has several properties, one can choose whether to have one as a definition and the other as a theorem, or vice versa. There's nothing fundumentally wrong with the definitions above of Pi an e, except their unpopularity. Defining them in the usual way would raise too many questions to be useful for this discussion. And as for e, I think both your definitions, and perhaps also the one as an infinite sum, appear in the literature. -- Meni Rosenfeld (talk) 15:27, 23 February 2006 (UTC)