Talk:0.999.../Archive 5

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Archive 5

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This archive page covers approximately the dates between 2005-12-18 and 2006-02-11.

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1 Rules of engagement
2 Infinitesimals
3 Explanation for removal of good-faith insertion
4 The debate resumes - Problems with Rasmus's Proof
5 Blah, blah, blah...
6 Rasmus' proof in all its glory
7 Rasmus' proof and its error
8 Proof that 0.999... < 1
9 What is wrong with the joke?
10 What is wrong with this discussion?
11 Take it from the top, shall we?
12 A few points to consider.
13 Levels of hypocrisy.
14 Apples and bannas: not enough explanation in the concept

Rules of engagement

Dear all,

I assume all of us are well-meaning people, wanting a good result for this article. That makes me ask the following favor to anonymous contributors:

Please make an account. It is not productive for us to deal with a person who always uses a different IP address. It is impossible for us to keep in touch with you this way. Making an account will take you five secods. Just choose an imaginary username which has nothing to do with your real name, and a password. No more. But so much gained.

Please sign your posts. Use four tildas for that, like this: ~~~~. I would really, really ask you to give it a try. You leave a lot of unsigned comments and nobody can tell which is what you wrote, and which is somebody else, and which is another anonymous user.

I truly appreciate you taking your time to read all this, and do us a the small favors I asked which will take you a very small amount of time (not infinitesimal, but close :) and will make it so much more pleasant for us to have constructive discussions. Sincerely, Oleg Alexandrov (talk) 06:04, 18 December 2005 (UTC)

As an aside, a discussion about this very topic has been hotly debated by an on-line society for people with a high IQ. Keep it going SM! —Preceding unsigned comment added by 68.148.229.166 (talk • contribs) 06:30, 2005 December 18

What?! You mean Hardy is discussing this without me?! How can this be? I too have a high IQ (over 140) -not that it means anything. The IQ concept was developed by a human who I believe had to wipe his arse every time he had a BM. Do you think Hardy wipes or washes? I wash only. Toilet paper is for those who have bad hygienne and stinky butts. Although I don't care to0 much for Islam, this is something we could learn from them. Wait, I think the ancient Greeks invented the bidet if I am not mistaken? Oops, I think Melchoir is going to censor this when he wakes up. Hopefuly a few people will get to read it before he does and have a good laugh if nothing else. Not that it's any of my business but I am infinitesimally (singular ONLY) curious, do you wash or wipe Oleg? 71.248.130.143 14:14, 18 December 2005 (UTC)

I am not going to argue about infinitesimal. As far as I am concerned, this (notice I will not even consider the plural form - it makes no sense whatsoever to me) does not exist and you most certainly cannot prove that x+x+x+... < 1 for an infinite number of terms in x unless x is zero. You can only show in a similar manner to Rasmus that n*10^(-n) < 1 using induction. Now as I stated, one can show that 0.999... < 1 and that 0.999... = 1 depending on how you approach the proof. I believe that this anomaly exists because 0.999... is not a finitely represented number. This is a problem with the decimal system and all other radix systems. What this means is that the Archimedean property applies only to reals that can be finitely represented. Rasmus's proof is no better than the induction proof that 0.999... < 1 since it uses a result of induction to arrive at the conclusion that nx < 1. Rasmus justifies his argument by stating that because the Archimedean principle cannot be applied, x must be zero. However, I maintain that the Archimedean property can only be applied to finitely represented numbers. For x > 1/n, n can take on the value of a suitable natural number but not infinity. To use the fact that 0.999... has an upper bound in a proof such as Rasmus's defeats the purpose. So what should be believed? I think that 0.999... should be considered less than 1 because it has to be considered in the context of the decimal system. If the full extent of 0.999... were known, there would be no problem with the Archimedean property or any of its corollaries. 158.35.225.229 18:49, 21 December 2005 (UTC)

Infinitesimals

I have not yet given up on the "infinitesimal" proof, especially since anon agrees there are no infinitesimals in the field of real numbers. (Compare the 0:16 post of 18 December 2005. By the way, meaningful concepts of infinitesimals in larger number sets can be found in Winnig Ways for your Mathematical Plays, part 1, which is written on a rather basic level.)

0.999... and 1 are real numbers. Let x = 1-0.999... be their difference. Note that I do not care whether I can give a decimal representation for x. Note also that I do not make claims about the existence of numbers between 0.999... and 1. All I claim is that I can subtract, and that the difference of two real numbers is again a real number. And that is due to the fact that the real numbers are a field. By definition, a number y ≠ 0 is an infinitesimal if every sum |y|+...+|y| of finitely many terms is less than 1, no matter how large the finite number of terms. We agreed such a thing does not exist in the reals. Now form any sum |x|+...+|x| of finitely many terms, say n terms. Obviously, |x| = 1-0.999... < 1/n. Thus, |x|+...+|x|<1. So if x were greater than 0, it would satisfy the definition given above. That can't be, since we agreed that no real number is an infinitesimal. Our only way out is x=0. Thus, 0.999...=1.

If there are problems with this proof, please be precise in denoting them.--Huon 00:00, 20 December 2005 (UTC)

The proof would be flawless if your definition of infinitesimal is true. Only problem is it is not true because it does matter how large the finite number of terms become. If you feel comfortable that the sum of these terms will always be less than 1, how is it that you do not feel the same way about the sum of 9/10+9/100+9/1000+... ? I can make the same statement here, i.e. for finitely many terms, this sum will always be less than 1. So what?! 71.248.130.208 02:54, 20 December 2005 (UTC)

Hmmm, I had sort of given up on this discussion, but one last try: Let x = 1-0.999... as above. Consider the set S = {x, 2x, 3x, 4x, ...} = {nx|n in N}. (If you don't agree with me setting x = 1-0.999... [1], just consider the set S = { (1-0.999...), 2(1-0.999...), 3(1-0.999...), ... } = { n(1-0.999... ) | n in N} instead ). Could you answer these questions?

Does S have an upper bound?
Does S have a least upper bound?
If S has a least upper bound, what is it? If you can't give an exact answer, can you give an interval (ie. 0.5 < sup S < 1)?
If 1-0.999... != 0, then 1/(1-0.999...) must be a real number. Can you describe the properties of 1/(1-0.999...)? For instance is there any natural number n, so that 1/(1-0.999...) < n ?

Rasmus (talk) 07:23, 20 December 2005 (UTC)

Answers:

 1. S does not have an upper bound therefore it cannot have a least upper bound.
    So questions 2 and 3 are not relevant.

 4. 1/(1-0.999...) is a real number. Properties: all we can say is that it is a
    very large indeterminate number comparable with infinity. There is no natural
    number n, so that 1/(1-0.999...) < n.

So now you are going to conclude that since 4 is true that 1/(1-0.999...) is not a real number - yes? What about 1/(3.15-pi)? Is there a natural number n so that 1/(3.15-pi) < n ? 158.35.225.231 13:09, 20 December 2005 (UTC)

All members of S are of the form nx. Consider one such member, nx. As we saw a month ago, we can choose a natural number

m > log 10 (n)

, and use the fact that $0.999...>\sum_{i=1}^{m}\frac{9}{10^i}$ to show that $x = 1-0.999... < 1 - \sum_{i=1}^{m}\frac{9}{10^i} = \frac{1}{10^m} < \frac{1}{10^{\log_{10} (n)}} = \frac{1}{n}$ , and thus $nx < n \frac{1}{n} = 1$ . So all members of S are less than 1, yet you claim it has no upper bound?

As for 4, you claimed here that you accepted Planet Maths definition of the Archimedean property [2] (or was that another anon?). You don't feel this is a contradiction? Planet Math claims: "Let xbe any real number. Then there exists a natural number n such that n>x".

And finally, of course there is a natural number n so that 1/(3.15-pi) < n. pi<3.142, so 1/(3.15-pi) < 1/(3.15-3.142) = 125.

Rasmus (talk) 15:28, 20 December 2005 (UTC)

You have just proved that 0.999... < 1: nx < n*1/n = 1 => x < 1. If x is a real number greater than 0, there exists a natural n such that 0 < 1/n < x or nx > 1. So if x = 0 (which is what you would require for having 0.999... = 1) then no n exists such that nx > 1. Hence x must be greater than 0 and if x is greater than 0, then 0.999... must be less than 1. Now do the right thing and delete this garbage article. [User:158.35.225.231|158.35.225.231]] 17:35, 20 December 2005 (UTC)

Could you clarify that argument please? In the above I showed how for all natural numbers n, nx < 1. If you accept that "If x is a real number greater than 0, there exists a natural n such that 0 < 1/n < x or nx > 1", the conclusion must be that x is not a real number greater than 0.

You also didn't comment on the Archimedean property (are you the same person as 192.67.48.22?)

Rasmus (talk) 22:28, 20 December 2005 (UTC)

You showed nx < 1. The Archimedean property says there exists an n s.t. nx > 1. There is no x that satisfies nx < 1 and nx > 1. So how do you reach the conclusion that x = 0? You are looking only at nx < 1 and thus drawing the conclusion that x must be zero? Okay, let me try to understand what you are saying:

The Archimedean property shows the relationship between a natural number n and a number x greater than 0 such that nx > 1. This means that x and n must be greater than zero. Your proof demonstrates that a number x and some natural number n have the property that nx < 1. The only n that satifies this is n=0 for otherwise x must be zero. How do you associate the Archimedean property with your proof? They both state different facts. So what I am trying to say is this: if you are to draw any conclusion that is backed by the Archimedean property, then your proof must result in a form that resembles it, i.e. nx > 1 and not nx < 1. You cannot arrive at the conclusion that nx < 1 and then state by the Archimedean property that x is not a real number greater than 0. By demonstrating that x (1-0.999...) < 1/n for any n, you have proved conclusively that x is greater than 0 because for whatever 1/n you give me, I can always find an x that is smaller. This x is greater than zero and sounds very real to me. [User:71.248.134.241|71.248.134.241] 00:26, 21 December 2005 (UTC)

What I showed before were that for x=1-0.999... and all natural numbers n: nx<1. (I actually only wanted to use it for showing that S had an upper bound, since you had earlier rejected the application of the Archimedean property). Since the Archimedean property state that for all real x>0, there exists a natural number n, so that nx>1, we have a contradiction unless x is not a real number greater than 0.

I can't make much sense of your last argument. You claim that "x (1-0.999...) < 1/n for any n" => "x is greater than 0"? I assume the parenthesis is just a clarification and not a multiplication, so that it is actually (for all n in N: x < 1/n) => (x > 0) ? Your argument for this seems to imply that you can change the x as you go?! Anyway x=-1 (or even x=0) is a counterexample, which, frankly, you ought to have been able to see for yourself.

Rasmus (talk) 07:32, 21 December 2005 (UTC)

Fine. I see your argument now. It's always been confusing because for any 1/n, I can always find an x that is smaller but not zero. In an earlier discussion, you maintained that the induction proof was incorrect because it does not show P(infinity). Do you realize that one can say the same to you regarding this argument? You may say that the lowest x one can find is zero but then you are assuming P(infinity) is true. So although your argument is valid, you have not shown P(infinity). It seems to me that one can show equally well by induction that 0.999... < 1 and using your method that 0.999... = 1. How can 0.999... be both less than and equal to 1? This is strange... 158.35.225.229 13:14, 21 December 2005 (UTC)

Well, the difference is that I don't need to go to the limit. To use the Archimedean property, I only need to show that for all finite natural numbers n: nx<1. I do not need to show that " $\infty x < 1$ " (whatever meaning one would assign to that statement). Rasmus (talk) 14:07, 21 December 2005 (UTC)

One can say exactly the same for the induction proof, i.e. only need to show that 0.9999xn < 1. Same thing. Let me get one thing straight: you are also saying that if the Archimedean property does not apply, then x cannot be a real number, right? If your answer is 'yes', then the Archimedean property only applies to finitely represented reals in any radix system. 0.999... is not finitely respresented. 158.35.225.229 14:20, 21 December 2005 (UTC)

Rasmus is going to say that

the Archimedean property is a property for the entire set of numbers, and it applies.
x is indeed a real number, but not one that is greater than 0.

After all, what we are trying to show is just x=0. If you are now willing to sacrifice the Archimedean property for your brand of "real" numbers, you will probably agree that yours are not what mathematicians usually call the real numbers.

Concerning the definition of infinitesimals I gave above: That was the Wikipedia definition; I just copied it. If you don't believe that definition to be correct, look it up in, say, Winnig Ways.

Finally, of course 0.999...9 with a finite number of nines is less than one - by 10^{-n}, if n is the number of 9's. Now if you truly were going to use a limit argument for the case of an infinite number of nines, then the difference between 1 and .999... would have to be $10^{-\infty}$ , whatever that is. I personally do not endorse the following reasoning, but you might still find it interesitng: In order to show that x:=1-0.999... is an infinitesimal or zero, I can also show that x+x+x+... < 1 for an infinite number of terms: For every natural n, n*10^{-n} < 1. Thus, by your own methods, $\infty\cdot 10^{-\infty}<1$ . Thus, even with a stronger (and more strange) definition and with your methods of reasoning, x is an infinitesimal or zero, and infinitesimals don't exist. Thus, x=0 and 0.999...=1. What now? If you still doubt that x is either zero or an infinitesimal, please give a definition of infinitesimal you are willing to accept (keeping in mind that in the reals, there are no infinitesimals). --Huon 17:15, 21 December 2005 (UTC)

We agree that I can't show x+x+x+... < 1 for infinitely many summands unless x=0. But I definitely can "show" that (1-0.999...)+(1-0.999...)+(1-0.999...)+... < 1 for infinitely many summands. Let's do it step by step:

1-0.9 = 0.1 < 1
(1-0.99)+(1-0.99) = 0.02 < 1
(1-0.999)+(1-0.999)+(1-0.999) = 0.003 < 1 ...

Similarly, for every n, the sum of n terms of the form (1-0.999...9) (n nines) is less than 1. Thus,

(1-0.999...)+(1-0.999...)+(1-0.999...)+... < 1 for infinitely many summands (using methots not endorsed by me).

Thus, we have 1-0.999... = 0. To be precise, this "proof" is not mathematically rigorous (that's why I employ all these quotation marks), but it is just as good as the "induction proof" claimed to show that 0.999...<1. If one of these "proofs" is correct, then so is the other. Thus, in a way, I have disproved the induction proof, since using its methods leads to a contradiction. --Huon 19:42, 21 December 2005 (UTC)

I am not sure this method works but this is not relevant to what I said. Anyway, I agreed one could prove this but I stated that it is a result of induction. I said that Rasmus's proof is also a result of induction. 158.35.225.229 19:51, 21 December 2005 (UTC)

Actually, none of the proofs here are using induction. Induction is a special technique to show that a statement is true for all natural numbers. An induction-proof is easily recognized by being split into two parts: The basis (showing that the statement is true for n=0) and the inductive step (showing that if the statement is true for n, it is also true for n+1). Neither mine, nor your or Huons have this form.

As for infinitesimals, they are not defined by any infinite sums. Rather they are defined by the property that any finite sum is less than 1. Ie. if for all natural numbers n, nx<1, we say that x is an infinitesimal. But it is just a name, if you don't like it, we can call them for very-small-numbers. We also found out that if 0.999...<1 then 1/(1-0.999...) is a ~~illimited number~~ very-large-number. The existence of very-small- and very-large-numbers is in contradiction with the Archimedean property and the Least-upper-bound-property. What we call the real numbers is (uniquely) characterized by being a complete ordered field that has the LUB-property. You can of course define another set of numbers without the LUB-property and choose to call them the real numbers (but then you need to convince everybody else to follow your naming-convention...). Let us take a look at your options:

You can follow Fred Richman [3] and use the Decimal Numbers. Then by definition 0.999...<1, but you not only lose LUB-property, you also lose negative numbers (since 0.999... + 0.999... = 1.999... = 1 + 0.999..., 0.999... doesn't have an unique additive inverse) and division (since there is no Decimal Number x, so that x(1-0.999...)=1).

You can extend the real number field with ~~infinitesimals~~ very-small-numbers (using the transfer principle). The only important property you lose is the LUB-property (and thus the Archimedean property). Of course you cannot express these numbers using decimal numbers, and most people don't like the concept of very-small-numbers. Also, even here there is no real reason not to define 0.999...=1. But these numbers are actually rather interesting. They are called the hyperreals and are the subject of non-standard analysis.

Rasmus (talk) 22:17, 21 December 2005 (UTC)

Actually all the proofs here use induction. You have also used induction whether you like it ot not. How did you arrive at x < 1/n ? You started by assuming a finite sum and then continued to show how it is always less than 1. This is true and it proves that 0.999... is always less than 1. However, you took this result and then tried to explain it away with the Archimedean property. You are defining real numbers using the Archimedean property. What you don't seem to understand is that the Archimedean property does not allow for very small numbers or very large numbers. It just so happens 0.999... is a number that is misunderstood because it is very close to 1. However in the decimal system, there is no way to represent many numbers exactly, so you resort to the LUB property to reach conclusions about numbers. In my opinion it is just as easy to have 0.999... < 1. You do not lose any of the properties you mentioned. Look, you could easily define the real number system in terms of the decimal system if you were to use finite representation of numbers. In fact, this is how we use the decimal system. By doing so, all the properties of the real numbers hold, including very small and very large numbers and the Archimedean property.71.248.139.119 23:01, 21 December 2005 (UTC)

Rasmus and I do not use induction, since we do not use that the result holds for n in order to show it is true for n+1. Rasmus arrived at x<1/n by an explicit calculation; he did not use x<1/(n-1) in order to show it. (By the way, x was taken as 1-0.999... How does x<1/n imply 0.999...<1 ?)

Using only finite decimal representations as numbers sacrifices several properties:

Without division (such as 1/3, which has no finite representation, if I understand you correctly), the reals are no longer a field.
Completeness is also lost, since there are Cauchy sequences of finitely represented numbers whose limit has no fintite representation.
That's not one of Rasmus' properties, but surely an orthogonal triangle with two sides of length one should have a third side whose length is again a real number? After all, to the ancient greek mathematicians numbers were objects of geometry, be it lengths, areas or volumes.

Finally, then our whole discussion would be rather empty, since 0.999... is not finitely represented and would not even be a real number (whether it is less than 1 cannot be answered using a set which does not even contain 0.999...). Probably I misunderstood the statement about "defining the real number system using only finite representations of numbers"; please clarify it. --Huon 00:13, 22 December 2005 (UTC)

Very well. I concede that Rasmus's proof is quite solid. I can't argue against it. I still don't think you can rule out the design of the decimal system contributing to these anomalies, i.e. you have Rasmus's proof on the one hand and a simple proof by induction on the other hand that says exactly the opposite. Maybe you should include both proofs in an article about whether 0.999... equals 1 or not. Remember the decimal system is a model of the reals. 71.248.139.119 01:39, 22 December 2005 (UTC)

Explanation for removal of good-faith insertion

An anonymous IP added a proof idea in the "Elementary" section that essentially duplicated a proof in the "Advanced section". While a good-faith edit (thanks), it doesn't seem to offer any improvement. The "squeeze play" idea is a good one, and does sometimes appear in informal explanations. The point here is that we already cover the same ground more rigorously. --KSmrq^T 18:56, 10 January 2006 (UTC)

The debate resumes - Problems with Rasmus's Proof

The article is still biased and incorrect. It shows only one view of this problem - Rasmus's proof that 0.999... = 1. This is an induction proof contrary to what Rasmus states. Rasmus also states that the difference between his proof and the opposing induction proof is that he does not need to run the limit to infinity. Well, the opposing proof does not require this either. To say that 0.999... > Sum (i=1 to m) 9/10^i is a result of induction. Is this still true if we run m through to infinity? The answer is no. All of Rasmus's remaining proof is based on this first induction result. Seems like Rasmus's proof is not as solid as once thought. I am inclined to have 0.999... < 1. 158.35.225.229 18:55, 11 January 2006 (UTC)

That 0.999... is greater than $\sum_{i=1}^m\frac{9}{10^i}$ for all natural numbers m might indeed be proved by induction; one can also prove it directly. I assume Rasmus believed it to be obvious. And indeed this result becomes wrong if we proceed to the limit for m tending to infinity. But Rasmus need not do that. On the other hand, the "opposing proof" states quite analogously that $\sum_{i=1}^m\frac{9}{10^i}<1$ for every finite m (which is undisputed), and then "shows", supposedly by induction, that the same must be true for the limit - just the argument which we now see to be false. That "proof" makes an even more general (false) claim; it can be found in the Archive.

Still, this gave me just another idea for a proof that 0.9999...=1. I will stop making any assumptions about the nature of 0.9999... but the following:

For every natural number m, $\sum_{i=1}^m\frac{9}{10^i}<0.9999...$ . Here I make no claims about limits.

Now we use two facts of analysis:

Let a_m be a convergent sequence of real numbers with $\lim_{m\to\infty}a_m=a$ . Let b be any real number with a_m < b for all m. Then $a\leq b$ . Note that not necessarily a<b; a proof should be found in almost every undergraduate textbook on analysis.
We have $\lim_{m\to\infty}\left(\sum_{i=1}^m\frac{9}{10^i}\right)=1$ . This is a statement about limits only; it does not depend on (or even use) the definition or properties of 0.9999... A proof would be a slight generalisation of the proofs used to show convergence of the geometric series.

Now let us combine these facts. Take $a_m=\sum_{i=1}^m\frac{9}{10^i}$ , b=0.9999... Then by using the second fact of analysis, the sequence of the a_m's converges to a=1. Now using the first fact of analysis and the assumed property of 0.9999..., we conclude that $1\leq 0.9999...$ . Since 0.9999... is not greater than 1, we conclude 1=0.9999... --Huon 14:10, 20 January 2006 (UTC)

By the second fact on analysis, the sequence of the a_m's converges to b=1 and not a=1 as you have written. So you end up concluding that $0.999...\leq1$ and not $1\leq 0.999...$ . So in essence you were unable to show that it is equal. Finally since we can show by induction that $\sum_{i=1}^m\frac{9}{10^i}<1$ for any m (except infinity), it would be correct to say that $0.9999... < 1$ . 68.238.101.241 22:17, 22 January 2006 (UTC)

I'm sorry, but I can't follow any of your arguments. I defined a to be the limit of the a_m's. Since you agree that the a_m's converge to 1, we have a=1. By comparison, b is a number which is larger than all the a_m; 0.9999... satisfies that condition.

On the other hand, I agree to $\sum_{i=1}^m\frac{9}{10^i}<1$ for any m (except infinity). But why does that imply anything about 0.9999..., which has an infinite number of nines? By comparison, I could state: $\sum_{i=1}^m\frac{9}{10^i}$ is rational for any m (except infinity). Is 0.9999... still rational? If so, please give numerator and denominator; if not, then why should "being less than 1" be preserved if "being rational" is not?

By the way, please create an account. --Huon 19:51, 23 January 2006 (UTC)

No, since the a_ms converge to 1, you have that b=1, not a=1. You do not know what the last a_m is, do you? Rationality is preserved with 0.999... < 1. There is no problem with any facts of analysis. 68.238.108.20 11:52, 24 January 2006 (UTC)

Excuse me? a is defined to be the limit of the a_m's. Thus, when the a_m's converge to 1, we have a=1. Or, as a formula: $a:=\lim_{m\to\infty}a_m=1.$ Why should a not be 1? Why does b even enter this part of the discussion?

The "last a_m" comment is rather confusing. There is no "last a_m"; how should I know it? Why should I even want to know it?

Finally, if 0.9999... really is rational, give numerator and denominator. --134.76.82.144 13:13, 24 January 2006 (UTC)

Oops, I was in a hurry and forgot to sign in; that was me. --Huon 15:07, 24 January 2006 (UTC)

It matters a lot that b enters the discussion because you have made several erroneous assumptions based on b being the LUB: you stated that a_m < b for all m. Now since b=1 it must follow that any of the a_ms are less than 1 for a_m < b. You are getting yourself horribly muddled up. I did not say 0.999... is rational. I said there is nothing irrational about 0.999... < 1. Please do not misquote me. 158.35.225.231 17:35, 24 January 2006 (UTC)

I made no assumptions based on anything being an LUB. I don't even talk or care about LUBs here. All I assumed about b was that it is greater than all of the a_m's. I did not claim b=1; but indeed 1 would fit that description, and if I had chosen b=1, then I would have proved 1 being less than or equal to 1, which is correct. But 2, for example, would also fit that description, and if I had chosen b=2, then I would have proved 1 being less than or equal to 2 (which is, of course, also correct). But 0.9999... is also greater than all of the a_m's (or do you doubt that? It was the one property of 0.9999... I assumed). Thus, b=0.9999... is a valid choice. And then I prove that 1, which is the limit of the a_m's, is less than or equal to 0.9999...; that's what I wanted to show.

On the other hand, concerning rationality: You said, more or less: "All of the a_m's are less than 1; thus, 0.9999... is also less than 1." That is the best "proof" we have for 0.9999... being less than one, but it contains a serious gap when you say that a property which is shared by all the a_m's (being less than 1) must also be shared by 0.9999... Now I chose another property all of the a_m's have in common - being rational numbers. By an analogous argument, 0.9999... would also have to be a rational number, would it not? If you claim this kind of reasoning is sound, then you claim 0.9999... to be a rational number, and you should be able to give numerator and denominator. If, on the other hand, you do not believe that this kind of reasoning is sound for the property "being a rational number", then why should it be true for "being less than 1"?

I am sorry for misinterpreting your remark about rationality of statements instead of numbers, but I had assumed it to be an answer to my argument about rational numbers (which, maybe, was a bit too short itself).

Finally, I would ask you to be more precise. I made erroneous assumptions? Maybe, but which ones? You say b=1? Why, if I never stated it to be so? --Huon 23:31, 24 January 2006 (UTC)

Actually you wrote: "Let b be any real number with a_m < b for all m." So you made an assumption that b is a LUB. In your last post you claim that b=0.999... is a valid choice. It is not a valid choice because you do not know what it is. I think you need to leave analysis aside because it is confusing rather than helping you. You also write: "By an analogous argument, 0.9999... would also have to be a rational number, would it not?" To which I respond: No, this reasoning is in error, 0.999... would not have to be a rational number. Just consider the set of all rationals less than srqt(2). Would sqrt(2) have to be rational? Analysis is full of errors and contradictions. Rather than change it, most prefer to keep it along with the confusion and difficulties it poses for all learners. Math Professors (even those who have taught analysis over 30 years) are still unable to demonstrate flawless proofs. How is it that poor students who are complete novices are expected to learn and master this nonsense in just one course? Almost every student I have talked to who has passed an analysis course is still unable to completely grasp what it's all about. Most will gradually forget everything after the course because they 'learn' it parrot-fashion. Just ask a graduate if he/she can tell you exactly what the completeness property is - you will be surprised to find that most will not know. In fact many professors do not quote it correctly either. 158.35.225.231 15:35, 25 January 2006 (UTC)

Thanks for these detailed remarks. First of all, b need not be the LUB. It is an upper bound, but not necessarily the least. And indeed 0.9999... is an upper bound for the sequence of the a_m's. Or do you doubt that? If it were not, there would be a number 0.9999...9 (with a finite number of nines) which would be larger than 0.9999...! For this proof, I don't have to know what 0.9999... is; all I need to know is that it is greater than all of the a_m's.

Now you seem to agree that just because all of the a_m's have a certain property (being rational numbers, in my example), 0.9999... need not share this property. Thus, even though all of the a_m's are less than one, why should 0.9999... be? In all these discussions I cannot remember a single argument for this step.

Finally, concerning your remarks about analysis, I agree that it's an intricate subject and that mathematicians, professors included, may sometimes make flawed "proofs". But that's rather off-topic here (unless you find flaws in my proofs), and I invite you to discuss it on my talk page instead. Huon 17:55, 25 January 2006 (UTC)

I'm going to try and break user:Huon's proof down into a few steps to try and clear up the misunderstandings, because it makes perfect sense to me:

Lemma 1: Let a_m be a convergent series of real numbers with limit a. Let b be a real number such that a_m < b for all m, ie. b is an upper bound for the set {a_m} (but not necessarily the LUB). Then $a\leq b$ . (Proof omitted, but definitely standard university-level).

Lemma 2: $\lim_{m\to\infty}\left(\sum_{i=1}^m\frac{9}{10^i}\right)=1$ , ie. the limit of the series (0.9, 0.99, 0.999, ...) is 1. (Again, proof omitted but again university-level at most.)

Theorem: 0.999... = 1. Proof:

Let $a_m=\sum_{i=1}^m\frac{9}{10^i}$ , so $a=\lim_{m\to\infty}\left(\sum_{i=1}^m\frac{9}{10^i}\right)=1$ (By Lemma 2). Also let b=0.999...

Then a_m < b for all m. (As user:Huon has pointed out, b is not necessarily the least upper bound.)

By Lemma 1, $1=a\leq b$ .

The only tricky bit here is proving that $0.999...\leq 1$ , and to be frank I can't quite work it out myself and end up proving myself in circles, and about the only proof that seems to work is the rather pointless "by inspection". I think the whole discussion about stating that it's rational or not is based on the fact that while it's easy to see that a_m < 1, there's nothing to say that 0.999... should possess any of the properties common to the a_m, be it "rational" or "less than 1".

Oh, damn. Okay, Huon, here's a problem in your proof: As far as I can tell, you can only make your statement that a_m < 0.999... by the same reasoning that $0.999...\leq 1$ - by inspection of the digits. Explain to me how else that works and the proof works.

But then again, I'm still trying to work out how 0.999... has any meaning unless it's defined as the infinite series $\sum_{i=1}^{\infty}\frac{9}{10^i}$ which, as far as I can recall, is defined as the limit in any case. How do you get around that? (Bleh, in trying to clarify something I've gotten myself even more confused)Confusing Manifestation 18:44, 25 January 2006 (UTC)

ConMan, of the steps you want clarification on, neither requires inspection of digits. They both rely on the general principles, which are theorems in any order topology:

If a sequence is eventually >= some number, so is its limit.
If a sequence is eventually <= some number, so is its limit.

Does that help? Melchoir 19:23, 25 January 2006 (UTC)

The history of this series of proofs is that an anon disagreed with defining $\sum_{i=1}^{\infty}\frac{9}{10^i} = lim_{n \rightarrow \infty}\sum_{i=1}^{n}\frac{9}{10^i}$ . Various proof using constructions of the real numbers (Dedekind cuts etc.) had also failed to convince her. So I tried a proof that relied on as few properties of 0.999... and of the real numbers as I could manage. That proof used the Archimedean property rather than the property of limits that Huon uses; but otherwise the principle used is similar: If we refuse to define an infinite sum as the limit of the partial sums, we don't really have much else to fall back on. We might define 0.9999... = 2, and we wouldn't really have much of a contradiction. However, Huan and I argue, any meaningful definition of 0.999... would need to have $\forall n \in N : \sum_{i=1}^{n}\frac{9}{10^i} < 0.999...$ and $0.999... \le 1$ ; and from that we conclude that 0.999... = 1. Rasmus (talk) 20:49, 25 January 2006 (UTC)

Indeed I don't prove that 0.9999... is either larger than all of the a_m's or less or equal 1. While I explicitly stated that I assume the first property, I was a bit sketchy on the question of 0.9999...>1. Rasmus sums up my position quite nicely. I believe the anon would disagree with every definition of 0.9999... (possibly claiming we "cannot know what it is"); thus Rasmus and I try to give proofs using not a definition, but only "obvious" properties of 0.9999... Yours, Huon 21:41, 25 January 2006 (UTC)

Rasmus wrote: "If we refuse to define an infinite sum as the limit of the partial sums, we don't really have much else to fall back on." I completely agree. It is this definition that I cannot accept. In my understanding an infinite sum is not computable. All we can do is determine whether or not an infinite sum has a limit. To define an infinite sum as the limit of the partial sums is completely illogical. Whatever 0.999... is does not matter that much as long as we know it is less than 1. If you accept Dedekind cuts, then you should have no problem seeing that 0.999... < 1 by 'definition'. So all we can tell about 0.999... is that it is less than 1. Nothing else. 71.248.144.149 22:02, 25 January 2006 (UTC)

Unfortunately, your understanding of infinite sum seems to be at fault. Please define what, for you, an infinite sum is. If you cannot do that, it might be wisest to consider an infinite sum as a series, and thus the limit of a sequence of finite sums. By the way, I also see no reason at all why 0.9999... should be an infinite sum in the first place. And unless you want to state that there is a number 0.9999...9 with finitely many nines which is greater than 0.9999..., my proof holds no matter what 0.9999... really is.

Concerning Dedekind cuts: If I understand you correctly, you mean that the Dedekind cuts given by the pairs of sets $(-\infty,\alpha],(\alpha,\infty)$ and $(-\infty,\alpha),[\alpha,\infty)$ respectively correspond to different real numbers. That is not the case. In effect, Dedekind cuts are used as follows: Let X and Y be sets of real numbers such that the union of X and Y contains all real numbers, and that for elements x, y of X and Y, respectively, x<y holds. Then there is exactly one real number z such that for all elements x of X and y of Y we have $x\le z\le y$ . That number z would be the number corresponding to the Dedekind cut (X,Y). It does not matter wether z is an element of X or of Y. You can check that the two Dedekind cuts above both correspond to the same

α

If you are not convinced, have a look at the Dedekind cut gven by the sets of negative and nonnegative numbers, and the Dedekind cut given by the sets of nonpositive and positive numbers. At least one of them should correspond to 0. Which one? And to what does the other correspond? --Huon 23:16, 25 January 2006 (UTC)

The smaller set you described (X) does not include alpha, the larger one (Y) does. There is no number such that x <= z <= y for there are no numbers between X and Y. 158.35.225.231 18:45, 26 January 2006 (UTC)

So you say that the sets $(-\infty,\alpha],(\alpha,\infty)$ do not form a Dedekind cut? Of course there is a z such that x <= z <= y; such a z need not lie somewhere between X and Y; it may be an element of one of them. In the example given above, X={x real with x <= alpha}, Y={y real with y > alpha}, alpha does satisfy just that: We simultaneously have x <= alpha for all x in X and y > alpha for all y in Y, both by definition of X and Y, respectively. Thus, x <= alpha < y, which is even more than required. Since every real number is either greater than alpha (and thus an element of Y) or not greater than alpha (and thus an element of X), the sets X and Y given above also form a partition of the reals. Thus, (X,Y) is a Dedekind cut, and alpha takes the role of z.

Besides, if you do not accept Dedekind cuts where the "smaller set" includes alpha, then I seem to have misunderstood the remark about 0.9999...<1 by 'definition'. Maybe I miss something; please give more details. --Huon 19:10, 26 January 2006 (UTC)

You are saying the z is alpha. So if this is true, then (-oo,1] (1,oo) is the cut that represents 1. If you say this cut represents 0.999... also, then you are mistaken. 70.110.87.205 23:20, 26 January 2006 (UTC)

Why am I mistaken? And what other Dedekind cut represents 0.9999...? I am getting weary of asking for the same things again and again. Please be more precise; give more details. Please create an account. Is either so difficult? Besides, I still wait for any detailed challenge to my latest proof, as detailed by Confusing Manifestation. --Huon 13:58, 27 January 2006 (UTC)

This cut represents 1. So does (-oo,1) [1,oo). If you say this cut is the same for 0.999.., then you are assuming that 0.999... = 1. How do you arrive at this assumption? 158.35.225.231 18:57, 27 January 2006 (UTC)

I arrive at the assumption that 0.9999...=1 by the proof I gave at the beginning of this subsection. ConMan gave a more detailed version of this proof a little later, pointing out the one gap it contains: I did not explicitly state that I assume 0.9999... to be not greater than 1 (but on the other hand, nobody ever proposed 0.9999...>1).

Somehow, we seem to be running in circles. I am still waiting for the opponents of 0.9999...=1 to answer the following questions:

If you claim to have a proof that 0.9999... is not equal to 1, please give it in full detail. Above I pointed out where I saw a gap in previous attempts at a proof.
Give a detailed account why my proof above is not accepted.
Give a Dedekind cut corresponding to 0.9999...

Without either of the first two, I see no need to continue these discussions. Yours, Huon 22:14, 27 January 2006 (UTC)

The onus is on you to prove that 0.999... = 1, not the other way around. Neither you nor anyone else has been able to do this satisfactorily. As for a Dedekind cut for 0.999..., I could just as well say: (oo,0.999...] (0.999...,oo) - this corresponds to 0.999... Now if you can give a Dedekind cut for pi or e (Melchoir's Dedekind cut for pi is a joke), then I will give a better Dedekind cut for 0.999... 71.248.128.176 12:09, 30 January 2006 (UTC)

As I said before, I proved 0.9999...=1, and there still has been no detailed account of why that proof should be false.

Concerning Dedekind cuts, first of all I must admit that I was careless in reading up the definition; I missed the provision that the "smaller" set may not contain a largest element. Thus, when I wrote about (-oo, alpha], (alpha, oo) being a Dedekind cut, that is indeed false. Unfortunately, you seem to have fallen in line with my faulty notation (in retrospect I see you tried to warn me on this point). And since it does not really matter wether the "smaller" set does not contain a largest element or the "larger" set does not contain a smallest element, I won't change notation in mid-discussion.

Now on to your Dedekind cut. You will probably claim it is not the same as (-oo,1] (1,oo), but failed to give any arguments. Then we could form the intersection $(-\infty,1]\cap(0.9999...,\infty)$ . This intersection should be non-empty, since else we would have 0.9999...>=1. Does it contain elements besides 1? If so, how many? Give at least one example. Anything you could say about the properties of these elements would also be appreciated.

I still do not see why this Dedekind cut should lead to the result that 0.9999...<1 "by definition", as was claimed above. If I assume that 0.9999... and 1 are distinct, then I have found distinct Dedekind cuts, but if they are the same, then so are the cuts.

Finally, concerning Dedekind cuts representing pi or e, Melchoir's cut in the archive is not at all a joke. Melchoir explicitly shows why his partition of the rationals (he did not speak of Dedekind cuts of the reals, as I do here) is a Dedekind cut, and it is obvious that it indeed represents pi. The problem at that time seemed to be that the cut "says nothing about the value of pi". Indeed it does not, but why should it?

Please remember the entire discussion about Dedekind cuts (including my mistake) is beside the point. I gave a proof of 0.9999...=1 above. --Huon 14:45, 30 January 2006 (UTC)

So now that we agree that Dedekind cuts say nothing about the value of a number, we can continue to discuss your proof without these. As I stated, your proof is incorrect from the very first step, i.e. you are first assuming that sum (i=1 to m) 1/9^i < 1 for finite m and then using this to show 0.999... = 1. See, if the first step in your proof(in fact it is Rasmus's proof) were correct, the rest of the proof would be true. Unfortunately, you cannot assume this is true for finite m and then proceed to arrive at the result that 0.99... = 1. It is a contradiction.

As for your second proof above: Again your proof fails badly: In Lemma1 you write that a_m < b for all m. i.e it is an upper bound. Now if it is an upper bound it must at the very least be equal or greater than a=1. Then in step 2, you erroneously proceed to say: Also let b=0.999... Problem is that you have already assumed it is at least 1. So you have contradicted yourself. No need to go any further. 158.35.225.231 16:32, 30 January 2006 (UTC)

Concerning Rasmus' proof: Do you agree that all finite sums sum (i=1 to m) 9/10^i are less than one? I assume you do; if not, please correct me. Then why should 0.9999... share any properties with the finite sums sum (i=1 to m) 9/10^i? Especially why should it be a contradiction that all those finite sums are less than 1 and 0.9999... is not? 0.9999... obviously is no finite sum. Besides, I cannot see where Rasmus uses 0.9999...=1. He proves it, but in the proof he does not use it.

Now on to my own proof. I stated that b is an upper bound for the a_m's and that b=0.9999... You say that's a contradiction. Do you imply 0.9999... is not an upper bound for the a_m's? That there is an m for which 0.9999... with infinitely many nines is less than 0.9999...9 with m nines? The difference 0.9999...-0.999...9 = 0.000...09999... with m zeroes after the decimal separator followed by infinitely many nines does not look negative to me.

You seem to have gotten my proof backwards. First of all I claimed that 0.9999... is an upper bound, more or less "by looking at it", because the notion that a number with infinitely many nines should be less than one with only finitely many nines seems rather strange. It was the one property of 0.9999... I assumed. Then, I indeed reason that since 0.9999... is an upper bound, it must be equal or greater than 1.

By the way, I would only get a contradiction by claiming both b=0.9999... and b>=1 (as you said I did) if I knew that 0.9999...<1. I don't know that; else I would not set out to prove 0.9999...=1. Thus, your criticism is in effect one of circular reasoning (assuming 0.9999...>=1 in order to show it), not of self-contradiction. Of course, if you were right, my proof would still be faulty. Yours, Huon 17:57, 30 January 2006 (UTC)

First of all, you cannot claim that 0.999... is an upper bound because you start off not knowing exactly what it is and you cannot claim this more or less "by looking at it". What kind of math is this? You think you can tell by looking alone? Hmmm. I need to have my eyes checked I think... Secondly, Rasmus's proof uses the fact that 0.999... < 1, not 0.999... = 1 as you write. You seem to have gotten it backward. And finally, if 0.999... is an upper bound as you claim, then what makes you think it is equal to 1? After all, it is just one of the upper bounds... 158.35.225.231 19:29, 30 January 2006 (UTC)

Concerning the upper bound: Unfortunately, we did not agree on a definition of 0.9999... The article defines it as a recurring decimal, and thus as a limit. You disagree, bud did not provide another definition but "an infinite sum" (whatever that is), iirc even claiming we "cannot know" what it is. I had hoped we could agree on two properties of 0.9999...:

For every natural number m, 0.9999...>0.999...9 (with m nines).
0.9999... is not greater than 1.

Now you seem prepared to sacrifice either of these properties. If the first holds, 0.9999... is an upper bound, and thus >=1 (that's what I prove); if the second holds as well, then it must be equal to 1 (that conclusion seemed obvious to me).

What I meant by "looking at it" was that in order to be compatible with the usual ordering of finite decimals, 0.9999... would have to be larger than all of the a_m's, since those numbers share the first m digits after the decimal separator, and thereafter all of 0.9999...'s digits are greater than those of a_m. If, on the other hand, you truly mean by 0.9999... "some number with arbitrary properties except being less than 1", then it would indeed by definition be less than 1, but why should such a number be denoted by 0.9999... and not, say, by 0.1234567... (which would, to you, have equally unknown properties)? --Huon 09:45, 31 January 2006 (UTC)

Blah, blah, blah...

Quack, quack, quack... Revolver 22:27, 26 January 2006 (UTC)

Seems like the rest of the contributors on this site are all ducks like you? Do I hear an eagle anywhere? 158.35.225.231 19:31, 30 January 2006 (UTC)

Rasmus' proof in all its glory

I could not find a short, complete version of Rasmus' proof, so as a basis for detailed discussion I will repeat it in full detail. First of all, Rasmus also uses two properties of 0.9999...:

For all m, $0.9999... > \sum_{i=1}^m 9/(10^i)$ ,
0.9999... is not greater than one.

He arrives at both of these by inspection of the decimal representations, and he and I agree any meaningful definition of 0.9999... should satisfy these properties: [4].

Now let x be defined as x=1-0.9999... This is just to shorten notation; I could always write 1-0.9999... instead of x. By the second of the properties above, we have x>=0.

Now let n be any natural number, and let m be a natural number greater than log_{10}(n). Such an m exists by the Archimedean property. Then comes Rasmus' major formula: $x=1-0.9999...<1-\sum_{i=1}^{m}\frac{9}{10^i}=\frac{1}{10^m}<\frac{1}{10^{\log_{10}(n)}} = \frac{1}{n}$ . The first inequality is the point where the first of the properties from above is used.

Thus, for any natural number n, we have 0 <= x < 1/n. We conclude: 0 <= n*x < 1 for any natural number n. Thus, x must either be zero, or it is by definition an infinitesimal. Since by the Archimedean property the real numbers contain no infinitesimals, we can disregard that possibility. Thus, we finally conclude: 0 = x = 1-0.9999..., so 0.9999... = 1.

This proof makes no assumptions whatsoever on 0.9999... being less than 1 or not. But indeed one could turn it into a proof by contradiction: Let us assume x>0 (which amounts to 0.9999...<1). Then the calculation above shows that x is an infinitesimal. But there are no infinitesimals in the reals; especially x cannot be one. Thus, our assumption leads to a contradiction; it must therefore be false. Thus, x<=0 (and since we know by the second property that x>=0, we arrive at x=0 as desired). --Huon 10:23, 31 January 2006 (UTC)

Look at the first line: 0.999... is not greater than sum (i to m) 9/10^i if m goes to infinity. The major flaw with this proof is that it considers something different to 0.999... - it starts with a partial sum that is not the same as 0.999... Thus it is incorrect because Rasmus uses this as the starting point of his proof. Sorry, but you will have to come up with a valid proof to convince me. By definition, 0.999... < 1. Would 1 - [sum (i to m) 1/10^i as m->oo] be greater than 1 - 0.999... ? If you say 'yes' then you are indeed saying that 0 < 0 which does not make any sense. If you say 'no', then you contradict the proof. Either way you are trying to prove something that is entirely 'untrue' and has always been a load of rubbish. 158.35.225.231 14:13, 31 January 2006 (UTC)

I thought quite some time about your remarks, but I'm not sure I understand them. I'll go through them step by step:

What do you mean by "if m goes to infinity"? If you mean a limit, then I agree that 0.999... is not greater than lim (m->oo) sum (i to m) 9/10^i, but Rasmus' proof makes no mention of limits, nor does it use them implicitly. If, on the other hand, you mean "for some very large m", then I disagree; more on that later.
You say the proof considers something different to 0.9999... Indeed it does, but a connection between 0.9999... and the "partial sums" is drawn via the claim that 0.9999... is greater than all those sums. If 0.9999... has that property, then we can use the properties of the sums to deduce further properties of 0.9999...
You claim: "By definition, 0.999... < 1." If that were true, then indeed any attempt at a proof of 0.9999...=1 would be futile. But what definition do you use to arrive at this result? Please give a definition of 0.9999... If that definition uses the term "infinite sum", please also define what, to you, an infinite sum is. By contrast, I would be content to use the article's definition of 0.9999... as a recurring decimal and thus a limit, but you disagreed.
Would 1 - [sum (i to m) 9/10^i as m->oo] be greater than 1 - 0.999... ? The answer again depends on what "as m->oo" is supposed to mean. If the limit is meant, then the answer is no; if it means "for every m no matter how large it is", then the answer should be yes. Let me clarify why I have problems with the notation. Would 1 - [sum (i to m) 9/10^i as m->oo] be greater than 0? In the limit case, lim (m->oo) [sum (i to m) 9/10^i]=1, and the question becomes whether 1-1>0: Surely not. But for every m, no matter how large it may be, we have 1 - sum (i to m) 9/10^i = 1/10^m, which is just as surely greater than 0.

I am not quite sure what your main point of critique is. Either you mean that there is indeed some m for which 0.9999... is less than sum (i to m) 9/10^i. That could be clarified if you give a definition of 0.9999..., and the question would become one of which definition is best suited to have those properties we exect from 0.9999...

Or you mean that when for every m the sum (i to m) 9/10^i is less than 0.9999..., then the limit would also have to be less than 0.9999... This kind of reasoning is unsound, as can be seen by replacing 0.9999... by 1. If that was the problem you saw, I can discuss it in more detail, if you wish.

Maybe you mean something completely different. Then I obviously did not understand you at all; in that case please clarify your remarks and add more details. Yours, Huon 13:13, 1 February 2006 (UTC)

Definition of 0.999...:

 0.999... = 9/10 + 9/100 + 9/1000 + ....

1) There is no m for which 0.9999... is less than sum (i to m) 9/10^i.

2) Yes, I do mean the following: If for every m the sum (i to m) 9/10^i is less than 0.9999..., then the limit would also have to be less than 0.9999... - there is nothing unsound about this. In fact it demonstrates how *unsound* your argument is when you proceed to replace 0.999... by 1 for then you arrive at absurdities such as 0 < 0 or obvious contradictions. It points out serious errors in Rasmus's proof. 158.35.225.231 16:07, 1 February 2006 (UTC)

If I could interject I think I see the misunderstanding. The "for all m" would be clearer if it read "for any natural number m". The idea behind the first point is not to say anything about the limit of those finite summations as m tends to infinity but simply to state the property of 0.9recurring that it is greater than any of those finite summations.

As a logical consequence, 1 - (any of those summations) is greater than 1 - 0.9recurring. This is what justifies changing from 1-0.9recurring to 1-the partial sums later. If something is greater than 1-0.9recurring it must also be greater than 1-(any of the summations to a natural number). These summations are then shown to be less than some 1/n which shows that for any natural number n, 1/n is greater than 1-0.9 recurring. Ergo 1-0.9 is less than or equal to zero. And we already know it is not less than zero by point 2). So it must be equal to zero. So 0.9reccuring = 1 Mikekelly 18:13, 1 February 2006 (UTC)

Actually the idea is to show that it is true for any natural m as m approaches infinity. After all we are dealing with an infinitely represented quantity 0.999... To say that 0.999... is greater than any of the partial summations is obvious and meaningless. In fact the idea of m approaching ifinity has *everything* to do with it. If m does not approach infinity, then what are you talking about when you write 0.999... ? Ergo 1-0.9 is less than or equal to zero. We know it is not equal, therefore it must be greater than 0. Look, by the structure of the decimal system, 0.999... is less than 1. 158.35.225.231 20:25, 1 February 2006 (UTC)

I would make a comment about this statement above by 158.35.225.231:

"If for every m the sum (i to m) 9/10^i is less than 0.9999..., then the limit would also have to be less than 0.9999... - there is nothing unsound about this."

This is indeed unsound. To say that "it is true for any natural number m, therefore it is true even when m goes to infinity" is like saying, "every positive number 1, 2, 3, ... is finite, therefore positive infinity is finite". This is because what is true for finite m does not carry over to infinite m. Here's another example: 1 + 1/4 is rational, 1 + 1/4 + 1/9 is rational, 1 + 1/4 + 1/9 + 1/16 is rational... but $\sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}$ is not. -- KittySaturn 23:23, 1 February 2006 (UTC)

158.35.225.231 - Rasumus' proof does not involve limits. It does not need to. Maybe I could set it out in a slightly different way...

$x = 1 - 0.9999...$

We already agreed that 0.9999 is not greater than 1 so x is not less than 0. Thus also showing it cannot be greater than zero is sufficient to prove that x =1 which implies 1 = 0.9999...

Now to show it is not greater than zero, take any ε arbitarily close to zero. I will prove that x must be less than ε. This proves x cannot be greater than 0...

by archimedian principle $(\forall\ \epsilon\ > 0), (\exist\ m,n \isin\$ .{ natural numbers } $)$ such that ..

$\epsilon\ > \frac{1}{n} = \frac{1}{10^{\log_{10}(n)}} > \frac{1}{10^{m}} > x$ .

QED. Hope that makes sense..

Note

$1-\sum_{i=1}^{m}\frac{9}{10^i}=\frac{1}{10^m}$ Mikekelly 00:24, 2 February 2006 (UTC)

Kitty Saturn: Your analogy is not even comparable. There is no similarity between what I said and your analogy. The proponents of the idea that 0.999... = 1 base this false notion on the fact that the limit of sum (i=1 to m) as m approaches infinity = 1. This false notion is unsound, there is nothing unsound about my argument. Your epsilon-delta argument shows only that you learned analysis parrot-fashion and demonstrates that you really never understood anything. And I ask you again: If m does not approach infinity, then what are you talking about when you write 0.999... ? 71.248.131.234 11:54, 2 February 2006 (UTC)

My epsilon-delta argument does not rely on limits of the partial sums. All it relies on is that 0.999... is greater than any of the partial sums, allowing the step $\frac{1}{10^{m}} > x$ . If you do not understand that you do not understand the proof.

Seeing as I composed that formulation of the argument myself I am hardly parroting anything. Perhaps if you understand Rasmus' proof you would see that it does not rely on the limit of the partial sums. As it is, I am inclined to say the reason you do not agree with it is because you do not understand it. I have proved that x can be shown to be less than any value arbitarily close to zero. How then can it be greater than zero? Mikekelly 15:47, 2 February 2006 (UTC)

Rasmus' proof and its error

Rasmus starts his real argument for 0.999... = 1 with:

$x=1-0.9999...<1-\sum_{i=1}^{m}\frac{9}{10^i}=\frac{1}{10^m}$ .

I was almost fooled by this proof until I considered what would happen if m indeed were allowed to run through infinity and one accepts that 0.999... = 1. What happens is that it results in an absurdity: 0 < 0 = 0. Let's suppose that 0.999... = 1. Then 1-0.999... = 0 and $1-\sum_{i=1}^{m}\frac{9}{10^i}= 0$ (as m approaches infinity).

This is the evidence that this proof is blatantly incorrect. Crafty fellow that he is (or not, I don't know), Rasmus almost convinced me. However, I know that by definition 0.999... is less than 1. It is part of the architecture of the decimal system. It is not only idiotic to try to prove otherwise but also an exercise in futility.

71.248.131.234 12:12, 2 February 2006 (UTC)

I never claimed the inequality held for "m = ∞". I claim, however, that the inequality holds for all natural numbers m. Please make sure you understand the difference - it seems to be fundamental to this discussion. Rasmus (talk) 12:23, 2 February 2006 (UTC)

And I maintain that if the inequality does not hold for "m = ∞", your proof is null and void. Are you not interpreting the value of 0.999... to be equal to 1 based on the fact that 1 is the limit of 0.999...? If yes, then why do you dismiss any proof that needs to use this fact? All the fuss is about having 0.999... = 1, is it not? On the one hand, you quite readily feel comfortable accepting the limit is equal to the actual number whilst in your proof you decide to shy away from it. Look, every proof needs reality checks each step of the way. Your proof fails these checks misereably as described above. Please make sure you understand what reality checks are. 158.35.225.231 14:39, 2 February 2006 (UTC)

The inequality does not hold for "m = ∞", it does not need to. At no stage does the proof rely on 1 being the limit of the sequence of partial sums. I rely on 0.999... being greater than the partial sums to justfiy the step $\frac{1}{10^{m}} > x$ . If you do not understand that you do not understand the proof. Mikekelly 15:47, 2 February 2006 (UTC)

Proof that 0.999... < 1

For any natural m, $\sum_{i=1}^{m}\frac{9}{10^i} < 1$ .

Simple Proof by Induction: The result is true for m = k. The result is true for m=k+1. Conclusion: the result is true for any k. 158.35.225.231 14:52, 2 February 2006 (UTC)

Please point out exactly where you think I use the inequality for "m = ∞". There is a subtle, but very important difference, between something being true for all natural numbers and something being true "in the limit". Your attempt to prove that 0.999...<1 by using induction makes the same mistake: Certainly $\sum_{i=1}^{m}\frac{9}{10^i} < 1$ for all natural numbers m, but to derive your conclusion from that, you would have to find a natural number m, so that $\sum_{i=1}^{m}\frac{9}{10^i} = \sum_{i=1}^{\infty}\frac{9}{10^i}$ . Rasmus (talk) 15:34, 2 February 2006 (UTC)

When you write 0.999... don't you mean Limit sum (i=1 to m) 9/10^i as m->oo ?? When you assume that 1 - 0.9999... < 1 - sum (i=1 to m) 9/10^i, don't you mean as m->oo ?? Why, of course you do. I do not make the same mistake for m = ∞ as you do because I do not assume that 0.999... = 1. 158.35.225.231 17:52, 2 February 2006 (UTC)

I carefully avoided defining $0.999...=\lim_{m->\infty}\sum_{i=1}^m \frac{9}{10^i}$ , since you apparently don't accept that definition. When I claim that $\forall m \in N: \sum_{i=1}^{m}\frac{9}{10^i} < 0.999...$ , I claim just that - no limits are involved, and m does not approach anything. By the law of the excluded middle, either $\forall m \in N: \sum_{i=1}^{m}\frac{9}{10^i} < 0.999...$ or $\exist m \in N: \sum_{i=1}^{m}\frac{9}{10^i} \ge 0.999...$ . I assume we agree that the latter does not hold. Rasmus (talk) 18:18, 2 February 2006 (UTC)

I know what you have done and how you reason. You almost had me convinced. However, you cannot start with this fact and deduce a result about another fact that involves m approaching infinity. 158.35.225.231 18:45, 3 February 2006 (UTC)

Shorter answer : proof by induction doesn't here because "infinity" is not a natural number. Mikekelly 15:47, 2 February 2006 (UTC)

Why don't you just hold your two cents worth? 158.35.225.231 17:52, 2 February 2006 (UTC)

Please read [[WP:CIVIL|this]! Rasmus (talk) 18:24, 2 February 2006 (UTC)

Actually he only has to find a natural number m such that $\sum_{i=1}^{m}\frac{9}{10^i} \geq 0.9999...$ . Besides, I doubt he would agree to $0.9999...=\sum_{i=1}^{\infty}\frac{9}{10^i}$ , at least if the expression on the left shall denote the limit of a sequence of partial sums. He tried to define: 0.9999...=0.9+0.09+0.009+... But what shall "+..." mean, if not the limit (which would be 1)?

By the way, I must admit that I do not know what "reality checks" are. None of Wikipedia's definitions seems to be correct (maybe the lucid dreams...?). Yours, Huon 15:51, 2 February 2006 (UTC)

I do agree that:

What I do not agree to is:

The only nonsense is what you have been writing. 158.35.225.231 17:52, 2 February 2006 (UTC)

Usually, by definition, $\sum_{i=1}^{\infty}\frac{9}{10^i}:=\lim_{m\to\infty}\sum_{i=1}^m\frac{9}{10^i}=1$ . In contrast, $\lim_{i\to\infty}\sum_{i=1}^{\infty}\frac{9}{10^i}$ seems ill-defined, since $\sum_{i=1}^{\infty}\frac{9}{10^i}$ does not depend on the summation index, thus there is no limit of the sum for i (or m, which does not even appear, or whatever) tending to infinity.

I am prepared to bet that every standard textbook of university level analysis agrees with my definitions; especially, Wikipedia does. If you disagree with these standard definitions, you should at the very least state your own in full detail.

By the way, you yourself said: "There is no m for which 0.9999... is less than sum (i to m) 9/10^i." That alone is enough to prove that 0.9999... >= 1, without using "all m", limits, or infinity. I'll do it by contradiction:

Let us assume that 0.9999... < 1. Let x := 1-0.9999... be the difference; by our assumption, x > 0. We can thus consider 1/x > 0: By the Archimedean property, there is a natural number m such that m > 1/x. Furthermore, this natural number satisfies 10^m > m; thus, we have 10^m > 1/x. Now considering the reciprocal values, we obtain 1/(10^m) < x. But for this m, we also have 1 - sum (i to m) 9/10^i = 1/(10^m). Thus, using this equation and the definition of x, we get 1-sum (i to m) 9/10^i < 1-0.9999...; now subtracting 1 and adding both the sum and 0.9999... gives: 0.9999... < sum (i to m) 9/10^i. By your statement cited above, that can't be true. Our assumption has led to a contradiction and must therefore be false. So the opposite must be true: 0.9999... >= 1. Q.e.d. Yours, Huon 20:27, 2 February 2006 (UTC)

Nice try. However when you add 0.999... and subtract 1 you have a very small negative number and your contradiction arises because you use this very small negative number as if it were a real number. If you use (0.999... - 1) in your proof, you are assuming it is a real number. 70.110.93.192 05:52, 3 February 2006 (UTC)

When I subtract 1 and add 0.9999..., I have, consecutively, -sum (i to m) 9/10^i < -0.9999... and 0.9999...-sum (i to m) 9/10^i < 0. But 0.9999...-1 does not appear.

By the way, if you don't like negative numbers, I can of course add first and subtract later: First adding 0.9999..., then adding the sum and finally subtracting one gives, consecutively, 1-sum (i to m) 9/10^i +0.9999... < 1, then 1+0.9999... < 1+sum (i to m) 9/10^i, and finally once more 0.9999... < sum (i to m) 9/10^i.

More interesting is the claim that 0.9999...-1 is not a real number. 1 surely is a real number. If 0.9999... is one too, then so is their difference, i.e. both 1-0.9999... and 0.9999...-1. So you effectively claim that 0.9999... is not a real number. What else should it be? And even if it were not a real number, why should my calculations be wrong? Yours, Huon 09:39, 3 February 2006 (UTC)

What I am trying to say is that you cannot do any arithmetic with 0.999... whilst you do not know what it is. Proofs are faulty when you ignore the fact that you started out trying to establish what 0.999... is and then halfway through your proof, you make assumptions (that are wrong) about the thing you are trying to prove and consequently you end up with contradictory results. Once you let x = 1-0.999...; you have to treat it as a whole throughout your calculation. You cannot just start separating the parts and doing arithmetic as you would other real numbers. Why? Well, 1-0.999... may not be a real number. Even if both 1 and 0.999... are real numbers, this particular difference may not be 'real'. As an analogy consider -1 which is accepted to be a real number. If we try to take the square root, it is no longer real. So what I am saying to you is that the difference 1-0.999... is not a real number. It can only be approximated in the real number system by 0. However, since it is not quite 0, 1-0.999... is not equal to 0 and hence 1 and 0.999... are not equal. Ergo, 0.999... < 1. 158.35.225.231 17:37, 3 February 2006 (UTC)

By saying 0.999...<1 you are asserting 0.999... is a real number.

also the difference of two real numbers most certainly is a real number, by the definition of subtraction on reals.Mikekelly 19:21, 3 February 2006 (UTC)

You would be correct if this is what I said but I did not say this. I reached the conclusion that 0.999... is a real number (in fact a special kind of real number) and then realized it must be less than 1. My proof is simple:

For any natural m, $\sum_{i=1}^{m}\frac{9}{10^i} < 1$ .

Simple Proof by Induction: The result is true for m = k. The result is true for m=k+1. Conclusion: the result is true for any k. I do not need to worry about P(infinity) because the Archimedean property that determines (positive) real numbers does not require this. Also, 0.999... is not actually a proper real number. It is only real in the sense that is is bounded above. 158.35.225.231 20:48, 3 February 2006 (UTC)

I still do not understand your proof. What you wrote here was mostly copied from above, and that part I understand (until "... true for any k"). But I fail to see any connection between your result about all finite natural numbers and 0.9999..., probably still due to difficulties with your definition of 0.9999... You said it to be 9/10+9/100+9/1000+..., but what exactly is "+..." supposed to mean?

(As an explanation of this very point, you invoke the Archimedean property, but I see no connection between that property and the conclusion you draw. For example, if 0.9999... is not actually a real number, why should it satisfy the Archimedean property? And even if it does, how does that help?)

Probably you are considering non-standard analysis, enriching the field of real numbers with numbers which are no longer real, such as infinitesimals? You could do that, and then indeed have 0.9999...<1, but as the name suggests, it would be rather non-standard. Besides, your way of doing it seems to offer some problems. I suppose you still want to be able to tell whether one of your not-quite-real numbers is positive or not. Now consider 1/10, 1/100, 1/1000, ..., i.e. the sequence b_m with b_m := 1- sum (i to m) 9/(10^i) = 1-a_m. If "m tends to infinity", whatever that means in your version of non-standard analysis, we should get 1-0.9999..., which is claimed to be positive. Now let us instead consider the sequence c_m := (-1)^m (1- sum (i to m) 9/(10^i)) = (-1)^m b_m. The c_m alternate between positive and negative values. So if m tends to infinity, do I get a positive not-quite-real number? A negative one? Zero? Or is there no not-quite-real number corresponding to the sequence of c_m for m tending to infinity? Yours, Huon 23:18, 3 February 2006 (UTC)

The "+..." means it is an infinite sum that cannot be determined. I do not even try to determine this sum. All I care about is whether or not it equals or exceeds 1. Evidently, no matter how many terms I sum, this 'infinite sum' shall never equal to 1 and it shall never exceed 1. This is all I care about. Now when you talk about an infinite sum being equal to its limit - this is absolute nonsense, has always been absolute nonsense and will always be absolute nonsense. I don't care if it was published in the unholy Quran, it is still absolute nonsense. And you are all a bunch of stupid fanatics when you ignorantly accept such nonsense to be fact. How you cannot understand the above simple proof, is evidence you are intellectually lacking. The author of this article is nothing but an arrogant fool who knows shit.71.248.131.74 02:06, 6 February 2006 (UTC)

You claim 0.9999... is an "infinite sum that cannot be determined". I asked you before to define what you mean by "infinite sum", since that term is not standard notation. You should at the very least be able to say what kind of object it is. It is not a "proper" real number. What else is it? What set of object contains it? What properties does it have? Why can you compare such an object to 1? If you were right, and that sum could indeed not be determined, then how could you determine it to be less than one?

Anyway, your "definition" seems to be predestined to use what the article calls the "algebra proof". Whatever that infinite sum is, I can surely add and subtract such sums (if necessary, using "+..." or "-..."), and just as surely, I can multiply them by 10, multiplying all the summands in the process.

Finally, you have been asked before to be more civil. Don't you believe such insults weaken your cause? If you have arguments, feel free to contradict me, but if you don't, then uttering vulgarities won't help you. Yours, Huon 11:00, 6 February 2006 (UTC)

I think I defined it quite well. Let me try and make it even more simple: An infinite sum is one that has an infinite number of addends. This is very standard notation. There are many examples: 0.3333..., 0.666.., 0.999..., pi, e, etc. An infinite sum may or may not have a limit. Induction shows us that in the case of a number such as 0.999..., its sum can never equal or exceed 1. This is very self-evident. Although it can be shown by induction, it is axiomatic and by definition of the decimal system it is true that 0.999... < 1. 158.35.225.231 18:13, 6 February 2006 (UTC)

Anonymous editor, the reason I do not understand is your proof is that you are attempting to use proof by induction to prove something about the limit(value) of an "infinite sum". This does not work. Just because a summation has a property after any finite number of steps does not imply the "infinite sum" has that property. For example, the exponential function can be defined in terms of power series:

$e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots$

or as the limit of a sequence:

$e^x = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n.$

The summation after finite steps is never equal to the exponential function(except in special cases if x = 0 for example) and yet the "infinite sum" most certainly is equal to its limit which is equal to the exponential function. The first derivative of any of the partial sums of exp(1) is not equal to that partial sum and yet the limit does have this property. If you do not agree with this then I think you are using some non-standard form of mathematics I do not understand.

I am also still not sure what you mean by "a special type of real number". 0.9999... is either a real number or it is not. If it is then it can be manipulated arithmetically so your objection to Huon's proof does not apply. If it is not a real number than it makes no sense to try and place it on the real line by saying it is less than 1. Mikekelly 15:08, 6 February 2006 (UTC)

You are quite good at producing very bad analogies. The number e is not similar to 0.999... because e's representation (to infinity) is unknown. This is not the case with 0.999... - we can confidently state that we know the full extent of 0.999... even to infinity. We cannot say this about e or pi or any other irrational number. By special kind of real number, I mean it is an infinite sum whose limit is known exactly. You cannot say this about pi or e. You can only approximate the limit of pi or e. The decimal system does not allow us to represent most numbers exactly. In all arithmetic involving numbers such as pi, e or any irrational number, the best we can do is arrive at an approximation. Don't tell me that 4*pi is exact because you do not know exactly what pi is, i.e. its full extent. By extent I mean the complete decimal representation to infinity. It is so hypocritical of all on this forum to feel so cocky about 0.999... being equal to 1 when the best they can say for pi is that it is approximately 3.14... Please, tell me what the limit of pi and e is? I want the exact limit, not a formula telling me how to calculate it to any desired degree of accuracy. In fact, it can never be 100% accurate in any arithmetic. 158.35.225.231 18:13, 6 February 2006 (UTC)

pi is exactly the ratio of a circle's circumference to it's diameter. 4*pi is exactly 4 times the ratio of a circle's circumference to it's diameter. The exact limit of exp(1) is the non-zero real number that is equal to its first derivative. You are correct that these do not have a finite decimal expansion. The decimal system can only approximate the values of these numbers. This all seems completely irrelevant. The decimal system is not special or magic. I think it was agreed though that 0.9999... represents the infinite series

$\sum_{i=1}^{m}\frac{9}{10^i}$ .

Huon's proof shows that the limit of this series as the summation index tends to infinity is equal to one.

You say that 0.9999.... is a special number the limit of which is known exactly. What is this limit if it is not 1? Huon's proof shows that it must be equal to one. Your only objection was that one cannot do arithmetic with these "special real numbers" but you did not explain why. Why should this be? The difference of two real numbers is a real number, by definition of subtraction. You can do arithmetic with real numbers.

You have still not addressed the fact that your proof by induction is totally invalid because proof by induction only proves the property "is less than 1" is possesed by every partial sum. The limit of the sequence of partial sums need not have a property possesed by each of the partial sums. That was the point of my example of the exponential function, which I think you missed. Mikekelly 19:50, 6 February 2006 (UTC)

Actually none of this is irrelevant. You can say that pi is exactly the ratio of a circle's circumference to its diameter. What you cannot say is what is it's exact value in any radix representation or otherwise. 0.999... is a special number because its limit is 1. However, the sum of 9/10+9/100+9/1000+.... is never equal to 1. In fact it will always be less than 1. When most people talk about 0.999..., they do not mean the limit of the sum, they mean the actual sum. It is both unintuitive and stupid to think of the limit as the value of the sum. It is impossible to find the actual sum of 0.999... because it has an inifinite number of addends (terms). My proof by induction is *completely valid* (contrary to your statement) and it does prove the result for any number ot terms (including and infinite number of terms). Proof by induction does *not require* that you show P(infinity) - this is nonsense that was started by Rasmus and his sidekicks. My proof is the only valid proof in all the nonsense that has been contributed on this page. You are just as confused as the author of this article who is nothing but a pompous and arrogant fool. 71.248.142.241 23:23, 6 February 2006 (UTC)

Your "induction proof" shows the result for an infine number of terms, but not P(infinity)? What is the difference? Please also give any reference at all as to why induction shows your result for an infinite number of summands. And while I ask for explanations of unclear statements, what is the "limit of a number"?

By analogous "induction proofs", you can show "infinite sums" satisfy the distributive law and the law of associativity, at least to the degree required in order to arrive at the article's algebra proof. Yours, Huon 00:16, 7 February 2006 (UTC)

I don't talk about the limit of a number. I talk about the limit of a sum as being a number if it exists. It would help if you could read properly. And what on earth is a 'summand' ? There is no such word... 71.248.142.241 01:09, 7 February 2006 (UTC)

You said: "0.999... is a special number because its limit is 1." So you talk of a number which has a limit. As to a summand, according to Merriam-Webster, it is another word for "addend". You seem to lack mathematical arguments, you have to resort to insults and bickering about language...

By the way, I'll "prove by induction" that 0.9999... is a real number after all:

For any natural m, $\sum_{i=1}^m\frac{9}{10^i}$ is a real number.

Simple Proof by Induction: The result is true for m = k. The result is true for m=k+1. Conclusion: the result is true for any k.

Why should my "proof" be false if yours is true? Where's the differernce? Yours, 134.76.82.147 10:06, 7 February 2006 (UTC)

Oops, I forgot to sign in again, above thwat was me... Huon 13:23, 7 February 2006 (UTC)

That's what it means exactly. I do not talk of the limit of a number. 0.999... is an infinite sum that equals some number you cannot calculate and it is decidedly not 1. Merriam-Webster is not my definition of a good dictionary and if I recall correctly. you started bickering about language first. 158.35.225.231 12:53, 7 February 2006 (UTC)

So what you have been trying to say in a roundabout way is that pi and e are irrational numbers. Yes, they are. What relevence does this have? I was never trying to draw an analogy between the actual numbers 1 and e, rather to point out that a property held by an infinite summation need not be held by its partial sums and visa versa.

Please provide some reference for the validity of your proof by induction. It proves that the summation after finite steps is less than one. In all standard mathematics that I know of your method proves nothing about the infinite summation.

It may be unintuitive for the value of an infinite sum to be the limit of the sum but this is not the same as to say it is incorrect. In fact that is the definition used in all standard mathematics. What is your justification for saying this is stupid? "That is unintuitive" is not a mathematically sound argument. Read the Principa Mathematica and an introductory real analysis textbook and point out the error.

It seems very stupid or ignorant indeed to me to say that the exponential function is not equal to the limit of its power series expansion. Just because "it is impossible to find the actual sum of 0.999... because it has an inifinite number of addends (terms)" doesn't mean the value of the sum does not exist and cannot be found by other means. For example, Huon's proof, which doesn't even involve those troublesome "limits" you seem loathe to work with.

Incidentally can you restate your problem with Huon's proof? I keep asking but you seem unable to for some reason.. Mikekelly 10:16, 7 February 2006 (UTC)

Indeed the exponential function is not equal to the limit of its power series expansion. It is only approximated by the expansion. And I am not trying to say e and pi are irrational (any fool knows this) - what I am saying is that 0.999... is less than 1. As for Huon's proof, I stated my objections clearly, go back and reread it - you seem to have a problem with comprehension, not me. 158.35.225.231 12:53, 7 February 2006 (UTC)

The exponential function usually is defined to be equal to the limit of its power series expansion, or, equivalently, via the limit of a related sequence.

As to 0.9999..., I am still not sure I understand your point of view. Is 0.9999... a number? You said it was a "special number", which would indicate it is. You repeat it is an "infinite sum that equals some number". So again it seems to be a number. And you claimed it to have a limit ("its limit is 1"). The question remains: How can a number have a limit? This is, by the way, not bickering about language, but asking for mathematical definitions. References would be welcome.

Finally, you could also say something about the article's algebra proof. It seems perfectly well-suited to your "infinite sum" definition. Yours, Huon 13:23, 7 February 2006 (UTC)

Show me one reliable reference that backs up your claim that 0.999... = 1. Please, the reference must be before 1910. 'Mathematicians' born after this date generally tend to be a bunch of baboons. 158.35.225.231 16:48, 7 February 2006 (UTC)

An early reference for 0.9999...=1: Hans Carl Friedrich von Mangoldt, Einführung in die höhere Mathematik, Vol. 1, Leipzig 1911. Section 5, chapters 72-74 cover what you seek. This book has some obvious disadvantages: It is in German, and it doesn't quite make your 1910 deadline (but the author was born in 1854).

Unfortunately, it was the only early book on analysis I had available right now. Now it is my turn to ask for a refernence on your point of view - oh wait, I already did that, and you didn't provide any, nor did you answer my other recent questions... Yours, Huon 19:20, 7 February 2006 (UTC)

Not so fast. You found only one book and it's written in German? So why don't you translate what it says? Are you sure you understood it correctly? Judging by my experience with you, I have my doubts. Perhaps it was saying the limit of 0.999... = 1 in which case it is correct. If it was saying 0.999... = 1 then it is oviously nonsense. 71.248.135.32 01:15, 8 February 2006 (UTC)

In short, Mangoldt explicitly states 0.090909...=1/11 and proceeds to give multiplication formulas by which one easily sees that 0.9999...=11*0.090909...=11/11=1.

I could translate the relevant pages, but that would be too long for this talk page. And concerning "only one book": What did you expect? That book is almost a hundred years old. Would you agree I'm right if I could find another one? You still owe me a reference for your point of view, and an answer to my questions about the article's algebra proof. Yours, Huon 10:31, 8 February 2006 (UTC)

OK let me see what the objection to Huon's proof was...

"What I am trying to say is that you cannot do any arithmetic with 0.999... whilst you do not know what it is."

Why? An infinite summation bounded above by say 1 and below by say 0.9 certainly is a real number and can be maniuplated with arithmetic as such.

"Proofs are faulty when you ignore the fact that you started out trying to establish what 0.999... is and then halfway through your proof, you make assumptions (that are wrong) about the thing you are trying to prove and consequently you end up with contradictory results."

What assumption was made? That 0.999... is a real number? How could it be anything else? Why are you trying to place it on the real number line if it is not a real number?

"Once you let x = 1-0.999...; you have to treat it as a whole throughout your calculation. You cannot just start separating the parts and doing arithmetic as you would other real numbers."

Why?

"Why? Well, 1-0.999... may not be a real number. Even if both 1 and 0.999... are real numbers, this particular difference may not be 'real'."

By the definition of subtraction on reals the difference of two reals is certainly a real number.

"As an analogy consider -1 which is accepted to be a real number. If we try to take the square root, it is no longer real."

Hoo boy, I thought you said *I* came up with bad analogies? Different operations, different acceptables parameters. Subtraction is defined on all real numbers and yields a real number. "Square root" is a different operation and not relevent.

"So what I am saying to you is that the difference 1-0.999... is not a real number."

Justify this.

"It can only be approximated in the real number system by 0. However, since it is not quite 0, 1-0.999... is not equal to 0 and hence 1 and 0.999... are not equal."

Justify this. Your proof by induction does not do this because 0.999... is an INFINITE summation and proof by induction can only tell you about the FINITE partial sums.

Your objection is invalid. Huon's proof is correct. Yours, Mikekelly 19:57, 7 February 2006 (UTC)

Yes, the difference between two real numbers is real but is 0.999... real in the same way as other reals? Let me see, you have still not answered my questions: can you tell me what the value of pi is exactly? Let's say it approaches a limit of 3.14...7 in the 10^73429875276534 position. So does this mean pi = 3.14...7 or that it's limit = 3.14...7 ? As for my proof by induction: it tells me about the INFINITE sum (not the FINITE partial sums as you claim) being less than 1 since at no time can any of the partial sums produce a 'carry' such that 0.999... might even equal to 1. This is so simple, a child can see it. You do not need real analysis or any higher math to show this. If you are saying the limit of 0.999... = 1 then I will agree with you but as long as you talk about 0.999... being equal to 1, you are no better than most of the fools who teach mathematics at college. 71.248.135.32 01:15, 8 February 2006 (UTC)

What does "real in the same was as other reals" mean? Please provide some explanation before resorting to this handwaving again. A number is either real or it is not. If 0.999... is a real number than one can do arithmetic with it and your objection to Huon's proof is so much marsh gas. If it is not a real number then you cannot place it on the real number line. Which is it?

Again, pi is not exactly representable in a finite decimal expansion. It is exactly equal to the ratio of a circle's circumference to it's diameter, it is exactly the lowest positive solution to the equation sin(x) = 0, it is exactly equal to the square of the gamma function evaluated at a half it's equal to a bunch of other things. None of these are a finite decimal expansion but that is neither herre nor there. It does not make them any more or less exact. If you think otherwise you are letting your intuition based on elementary school arithmetic override mathematical reasoning.

By definition your proof by induction proves the property "is less than one" is held by every FINITE partial sum. That's what induction can do. Jumping from that to say that the INFINITE summation must also be less than one is wholly unjustified. Yes a child can see that after any FINITE partial sum the total cannot be 1. This does not mean that child's intuition has anything to say about the infinite summation. Proof by induction doesn't come from inuition, it is one of the Peano axioms. You are attempting to abuse it to prove something it cannot prove. I suggest this is because you don't understand induction. Mikekelly 09:47, 8 February 2006 (UTC)

Can you represent pi on the real number line? No. So who is producing a lot of marsh gas now? 0.999... is similar to pi in this respect. Now try and use Rasmus's proof or Huon's (it does not matter) and show me how you prove it is a real number. 71.248.146.204 21:53, 8 February 2006 (UTC)

By "place on the line" I was referring to you trying to form an inequality between 0.999... and 1. Nothing to do with representations.

You have two options : 0.999... is a real number, or it is not.

If it is, you can do arithmetic with it. In particular note than the difference of two real numbers is also a real number and can be manipulated as such. So Huon's proof is valid.

If it is not you cannot say "0.999... < 1" because you can't compare non-reals with reals like that. Also you would have to be claiming that the limit of an increasing sequence bounded above by 1 and below by 0.9 is not a real number which is patently false. Mikekelly 22:32, 8 February 2006 (UTC)

You can do arithmetic with it in the same way as you can do arithmetic with pi or e - only as an *approximation* because you do not know its full extent. Why do you have such an obssesion with setting 0.999... = 1 when you can't set pi or e equal to anything else? So once again, demonstrate (if you can) how pi and its predecessor (or successor if you wish) conform to the archimedean property in the same way as you are expecting 0.999... and 1 to conform to the Archimedean property. Your problem lies in the fact that you do not know that the Archimedean property does not work for all reals. Reals such as pi, e, irrational numbers and yes, 0.999... do not conform to the Archimedean property as do other reals. Using the same argument as Rasmus, show me how pi and its predecessor are real. 158.35.225.231 18:22, 9 February 2006 (UTC)

So now your objection to Huon's proof is that the Archimedian Principle does not apply to all real numbers? Well this just gets more and more ludicrous...

The Archimedean Property states that the set of real numbers contains no infinitesmials. From this can be derived...

The Archimedean Principle, which Huon invokes precisely once in his proof :

$\forall x \in R$ , $\exists n \in N$ st

n > x

I am absolutely at a loss as to how you are claiming the archimedian principle does not hold for, say, π. Here is a natural number that is greater than π : 4. Satisfied? Mikekelly 19:22, 9 February 2006 (UTC)

pi is real: pi can defined to be twice the smallest positive zero of the cosine. That such a zero exists (and is a real number) can be shown by the intermediate value theorem. That this definition of pi is equivalent to the "ratio of circumference and diameter" definition can be shown by integration.

On the other hand, pi's "predecessor" is ill-defined. Real numbers have no real predecessors (or successors) - the difference between a number and its predecessor would have to be both a real number and an infinitesimal, which is impossible. If you do non-standard analysis, then the predecessor may or may not exist, but definitely is not a real number.

Since this line of discussion gets more and more off-topic here, I invite you to continue on my talk page instead. Yours, Huon 22:12, 9 February 2006 (UTC)

What is wrong with the joke?

Why did Timothy Clamans remove the Mathematical Gazette story from the external links? It's a version of the "Fraction proof" and is relevant to the article. In fact I was thinking of mentioning it directly in the "fraction proof" section and was surprised it had been deleted. I'm going to put it back unless someone explains why I shouldn't. See: http://www.steve.bush.org/links/humor/pg001185.html 71.141.251.153 01:57, 4 February 2006 (UTC)

I like it. I think perhaps you were confused with one of the IPs causing a ruckus on this talk page. If you signed in (hint hint) it wouldn't be a problem. Melchoir 03:02, 4 February 2006 (UTC)

I put in an extended paraphrase of the story instead of just a link. Maybe that's better. Thanks for the hint about logging in but I like the egoless nature of editing anonymously. 71.141.251.153 03:19, 4 February 2006 (UTC)

Back on topic... I think the joke should say, but the description of the joke should not. It does not contribute any encyclopedic information towards the article. A simple description such as:

In 1954, the Mathematical Gazette published a humorous article about infinitesimals and limits. [5]

However, ==Popular culture== should be renamed ==Trivia==, to allow for more space for other humourous articles. No doubt there are more. x42bn6 Talk 09:02, 5 February 2006 (UTC)

Oh, I think it does contribute encyclopedic information. Now, I have to admit that my first reason for liking the joke is that it sets up a convenient straw man to poke fun at the absurdity of calling 0.999... less than 1. But! It also contains a unique argument for the equality, one that some readers might find more convincing than the mathematical arguments. It uses percentages, which the rest of the article doesn't, and I suspect that the laity might be more comfortable with percentages than with fractions. Melchoir 09:15, 5 February 2006 (UTC)

The only absurdity is claiming that 0.999... = 1. 71.248.131.74 02:08, 6 February 2006 (UTC)

This joke is only telling why the fact that 0.999 euals 1 could be important to agree on when working with someone who knows that it is ture, not that 0.999... indeed does equal 1. Suqqest this for fractions or something or better don't. Timothy Clemans 05:18, 6 February 2006 (UTC)

So, what's the difference? All true statements in mathematics are ultimately true by agreed-upon definitions. The meaning of 0.999... is something we agree on. The article must address why we agree on it, including the parts of mathematics that break without it. Melchoir 05:27, 6 February 2006 (UTC)

And please stop removing it. You have yet to convince anyone to support you. Melchoir 05:28, 6 February 2006 (UTC)

Why should any jokes be put in any mathematics encyclopedica articles? It is very clear that it is not the proof but the foundational parts infinitesimals and limits that is speaking to. There is mention of why the teacher uses 6 as the limit.Timothy Clemans 05:40, 6 February 2006 (UTC)

This is not a mathematics encyclopedia. The subject of this article is a mathematical topic, but the article itself need not-- and should not-- limit itself to mathematics. And the joke is not just some rambling by a Wikipedia contributor; it has been published twice, and it speaks to the topic. Melchoir 06:05, 6 February 2006 (UTC)

After reading it more carefully, shouldn't this joke actually be at Limit (mathematics)? x42bn6 Talk 08:29, 6 February 2006 (UTC)

Hmm... tough call. Yeah, I'd support that move. Melchoir 09:03, 6 February 2006 (UTC)

I think this article is the right place for it. Limit (mathematics) is pretty technical; this article (and the joke) are at a more popular level, and I thought the joke added some color to the article, plus the joke deals directly with the 0.999999... question. Renaming "Pop culture" to "Trivia" is ok (both those headings are used in other articles), or the whole thing could be moved to the "Fraction proof" section if rewritten slightly or labelled as a "Fiction proof" version of the fraction proof.

I put in the paraphrase and the tedious explanation of the punchline because I thought that non-native English speakers or people unfamiliar with mathematical limits otherwise might not get it (I thought maybe that's what had happened with Timothy at first). I also thought the long paraphrase was more encyclopedic than just having a link, especially since the linked page could go away at any time. 70.231.131.185 09:15, 6 February 2006 (UTC)

There are cases in which enthusiastic people add their favorite pet ideas to articles, and then those articles start containing a lot of somewhat off-topic (or either way not much interesting) cruft (see for example 1 (number)). But the section which Timothy Clemans was trying to remove does not qualify under such an assesment I think. The section is an interesting read, and relevant, so I think it should stay. Oleg Alexandrov (talk) 16:12, 6 February 2006 (UTC)

I like jokes, and make a point of finding ways in my formal writing to evoke smiles. (My aphorism: "When we laugh, we learn.") That said, I am uncomfortable with quoting this joke in an article devoted to proofs. I am even more uncomfortable explaining the joke. I am fairly comfortable citing the joke, either here or in a more appropriate article.

Son: Dad, will you do my mathematics homework for me tonight?
Dad: No, son, it wouldn't be right.
Son: Well, you could try.

--KSmrq^T 16:59, 6 February 2006 (UTC)

I don't think there's a more appropriate article for the joke. It certainly doesn't belong in Limit (mathematics), which is technically advanced. This article tries to explain things at a less technical level, and as such, some informality seems like the right thing. Also, looking at the article reminded me of the joke immediately, which is why I looked for a web copy and cited it. The cite got removed immediately, possibly (I thought) because its relevance wasn't clear enough, so I added the longer version.

As for the explanation, yeah, I felt a little bit odd writing it, but I think it can be of help to some readers (those unfamiliar with English-language idioms or mathematical limits) and so adding it seemed worthwhile. The example stories in the shaggy dog story article are similarly all explained (I resisted describing the joke as a shaggy dog story). 70.231.131.185 18:35, 6 February 2006 (UTC)

KSmrq, I respectfully submit that this article is de facto not devoted to proofs. (And that's a good thing.) There's background, generalizations, definitions, and justifications, to quote from the intro and the table of contents. In fact, the article's title is a misnomer, is it not? Melchoir 19:55, 6 February 2006 (UTC)

The title is correct; the article incorporates supplementary material to make the proofs accessible to a wide audience. Likely readers include pre-university students, their teachers, higher mathematics students, and the general public.

After discussion with the contributing editor, I'm inclined to delete the "Definitions and justifications" section, barring objection. It detracts from the focus of the article while adding minimal value. I'd also delete the "Popular culture" (joke) section, but retain a citation. The "Generalizations" section exists per the style guide, and because it's important to understand that the ideas and proofs are specific to neither 0.999… nor base 10. --KSmrq^T 21:10, 6 February 2006 (UTC)

I'm not going to defend "Definitions and justifications", since it kind of veers off topic; and without sources, easy come, easy go. And I see what you're saying about "Generalizations". But the joke is sourced, and it's relevant: if not to the proofs themselves, at least to the ideas already discussed in the article. In fact, since it's a joke, I don't think there's any danger that it might distract from the mathematical arguments, so what harm does it do? Melchoir 21:22, 6 February 2006 (UTC)

0.999... is not equal to 1. It is exactly less than 1. I know Greek fairly well and I can tell you that you used the word aphorism incorrectly. Don't use words whose exact meanings you don't know. Your brain is so soft that it is in advanced atrophy. In Greek there is a koine expression for this state of being; you would be rightly called a 'malaka'. I hate to use this word but you are such an annoying and arrogant fellow who would be better off masturbating with the likes of Hardy - another pompous and arrogant fool who knows nothing. What are your qualifications by the way? Did you receive a major in Poetry and Art with a minor in Mathematics? Drop off the face of the earth fool! Good grief!!!! 71.248.142.241 23:33, 6 February 2006 (UTC)

If it does stay in this article, I think it should still be reduced to one paragraph. Users can read the joke by clicking the link. There is no need to describe the joke, to me. Why? This article answers a question that baffles so many people. A joke is a nice touch, but should remain a touch and not a large section of its own. x42bn6 Talk 00:21, 7 February 2006 (UTC)

A small paragraph is fine with me; I just want to keep more than a bare citation. Melchoir 00:29, 7 February 2006 (UTC)

Of course! Otherwise it would just go into an ==External links== section. x42bn6 Talk 00:49, 7 February 2006 (UTC)

Unfortunately the article doesn't mention what limits are, so the joke may be hard to understand without explanation (though it could be shortened). See some of the shaggy dog story examples, as mentioned earlier, and keep in mind that most of the article's readers won't be that mathematically sophisticated. Maybe the joke could be moved to near the "limit proof" with a mention that it evokes the fraction proof. The limit (mathematics) article itself is way too technical and could use a rewrite or introductory section instead of launching off into topology and category theory. I may take a crack at that.70.231.131.185 04:07, 7 February 2006 (UTC)

What is wrong with this discussion?

Anyone reading the following paragraph knows that it is BS:

"The 9s case does surprise, perhaps because any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1. Thus infinity, a sometimes mysterious concept, plays an important role behind the scenes."

First of all it says a lot about the arrogant character of the author. His ego is so huge that it overpowers the remaining sentence. He makes the statement that where the '9s eventually stop, it is strictly less than 1.' Well, in this case, let's say they eventually stop at 'infinity behind the scenes' - in this case is it still strictly less than 1? On what grounds does the author make such a statement? ksmrq then proceeds to handwave even more furiously (and somewhat poetically) with the phrase 'Thus infinity' adding some academic flavour: '...a sometimes mysterious concept...' - to which I ask: Are there times when it is not mysterious? Hmmm? The crowning pinnacle of his brilliant paragraph is: 'an important role behind the scenes.' By this time my mandible is touching the floor with surprise, awe and suspense. Am I reading about arithmetic or a suspense novel? I wonder...

The user ksmrq has written an article that has no mathematical foundation, nor does it contain any logical conclusion. There are NO REFERENCES to respected authors who agree with what is written in this article. Furthermore, there is not a single reference in an acknowledged handbook that backs up any of his statements. Wikipedia continues to publish this absolute nonsense that is entirely untrue and misleading. 71.248.135.32 01:22, 8 February 2006 (UTC)

You are welcome to go find some references and add them yourself. Melchoir 01:56, 8 February 2006 (UTC)

There are no such references - this is why I challenged you to find these. 158.35.225.231 13:43, 8 February 2006 (UTC)

Actually, it is true. It is impossible to visualise infinity. You can view loads of 9s, but not an infinite number of 9s. That said, the paragraph does need to be reworded.

A lot of the problems understanding this proof is the fact that not a lot of people understand the concept of infinity. Infinity is not just a very large number, it is THE largest number possible. When you reach this value (which is not possible in real life!), you get the limit of 1.

Some may argue that because it is not possible in real life, then why should we accept it? The answer is simple: Mathematics deals with these types of things. Calculus is based upon infinitely small changes (not very small changes), and calculus surely must be correct, given its many, many uses. So, given that 1-0.999... is the smallest number closest to zero possible, in Mathematics, we say that is zero. We can't visualise it, so we use limits and algebraic proofs to prove it.

I doubt you can find references too... Because it is a proof. Proofs are made, rarely referenced. x42bn6 Talk 03:26, 8 February 2006 (UTC)

x42bn6: In fact, no one understands the concept of infinity, not even a few people, NO ONE. You write that when you reach this value, you get the limit 1. Now does this make any sense to you x42? First of all, you can never reach infinity and second of all you don't ever get a sum of 1. In fact what you notice is that the SUM IS ALWAYS LESS THAN 1 and this is how you reach the conclusion that 1 is the limit. Okay, now let's see where your calculus argument went horribly wrong: Just because calculus works and we have done lots of things with it, does this mean that we understand infinity? I hate to break it to you x42, but any progress we have made was reached through experimentation and observation. There are many things we do not understand fully, e.g. electricity. However, we are familiar with many of its properties and can thus use it in many different ways. Calculus is all about limits and it was born as a result of man's attempt to calculate tangent gradients (later this developed into instantaneous rates of change). Newton used actual calculations in his observations. Much of the theory that was developed later has lots of problems and is by no means complete. Real analysis (also known as basic analysis and analysis of reals) was born as a result to appease clerics. That's right, men of the cloth stuck their stinky noses where they did not belong. Well, to cut a long story short, you should never make assumptions based on what you learn parrot-fashion at college because chances are it is filled with contradictions and errors. 158.35.225.231 19:46, 8 February 2006 (UTC)

To say "Infinity is not just a very large number, it is THE largest number possible" is a mistake, in multiple ways. Depending on what we are doing we may have an infinity that is not a number at all, an infinity that is the cardinality of the natural numbers, a larger infinity that is the cardinality of the real numbers, an infinity that is the one-point compactification of the complex numbers, and so on. Non-standard analysis can give us a plenitude of infinities which are members of the "extended reals"; if b is an infinity, then b+1 and 2b are other (and larger) infinities. Confusion over whether to treat infinity as an actual number is rife on this very talk page. In most discussions here, it is a mistake to do so. --KSmrq^T 06:38, 8 February 2006 (UTC)

Indeed infinity is not a number. This is about all you got right. Oh wait, it is "something mysterious behind the scenes." 158.35.225.231 13:43, 8 February 2006 (UTC)

What would a "respected author" be doing publishing about something so elementary? If 0.999... represents the summation $\sum_{i=1}^{\infty}\frac{9}{10^i}$ . Then it is absolutely trivial to find the value by limits. Some people contributing here are clearly uncomfortable with how limits work but that doesn't make limits less valid. If you are unable to find the value I suggest you get any first year analysis "handbook". Next We'll be hearing how $\sum_{i=1}^{\infty}\frac{1}{2^i}$ isn't equal to 1.Mikekelly 10:07, 8 February 2006 (UTC)

If this were so elementary, why is there such confusion over it? And why do respected authors publish any of the other elementary material that they do? Finally, I am not disagreeing that the limit of 9/10+9/100+... = 1. I am disagreeing that the infinite sum is ever equal to 1. And yes, the limit of 1/2 + 1/4 + ... = 1 but 0.111...(base 2) is never equal to 1. There is a very clear difference between infinite sum and limit of an infinite sum. An infinite sum is NEVER computable but its limit may or may not be computable. Let me make this simple for you: if you sit down and start summing 9/10+9/100+9/1000+... you will continue to sum it until you die and if yuo could live forever, you would never stop summing the terms - this is what an infinite sum means. Now how is it that some infinite sums have limits? Very simple: this is where analysis comes in - the difference between the terms gets closer and closer to zero (but NEVER zero, not even at infinity because there is no such number) so that at a certain time the sum stabilizes at a certain value that is called the limit. So what does any of this tell us about 0.999... ? Well, for one thing it is a real number because it has a limit. However, just like pi, it can only be used as an approximation: 0.999999 and it makes no sense trying to apply the Archimedean property with a number such as 0.999... and 1 because these numbers are infinitely close. Can you find a number between pi and its predecessor or successor? If you tried to compare pi and its predecessor the same way as you did 0.999... and 1 with the Archimedean property, you would run into the same sort of problems, i.e you would end up concluding that pi is NOT real but we know full well that pi is real for it is the ratio of the circumference to the diameter of any circle and both the circumference and the diameter are VERY FINITE and real. 158.35.225.231 13:43, 8 February 2006 (UTC)

Actually a reference to one of those first year analysis handbooks might improve the article. Can one of you provide one where this subject is covered a little more thoroughly than usual? Yours, Huon 10:39, 8 February 2006 (UTC)

Nothing published in the 20th century is acceptable - this fallacy that 0.999...=1 was orginated by some idiot who was not thinking properly. 158.35.225.231 13:43, 8 February 2006 (UTC)

I found two 19th century references:

Ch. Sturm, Cours d'Analyse de l'école polytechnique, Paris (in French). The edition I found is of 1895, but since Sturm died in 1855, the lecture must have been held quite a lot earlier. Sturm does not use recurring decimals (or any decimal representations of numbers except for integers), but explicitly states that the sum 1+1+1/(1*2)+1/(1*2*3)+1/(1*2*3*4)+... is denoted by e, so the "+..." notation clearly means the limit (quatrième leçon, p. 51).
Oskar Schlömilch, Handbuch der algebraischen Analysis, Jena (in German). The second edition is of 1851. Schlömilch explicitly uses recurring decimals and states that 1/9=0.111...=1/10+1/10²+1/10³+..., allowing the conclusion that 0.9999...=9*0.1111...=1 (Capitel 2, p. 19).

Of course both are unsuited for the article, unless we want to write something on the history of the proof. Yours, Huon 10:54, 9 February 2006 (UTC)

Your references prove my point: when you write 0.999... = 1, you are talking about the limit of the sum 9/10+9/100+9/1000+..., not the actual sum of 9/10+9/100+9/1000+... 158.35.225.231 13:06, 9 February 2006 (UTC)

Those references prove something else: Sturm and Schlömilch make no distinction between what you call "the actual sum" (1/10+1/10²+1/10³+... in Schlömilch's case) and "the limit of the sum". Actually, excepting you, I cannot remember such a distinction ever being made. If you claim otherwise, then give a reference. Yours, Huon 15:01, 9 February 2006 (UTC)

The fact that Sturm did not make a distinction between the two does not mean there is none. It seems so blatantly obvious. Suppose you were asked to find the partial sums of the following finite series: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 The partial sums are: 10, 19, 27, 34, 40, 45, 49, 52, 54, 55. The limit of all these partial sums is 55. In this case because it is a finite series, the limit is equal to the sum. However if we stopped summing after 5 terms we would say the actual sum is 45. But this is not equal to the limit. Now in an infinite series, we cannot calculate the actual sum but given that the terms decrease we can state that it is bounded above by some limiting value. Of course this limiting value is not the same as the actual sum. If we sum 9/10+9/100+9/1000+ forever we will keep getting closer to 1 but never actually reach 1. The two are not the same thing. And seeing that this is about the only reference you could find (in German too), I will not take it seriously. The onus is on you to show me proof, not for me to find a reference. I did not make the claim that 9/10+9/100+... = 1. You did. 68.238.99.15 01:35, 10 February 2006 (UTC)

Jacob Bernoulli, Tractatus de seriebus infinitis, 1689:

$a+ar+ar^2+\cdots=\frac{a}{1-r}$

So old it's in Latin. I suppose you can't take that seriously either? Melchoir 01:44, 10 February 2006 (UTC)

I can't believe you are so deceptive Melchoir? You wrote it down incorrectly:

 a+ar+ar^2+ ... +ar^(n-1)= (a-ar^n)/(1-r)

And you say you have credentials? Shame on you! No wonder more morons are produced because of teachers like you. There is a very large difference between this and what you wrote. Look, what you wrote really means:

  Limit (a-ar^n)/(1-r) as (n->infinity) = a/(1-r)  for |r| < 1

This was understood from the context it was written in. Once again, you misunderstood it. In fact, this very formula proves conclusively that 0.999... is less than 1. How? First it tells us that the limit of 9/10+9/100+... = 1. Since this is the limit of the sum and we can never compute the actual sum, it follows with mathematical induction that the actual sum will always be less than 1. Of course the limit of the sum is not the value of an infinite sum.158.35.225.231 13:43, 10 February 2006 (UTC)

I just looked up Jakob Bernoulli's works; Melchoir's quotation is correct (actually I found an even more general result, but the notation is the same). I won't deny that Bernoulli probably meant the limit. That's just my point: The "+..." notation always means the limit. You are the only one to claim it doesn't, or at least not in the case of 0.9999...=9/10+9/100+9/1000+...

You say the "limit of the sum is not the value of an infinite sum". Can you give any reference as to if and why there is a difference? Oops, I forgot, you can make any claims you want without having to back them up, sorry... Huon 14:33, 10 February 2006 (UTC)

Sarcasm is unnecessary. So if we agree that he meant limit of the infinite sum, then what makes you think that this means it is equal to the actual infinite sum? Can you compute an infinite sum? Or is it something that happens mysteriously behind the scenes? 158.35.225.231 18:09, 10 February 2006 (UTC)

None of the references I found ever considered the "actual infinite sum". Especially whenever decimal representations were used, they were meant to be limits. In effect, what I claim is: "Actual infinite sums" don't exist. Limits are well-defined; if "actual infinite sums" are not limits, then the entire concept can be scrapped without any loss to standard analysis. If you disagree, a reference, any reference, where that distinction is explicitly made would be welcome. (And yes, my sarcasm maybe was inappropriate, I'm sorry.) Yours, Huon 19:44, 10 February 2006 (UTC)

I would say the onus to justify their claim is on the person disagreeing with a fundamental idea of all standard real analysis used by all mathematicians everywhere since before anybody alive was born. The value of an infinite sum IS the limite of that infinite sum, if it exists, by definition. What else would you define it to be? A "special" real number that can't quite be pinned down? Mikekelly 09:38, 10 February 2006 (UTC)

Let me give you a real number that can't be pinned down. PI cannot be pinned down. In fact it can be expressed as a ratio of non-real numbers, so does this mean it is unreal? pi = [ln (-1)] / i Both ln(-1) and i are not real numbers. 158.35.225.231 19:00, 10 February 2006 (UTC)

That example is not the best available, since ln(-1) is not well-defined (only up to multiples of i*pi). There are other ways to express pi as a quotient of complex numbers which are not real. But what's the point? Every real number can be expressed as a quotient of purely imaginary complex numbers: a= (a*i)/i. Should that mean that no real number can be "pinned down"? Or should that example mean that when forming quotients of non-real numbers may yield something real, then forming sums or differences of real numbers may yield something non-real? Puzzled, Huon 19:44, 10 February 2006 (UTC)

For those who are actually interested: I am not misquoting anyone. Bernoulli never used such notation as "ar^(n-1)" and he never used the limit symbol; those are modern worries from modern textbooks. See for yourself. Instead of ... he writes &c; and instead of = he writes some other symbol; I understand there were plenty around that period. His notation for equality of ratios is bizarre, but his symbols for infinity, <, and > are modern. The geometric series is the first and least of his concerns; even at that early period, mathematicians were summing series like mad. Bernoulli evaluted a number of more interesting sums, and he proved that the harmonic series diverges. By 1735, Euler would determine zeta(2); by 1744 he would have zeta(26). There was no such concept as a Cauchy sequence or a Dedekind cut, and these men were perfectly happy to manipulate infinite and infinitesimal quantities with abandon. Three centuries ago mathematicians were less careful than we are today, and they still arrived at the same answers. Over time the methods of real analysis have changed, but the results have not. Melchoir 20:46, 10 February 2006 (UTC)

In fact, the divergence of the harmonic series was known to Nicolas Oresme in the 14th century. Melchoir 21:09, 10 February 2006 (UTC)

Take it from the top, shall we?

I have yet another proof for our IP-identified friend(s) that what it says in the article title is true, but to make sure that I don't get shouted down because someone doesn't agree with my basic premises, let me set those out first, and then if there are no arguments I will proceed with the proof.

OK:

$0.999\ldots = \sum_{i=1}^{\infty}\frac{9}{10^i}$

$0.999\ldots < \infty$ , and in particular, $\exists M\in\mathbb{R}$ such that $0.999\ldots < M$ .

$\sum_{i=1}^{\infty}\frac{9}{10^i} - \sum_{i=1}^{n}\frac{9}{10^i} = \sum_{i=n+1}^{\infty}\frac{9}{10^i} = \frac{1}{10^n}\sum_{i=n+1}^{\infty}\frac{9}{10^{i-n}} = \frac{1}{10^n}\sum_{i=1}^{\infty}\frac{9}{10^i}$

Given a sequence $(a_n) = (a_1,a_2,a_3,\ldots,a_n,\ldots)$ , $\lim_{n\rightarrow\infty}{a_n} = L$ means (ie. is defined as) $\forall\epsilon > 0, \exists m\in\mathbb{N}$ such that $\forall n\in\mathbb{N}, n \geq m \Rightarrow |L-a_n| < \epsilon$ .

The first is something that everyone seems to agree on already, the second one that without which we're all screwed, the third simple manipulation of sums that (to my knowledge) still apply when the sum is infinite, and the fourth is (again, to my knowledge) a basic mathematical definition. Tell me now: do you disagree with any of the above statements? Confusing Manifestation 03:13, 9 February 2006 (UTC)

Fixed a typo for you, a<->L. Melchoir 03:35, 9 February 2006 (UTC)

First 3 premises correct and fourth is Cauchy definition of a real number. So now let's see your proof. 68.238.111.130 11:55, 9 February 2006 (UTC)

In simple words, the fourth premise (rational definition of real numbers) states that in order for the sequence to be a real number, the difference between the terms must be getting closer to zero and that the partial sums approach a limiting value. One of its consequences is that the tail end of the sequence is less than a certain epsilon say e. If we choose this e to be 0.00001 (or any other small value), we can show that the tail end of 0.999..., i.e. 0.00000...999.... is always less than this e. Given this is the case, it proves conclusively that 0.999... < 1 since only a tail end sum that is equal to e will result in 0.999... = 1. As this can never happen, we arrive at the conclusion that 0.999... < 1 and this article is still as false as when it was first written.158.35.225.231 13:14, 9 February 2006 (UTC)

Actually, the fourth premise is the definition of a limit, a limit being the notion that Cauchy uses to define real numbers. It does not have anything to do with having tail end sums equal to any e. Even if it did, the same arguments used for 0.999... = 1 work just as well for 0.000999... = 0.001 . JPD (talk) 13:50, 9 February 2006 (UTC)

Actaully, the fourth is not a definition of a limit and it has everything to do with tail end sums equal to some e. In fact, it is a limit used to describe the real numbers in terms of rationals. Look, you must be a rookie, so please save your two cents worth, okay? 158.35.225.231 14:48, 9 February 2006 (UTC)

Read it again. It says " $\lim_{n\rightarrow\infty}{a_n} = L$ means (ie. is defined as)".... It's the definition of $\lim_{n\rightarrow\infty}{a_n} = L$ . Confusing Manifestation hasn't used it at all yet, simply stated it as a definition. Yes, limits are used to define real numbers in terms of rationals, but using limits in this way only involves differences (in your words, tail end sums) being strictly less than epsilon, not equal to it. Your argument is very ciruclar - why don't you just accept the definition, and wait to see how it is used in the proof. JPD (talk) 15:08, 9 February 2006 (UTC)

I think you are misunderstanding what I wrote. Read my original post again. I know the differences have to be less and my point is exactly this: if the tail end sum were to equal e (it does not for it is always less) then you would arrive at the conclusion that 0.999... must be equal to 1. Since it is always less than e, 0.999... must be less than 1. Nothing circular about this. Just stating the facts. 158.35.225.231 17:20, 9 February 2006 (UTC)

Firstly, definition 4 says NOTHING about the difference between terms. So no, you are not just stating facts.

Secondly, you seem totally unclear on the concept of limit. In simple terms, the fourth premise states that if a sequence is said to have a "limiting value" L then for an aribtary positive ε, after a certain point in the sequence the difference between any term and L is less than ε. Mikekelly 17:37, 9 February 2006 (UTC)

Not you again! I am not referring to the same e as in 4 and the differences JPD and I were talking about have everything to do with definition 4. If you cannot see this, you need to study real analysis. Listen, I have a BS in math. Don't need you to try and tell me anything about limits. Now either say something constructive or stop cluttering up this space with your garbage please!!!158.35.225.231 18:27, 9 February 2006 (UTC)

I probably am misunderstanding what you meant, because what you wrote was too vague to make sense. It's true that the differences you were talking about a vaguely related to definition 4, but whether they are equal to anything has nothing to do with the definition. Your argument is circular because 0.000999... is not "clearly less than" 0.001 any more than 0.999... is "clearly less than" 1. JPD (talk) 11:49, 10 February 2006 (UTC)

I will try to put it in plain English this time, to make sure that we don't have any confusion. First, I am not trying to define "a real number", I am trying to define the limit of a sequence. All it says is that the number L is the limit if and only if for any given ε all terms of the sequence after the mth are within ε of L. That is it. I am not looking at tail end sums or anything similar. I have not stated what any of the

(a n)

are, nor have I given a value to L. This is merely a standard definition of a limit. If you disagree with this then I may have some justification in believing your BS stands for something a little different to my BSc (and not that I'm not waving my qualifications around to say "I have a degree so I'm right", especially since I would apparently lose all credibility to you if I were ever to attain a PhD). Confusing Manifestation 13:52, 10 February 2006 (UTC)

I don't have any problem with it. I simply added that it is also used to define real numbers. JPD decided to argue with me. Why don't you just go ahead and state your proof? Please, don't keep us in suspense any longer.... 158.35.225.231

Ok, if there are no problems with the definitions, let's hit the proof:

Let $a_n = \sum_{i=1}^n\frac{9}{10^i}$ .

$0.999\ldots - a_n = \frac{1}{10^n}\sum_{i=1}^{\infty}\frac{9}{10^i} = \frac{1}{10^n}0.999\ldots$ by point #3.

$0.999\ldots - a_n = \frac{1}{10^n}0.999\ldots < \frac{1}{10^n}M$ , where M is some finite number greater than $0.999\ldots$ (which exists by point #2).

For any given $ε > 0$ , set $m = \lceil log_{10}\frac{M}{\epsilon} \rceil + 1$ . Then:

$m > log_{10}\frac{M}{\epsilon}$

$10^m > \frac{M}{\epsilon}$

$10^{-m} < \frac{\epsilon}{M}$

$\frac{M}{10^m} < \epsilon$

Therefore, we now have that:

$|0.999\ldots - a_m| < \frac{M}{10^m} < \epsilon$ , and since $|0.999\ldots - a_n| \leq |0.999\ldots - a_m| \forall n \geq m$ , we then know that $|0.999\ldots - a_n| < \epsilon$ .

By point #4, $0.999\ldots = \lim_{n\rightarrow\infty}{\sum_{i=1}^n\frac{9}{10^i}}$ . It has already been agreed that $\lim_{n\rightarrow\infty}{\sum_{i=1}^n\frac{9}{10^i}} = 1$ , and therefore $0.999\ldots = 1$ . In other words, I have shown that, in fact, that "the infinite sum is equal to the limit of the partial sums" is not a definition, but a provable statement. Confusing Manifestation 13:03, 11 February 2006 (UTC)

A few points to consider.

The fact that 0.999... equals 1 is hardly something worth of a "proof", it follows directly from the definition of real numbers. I think the article should reflect that.

Dedekind cuts and Cauchy series are mentioned, which is good. Most proofs mentioned, however, start from some arbity point that you just have to accept. If you don't think 0.999... is 1 you might not accept that 1/3 is exactly 0.333... Same goes for the algebra proof. You might think that 10c − c = 8.9999...

If you just understand what it means when 0.999... is written, the subject becomes very clear. Therefore one should try to explain what it means, exactly, and dispel any unclear mental images such as "endless number of digits".

Briefly, "0.999..." means the limit of (0.9, 0.99, 0.999, ...) which is 1, exactly.

More generally, real numbers can be defined as cauchy series of rational numbers: x = (a1, a2, a3, ...). If the series approaches a rational number it can be identified with a rational number. x = (1, 1.0, 1.00, ...) => x = 1. The notation "0.999..." actually means the series (0, 0.9, 0.99, ...) that approaches 1. Therefore 0.999... and 1 are the same, in a real number sense that is.

Comments? Mistakes? English is not my native language so some math terms can be wrong...

194.251.240.114 17:46, 10 February 2006 (UTC)

You write: 'The notation "0.999..." actually means the series (0, 0.9, 0.99, ...) that approaches 1.'

I think this is part of the problem. Although I strongly disagree with you that a proof is not necessary. 9/10+9/100+... is not equal to 1. Its limit is equal to 1 but these are two different things. The actual infinite sum is not equal to the limit of the infinite sum. The two concepts are completely different. Now if you define 0.999... the way you did above, then YES, I agree that it is the same thing. However, if you define it as 9/10+9/100+9/1000+... then NO, 0.999... and 1 are not the same thing. 158.35.225.231 18:04, 10 February 2006 (UTC)

INFINITE SUM MEANS THE LIMIT! Well, you could say that the infinite sum is a formal statement and it may or may not approach anything. Infinite sums are however defined as limits.

--194.251.240.114 08:52, 11 February 2006 (UTC)

Actually, the main argument on the talk page and its archives seems to have been the idea that the infinite sum is somehow a separate entity from the limit of the partial sums. However, see the above section for my proof that they are equal. Confusing Manifestation 12:37, 11 February 2006 (UTC)

Levels of hypocrisy.

How is it that you all feel comfortable saying:

   -1) sqrt[6*[1+1/4+1/9+1/16+.... ]]  equals to some number pi

and

   -2)  2 + 1/2! + 1/3! +   equals to some number e

However, you do not feel comfortable saying:

    9/10 + 9/100 + 9/1000 +   equals to some number y   ??

No, you have to say y = 1. Well, hypocrites, do the same for pi and e then!

158.35.225.231 19:49, 10 February 2006 (UTC)

Apples and bannas: not enough explanation in the concept

I always see this concept and every time I see it I get hung up on a rash of bad assumptions. Because I don't do decimal math all the time, when I see .999 (rep) I don't recognize it. Actually, it's not a complex number nor magic bullet -- if you remember where it comes from:

1/3 x 1 = .333 (rep) = 1/3

1/3 x 2 = .666 (rep) = 2/3

1/3 x 3 = .999 (rep) = 3/3 = 1

1/3 x 4 = 1.333 (rep) = 4/3 = 1 1/3

1/3 x 10 = 3.333 (rep) = 10/3 = 3 1/3

I fall into the trap everyone else does of overcomplicating the entire thing. Our number .999 (rep) is nothing but the remainder created when doing math on some fractions with reoccuring decimals. If we do math in decimal format, we get a decimal answer, which we then have to convert to a integer answer. Converting .333 * 3 = 1/3 * 3 = 1 is so natural that I never even stop to figure out how to get from the first to the last.

More examples:

  1 / 9 = .111 (rep)
+ 8 / 9 = .888 (rep)
 --------------------
  9 / 9 = .999 (rep)

  2 / 11 = .1818 (rep)
+ 9 / 11 = .8181 (rep)
 -------------------
  11/ 11 = .9999 (rep)

Remember, .999 (rep) is the initial remainder. To convert reoccuring decimals into integer values, divide by an equal number of nines:

 .999   999
 ---  = ---  = 1
 .999   999

Thus we have a remainder of 1 full item to add to our total!

This is grade school decimal math.

I understand all the confusion because I fall for it every time. Here is the biggest false assumption I have to dispel for myself.

False Concept: .333 (rep) is .3 with inifnitely repeating digits. WRONG! This is "notation": it means literally: When we try to determine the decimal equivilent of 1/3 we kept getting a remainder. Because the remainder repeats every time, we can do the math as many times as we need to. At any point we can stop and sum up the rest as a remainder and have a finite number. When we do math we always have a remainder that is equal to the repeating decimals. Thus .333 (rep) means we have as many 3s as we need to handle any math job, and when we do math we have some form of .333 (rep) left over. For example:

100,000 x .333 (rep) = 33,333.333 (rep) = 33,333 1/3

10,000,000 x .333 (rep) = 3,333,333.333 (rep) = 3,333,333 1/3

The remainder of 1/3 is given by taking the digits after we stop and finding their integer values by using the division by 9 process.

And what does .999 (rep) mean? It too is a finite number. It's different, though because it means that we have a series of 9s. At any time we can stop and we have a remainder that is some form of 1. When we add our 9s and the remainder, we get 1. The following are examples where I stopped mathing out .999 (rep) and created a finite version at completely arbitraty points:

 .999 (rep) =  .9999 + .0001 
            =  .999 + .001
            =  .99 + .01
            =  .9 + .1
            = 1.0

I don't pluck the "1" format for the remainder out of the air. I used 1 because that results from the dvision by 9s rule. If I decide to stop at say the sixth place:

 I have   .99999 remainder .000009

                   .000009     
 Which is .99999 +    --
                       9

 Which is .99999 +  .00001

Summed up: all numbers have three notational formats (ways of expressing them): for 1 these are integer 1, finite decimal 1.0, and repeating decimal .999 (rep).

When you begin to differentiate between the various notations for numbers, then you understand what the proofs are saying.

208.252.26.3 21:04, 10 February 2006 (UTC) jdn

I like your point about notation, although the remainder stuff is pretty sketchy. Would you like to log in? Melchoir 21:19, 10 February 2006 (UTC)

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