Talk:Proof that 0.999... equals 1/Archive 2

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Contents 1 0.999... < 1 ? 2 intro 3 About the above discussion 4 Mathematicians who think 0.999... and 1 are different numbers 5 No, we're not going to do this merry dance. 5.1 The subsection in which Melchoir attempts to impose order: 5.2 Melchoir cordons the following off so that 70.110.81.253 is not tempted to respond: 5.3 While we're waiting: 5.4 To which yet another IP says: 6 Enough. 7 Algebra Proof 8 And now for something different: 8.1 Melchoir has cordoned off the following from the above: 9 Here we go again! 10 If I may speak to the article itself... 11 Mind the following:

0.999... < 1 ?

How can he be wrong? He is referring to sequences with positive sums. What are you talking about? He is correct in stating it will never exceed 1. In fact it will never reach 1 either. As for computing infinite sums, I also, know of no formula. All I learned in high school and university is how to compute the limit of an infinite sum. The two are quite different. Finally, the fact that you can take the sum as close as you want to 1 does not mean it is equal to 1. It means that you can take it as close as you like but you will never reach 1. Just sit down and start adding up the terms and I guarantee you that you will sum until your last breath and still you will not have reached 1. Someone can continue to sum after you and he too will die summing the terms because the sum will always be less than 1. Philosophical grounds - hmmm? No, I think he is just using simple high school math. —The preceding unsigned comment was added by 68.238.99.105 (talk • contribs) 00:12, 18 October 2005.

You, also, are wrong. Of course it is correct that it will never exceed 1, and also that it will never exceed 3, but that is not the meaning of the assertion that the infinite expansion equals 1. If it were, then it would also be equal to 3. What is essential is that 1 is the smallest number that the finite truncations will never exceed.

And you are wrong that you learned how to find the limit of an infinite sum. What is taught is how to find the limit of the sequence of finite sums, not the limit of an infinite sum.

And you must have meant, NOT that he was "referring to sequences with positive sums", but to sequences with positive terms.

You wrote that "the fact that you can take the sum as close as you want to 1 does not mean it is equal to 1". You're very confused: you can take the sum of the finite truncations as close as you want to 1, but no one said those are "equal to 1". It is the infinite sum, not the infinitely many finite sums, that were asserted to be equal to 1.

Your points are very childish. If you need help in math, you could ask me or some other professional for such help. Michael Hardy 02:09, 18 October 2005 (UTC)

Your logic is 'impeccable': would it really equal 3? You appear to be very confused. It cannot equal whatever you like. It will never equal, nor exceed 1 - that is the assertion. I am talking about the "limit of an infinite sum", not "limit of the sequence of finite sums". The formula he quoted is used in determining whether an infinite sum has an upper bound. There is no assertion that it is equal to this upper bound. Your assertion is plain wrong: there is a very easy way to check yourself - start adding up the terms and I can gaurantee you, you will always have a sum that is less than 1. Please don't tell me you are dealing with a finite sum because then your assertion that the infinite sum is 1 is absolute nonsense! You may be confusing yourself with the fact that the terms are getting closer and closer to zero (Cauchy sequence). This does not mean that any term will ever be zero. —The preceding unsigned comment was added by 68.238.97.2 (talk • contribs) 10:49, 18 October 2005.

Would you please sign your postings, even if only with an IP number, so that we can know whether two anonymous postings are by the same person or different persons?

I have taught mathematics at five different universities, including MIT for three years, so I am not ignorant of mathematics, and if you want to understand these matters, you would benefit from listening to what I tell you.

You wrote: "I am talking about the 'limit of an infinite sum', not 'limit of the sequence of finite sums'". I don't know what "limit of an infinite sum" means in this context, unless it just means "limit of the sequence of finite partial sums", in which case it's a confused and confusing way of saying that.

Look: The value of an infinite sum IS the limit of the sequence of finite partial sums. They're the same thing.

In particular, the value of an infinite repeating decimal expansion such as 0.33333... IS just the limit of the sequence of finite truncations of it. The limit of the sequence of finite truncations of 0.3333333... is 1/3, so the value of this decimal expansion is 1/3.

You appear not to understand what "Cauchy sequence" means. To say that the terms of a series are getting closer to 0, or even that they are approaching 0, does not imply that anything is a Cauchy sequence. "Getting closer to 0" does not imply approaching 0 as a limit, since the terms of the sequence (1 + (1/n)) get closer to 0 without approaching 0. Moreover, that the terms of a series approach 0 does not imply that its sequence of partial sums or any other sequence associated with it is a Cauchy sequence.

As I said, if you need help in these matters, you should ask me or some other professional. Michael Hardy 18:17, 18 October 2005 (UTC)

I think you know exactly what I mean when I talk about Cauchy sequence: the distance between the terms is getting closer to zero. That's the definition of Cauchy sequence, i.e. Lt d(p,q) = 0 as min(p,q) approaches infinity. You write: "The value of an infinite sum IS the limit of the sequence of finite partial sums." This is not true. An infinite sum IS *indeterminate*. Get that? The formula used in this article to prove that 0.999... equals 1 proves exactly the *opposite*, i.e. that 0.999... does not equal 1. All the formula shows is that the limit of any of the partial sums of this sequence is 1. To say this is the infinite sum, shows extreme ignorance. The limit of any partial sum is *not* the infinite sum. You taught at Mit? So what, do you think I ought to bow down and be intimidated? You are wrong. You have been taught wrong too. This article is non-sense. The fact that you can write what you do, displays a fundamental lack of understanding. This non-truth of 0.999... = 1 has taken hold because most people don't understand that 0.999... is *not* a rational number. Neither is 0.333... a rational number. Partial sums from these sequences are used to approximate 1 and 1/3 respectively. I can understand using 0.333 to approximate 1/3 in base 10 (only because it can't be represented finitely in base 10) but cannot understand why 0.999 should be used to approximate 1 which has an *exact* representation. Don't you think it's about time you started thinking for yourself? I know how you will respond: You will say a rational number is any number that can be expressed as a/b where a and b are integers (b not 0). This definition of rational number is part of the problem. If a number cannot be represented *finitely* in a base that is well defined, then the number is not rational. Pi, e, sqrt(2), etc are irrational because there is no well defined base in which these can be represented finitely. You can't suggest that Pi, e, etc be respresented in their own base since these numbers cannot be completely determined. i.e. pi is not equal to 1.0 in base pi because the extent of pi is unknown. Similarly for e or any other irrational number. 15H51 18 October 2005 —The preceding unsigned comment was added by 68.238.97.2 (talk • contribs) 21:07, 18 October 2005.

There is nothing sacred about writing numbers in "bases" like base 10, as opposed to fractions like 1/3. Writing "1/3" or "√2" does represent these numbers finitely. That doesn't say wether they're rational or not. "1/3" is rational; √2 is irrational, but both are represented "finitely" here. Michael Hardy 01:38, 19 October 2005 (UTC)

Not entirely true. You start with 0.333... and then you try to show that it can be expressed as a/b. Or you start with 0.999..., 3.14... or some other representation and then try to show it can be expressed as a/b. A number is rational if it can be expressed in the form a/b (b <>0) with a,b integers. I maintain this is insufficient, you also need to add that the representation must be *finite* in some radix form. If indeed 0.999... is rational (it's not), then so is pi since pi can be expressed in the form a/b (i.e. 3 + 1/10 + 4/100 + ...) But of course pi is not rational because there is no number system besides pi in which pi can be expressed finitely in radix form. In base pi, pi is *rational*, i.e. pi = 10 (i.e pi + 0 units). 0.999... cannot be expressed as a/b in any number system. Please don't tell me it's 1 or 1/1 - this assumes that it is equal to 1. You need to think really hard about this. 68.238.97.2

Firstly, to answer one of your questions above: no I did not expect you to be intimidated; I expected you perhaps to be grateful.

Secondly, base 10 is no more sacred than base 3. The number 0.333..., whose expansion in base 10 is infinite, has a finite expansion in base 3, and in that base, the number 1/10 does not have a finite expansion, but an infinite repeating one.

Thirdly, the number 0.9999... with "9" repeating forever, does have a finite base 10 expansion, since it is 1.0, and can be written as a/b, where a and b are integers, since it is 1/1.

Your notions about what is a "rational number" and what is not are merely an example of what many people like to call "mere semantics". Michael Hardy 19:16, 19 October 2005 (UTC)

You have told me nothing I did not know in your first and second points. In fact, if you read my posts, you would see that I said this. Your third point is false and I pointed this out in my previous response. You cannot prove that 0.999... = 1 because you do not know the difference betwen the limit of an infinite sum and an infinite sum itself. In fact, 0.999... 'is not' equal to 1. You do not understand the formula used to show that the sum of an infinite sequence is bounded from above. Maybe you should sit down and think about it again? You have been unable to refute anything I have said and you have not even tried to understand it. If I am incorrect in stating that finite representation is 'required' for the definition of a rational number, then pi, e and sqrt(2) are all rational seeing these are the sum of their respective expansions. Frankly it has nothing to with semantics, only simple logic that even a ex-professor from MIT can't see or won't see?. 68.238.97.2

That something is required for rationality does not mean that everything that satisfies it is rational. That confuses a necessary condition with a sufficient condition. A necessary condition for rationality is not a sufficient condition to guarantee rationality.

But I congratulate you on the large number of your words. Michael Hardy 20:26, 19 October 2005 (UTC)

Talk about a lot of 'words'.... Could your rebuttal possibly be a little more abstract. You know you are wrong and just can't admit it. ... sour grapes? 68.238.97.2

You must be a retired lawyer. Michael Hardy 00:01, 20 October 2005 (UTC)

Wrong again. Retired supermodel. More profitable and unlike teaching/(child minding), no fake power-trips: the runway is a 'real' power-trip. But don't quit your day job. If your posted photo is recent, I can't tell you won't make it. Sorry, don't mean to be rude, just realistic. 68.238.97.2

Did you mean 'can tell he won't make it' ? :-) He is probably still figuring out how to make 0.999... add up to 1. —The preceding unsigned comment was added by 192.67.48.22 (talk • contribs) 13:56, 20 October 2005 (UTC).

Yes, that should have read: "I can tell you won't make it." I see he has not responded to your rebuttal. Instead he chooses to be sarcastic and rude to a lady. Frankly, he skirts the rebuttals and tries to be cunning and humorous. 68.238.97.2

Gee, if I'd realized you were Paris Hilton, I'd have recognized your mathematical brilliance. Michael Hardy 01:59, 21 October 2005 (UTC)

I am sorry but I have no idea who Paris Hilton is? Is he a mathematician? Wait, I can goolge it. Hopefully there aren't too many with that name. Now I know you love chatting but I really wish you would give this subject more thought. I can answer any questions you might have. You ought to be grateful for this. 68.238.97.2

This is quite trivial... I think the problem stems from the fact that CANNOT be regarded as a number with a definitive value. It is just said to be the largest number possible (which, of course, does not exist). I mean, the number 0.999... can be regarded as 1 minus the smallest real number possible (which is or ). But since infinity does not have a defined value, this becomes 0. Therefore .

The same goes for arguments such as 'does the expression converge to a limit?' The answer: no. Certainly, the numbers get smaller and smaller and smaller until they become almost equal to zero, but no, THE SUM WILL NOT DIVERGE TO A LIMIT (I will not prove it here).

This debate has gone on for ages; the problem stems from the fact that non-mathematicians will not understand geometric series easily, nor will they understand easily that infinity has no defined value.

I know I've dragged in quite a bit of other problems here, but my main point is that non-mathematicians will never understand this article fully.

But I am a firm believer that tapping '9' on a calculator forever is not within physical limits.  :) x42bn6 Talk 07:21, 8 November 2005 (UTC)

I think you're assuming that all infinite series are divergent. However, there do exist convergent series (like the one used in the proof that 0.999... = 1). Take a look through that article and hopefully it will explain convergence well enough. --BradBeattie 14:46, 8 November 2005 (UTC)

No, I did quite a bit of work on ratio test so I certainly know that convergence exists. x42bn6 Talk 03:25, 9 November 2005 (UTC)

You must be a non-mathematician because evidently you do not understand this at all. You do not understand geometric series or even what the difference is between an infinite sum and the limit of an infinite sum. You are not alone - most mathematicians don't understand this either. Hardy is a fine example. —The preceding unsigned comment was added by 192.67.48.22 (talk • contribs) 2005 November 9.

Yes, I do understand what the difference is. And I understand that non-mathematicians will never understand this fully until they understand the concept of infinity. And please sign your comments in talk pages using ~~~~. x42bn6 Talk 06:32, 13 November 2005 (UTC)

You are obviously a fake. No one understands infinity (not as a concept or otherwise) and you don't have a clue of what you are talking about. You are a good example of the mindset erroneous thinkers have who believe that 0.999.. = 1. You have erred and contradicted yourself several times already: First you state that infinity cannot be regarded as a number, then you proceed to write that 0.999... can be regarded as 1 minus the smallest real number possible which *you* say is 1/infinity. How can you define the smallest real number in terms of a number that is not defined?! Contradiction. Next, you fail miserably with your child-logic: "But since infinity does no have a defined value, this becomes 0." How did you reach this conclusion?! You are way out of your league. Think carefully before you post again! 192.67.48.22

Actually X42bn6s argument can be nicely formalized as a proof using the Archimedean property of the reals:

Assume 0.9999... != 1. We will show that this implies that 1-0.999... is an infinitesimal. For any reasonable interpretation of 0.999... it must be larger than any finite-length 0.999...9. I.e. . Let us call the quantity 1-0.999... for x. To show that x is an infinitesimal we have to show that for all n: . So let n be given. Let m be the smallest integer larger than log₁₀(n). Obviously , so . But then . Thus we arrive at a contradiction: if 0.999... != 1, 1-0.999... is an infinitesimal (or 0.999... < 0.99...9 for a finite-length number, or 0.999... > 1). Since the real numbers possess the Archimedean property and thus possess no infinitesemals, 0.999... = 1.

Rasmus (talk) 19:53, 15 November 2005 (UTC)

You are assuming that an infinitesimal posseses the property that |x|+|x|+.... < 1 no matter how many |x|s we sum. This is untrue and constitutes your first error. An infinitesimal cannot be quantified. Your second error is you decide to *call* your quantity 1-0.999... some 'x' - you cannot reach a contradiction on a false premise and then assume that your conclusion is true. 192.67.48.22

x is an infinitesimal if and only if . That is the definition, not an assumption. Feel free to use another definition, but unless it is equivalent to this one, you are speaking about something else. The Archimedean property of the reals is, that it does not contain any numbers (except for zero) that can be summed arbitrarily many times and still be finite. You can conceive of fields that does not have this property, but it is not the reals (see Hyperreal numbers).

Calling 1-0.999... for x does nothing for the proof except improve the readability. Feel free to substitute (1-0.999...) everywhere, it makes no difference for the correctness: Let n be given and choose m > log₁₀(n). Then , and we have proven that either 1-0.999... is an infinitesimal, 0.999...<0.99...9 for a finitelength-number, 0.999...>1 or 0.999...=1.

Rasmus (talk) 21:25, 15 November 2005 (UTC)

Your definition of infinitesimal is untrue and it is based on an incorrect assumption. You state that whatever the value of n is, the sum will never reach, nor exceed 1. Both are false. If what you say is true, then why do you not concede that the sum of 9/10^i (from 1 to n) < 1? You state that the sum of |x| (from i to n) < 1 but in the same breath you are trying to show that the sum of 9/10^i (from 1 to n) = 1 ?! You are very *confused* my friend. infinitesimal has never been properly defined. How can you quantify the number that is greater than zero yet less than every positive real number? You can give it a name, which we have: 'infinitesimal'. However, in every other respect, it is exactly like pi, e and sqrt(2), i.e. its full dimensions are unknown. To say that the reals posses no infinitesimals and then claim that pi, e and sqrt(2) (just some examples) is in itself a contradiction. Mathematicians shoot themselves in the head when they make statements such as: 'As small as you like' or 'As close you like'. How small? How close? I know mainstream thought is that the reals contain no infinitesimal. If the infinitesimal does not belong to the reals, then pi, e or any other number with similar properties does not belong to the reals also. This includes 0.999..., 0.333..., etc. 192.67.48.22

You make a lot of hand-waving here. The definition of infinitesimal I have given is the one used everywhere. You may have a different conception about what an infinitesimal is and be unable to properly define it, but the one I talk about is well-defined and well-understood. Likewise the reals. Likewise most mathematicians have an agreement about what the real numbers are, and which properties they possess. You might have a conception of a number-field that does not include pi, e and sqrt(2), but it is not the same as the one the rest of us call the real numbers.

Anyway: the proof above is interesting because it avoids limits and infinite sums, only drawing on some intuitive correct assumptions about the properties of 0.999... and the Archimedean property of the real numbers (which again is a consequence of the least upper bound property). I just mentioned it since you attacked X42bn6s intuitive understanding. There is really no purpose of us discussing the properties of the real numbers here, since the content of the article is governed by WP:NOR. Unless you can produce reputable sources for 0.999... ≠ 1, the article will continue to assert that 0.999... = 1.

Rasmus (talk) 13:31, 16 November 2005 (UTC)

Talk about hand waving! Your proofs are all fine examples of hand-waving. Contrary to what you think, the definition you provide of infinitesimal is not used everywhere. How can I have a conception of a number field that does not include pi, e or sqrt(2)? If I did, it would be incomplete and thus erroneous for pi, e and sqrt(2) are all very *real* and finite. You state that the Archimedean property is a consequence of the LUB theorem. Actually, it's the other way round. I have provided sufficient proof that 0.999... is not equal to 1 on the pages you archived. Since you are making the statement that 0.999.. is equal to 1, the onus is on you to provide proof which so far you have been unable to do. As for your *proof* being interesting in that it does not use limits and infinite sums, I would more accurately say that it is not a proof at all but mere hand-waving. And what are *reputable sources* if they don't agree with Wikipedia's views?! 192.67.48.22

You wave your hands, when you claim that the definition of infinitesimals I used is untrue without supplying references. You claim: "If the infinitesimal does not belong to the reals, then pi, e or any other number with similar properties does not belong to the reals also." From that you must either believe that the reals contain infinitesimals or that pi, e etc. does not belong to the reals. Take your pick. (Btw. it is easy to define a number field that does not include pi, e and sqrt(2). The rational numbers are one example).

You claim that the LUB property is a consequence of the Archimedean property. You have also stated that there is no definition of infinitesimals. What is your definition of the Archimedean property? We obviously have different definitions, since it is easy to see your claim is false using my definitions (The rational numbers obviously have the Archimedean property since Q &subset; R, but they do not have the LUB property ()).

Read WP:NOR for a discussion about reputable sources. In this case it would be a mathematical text-book or a peer-reviewed article. Proving something in the context of Wikipedia consists of referring to a reputable source. Disproving is the same. In case of disagreement as to what a reputable source is, we go by consensus. Using this meaning, you have neither disproved that 0.999...=1 or that 0.999...≠1. In the context of mathematics, a proper proof consists of enumerating your definitions and using these to deduce your conclusion. This is done in my proof above and in the (advanced) proofs in the article. Refuting a proof consists of showing that the deductions were incorrect. All you have done is arguing about the definitions. Hand-waving consists of claiming that something is false, but not giving references, counterexamples or proofs. You have done plenty of that.

Rasmus (talk) 14:29, 16 November 2005 (UTC)

I do not need to supply any references to show that your definition of infinitesimal is false. All I have to do is set n = infinity and already I have problems. As far as infinitesimals belonging or not belonging to the reals, it is you who have to take your pick! Firstly, you use the plural form - this is a contradiction in itself. Are there infinitesimals that are smaller than other infinitesimals? Secondly, the completeness principle states that every non-empty set which is bounded from above has a LUB. If you are using this, which you are, then you have to make up your mind whether you are using infinitesimals or not. I am not ignorant of mathematics so your discussion about fields above reveals nothing that I did not already know. Had you read my response in context (which you did not), you would have understood that I am claiming pi, e and sqrt(2) are part of the reals and these numbers have something in common with 0.999..., 0.333... in that they can only be appromixated. You archived the previous posts in which I provided *valid* mathematical proof (nothing hand-waving about this). Go back and read the posts. I pointed out errors in your 'proof' and when you realized that you had in fact written rubbish, you resorted to Wiki's NOR policy. An easy way out for you? 192.67.48.22

Sigh. Obviously n has to be a natural number. But even if we assigned some meaning to n = infinity, the proof would still work. There are no real x≠0, so that . As there are no infinitesimals in the reals, we cannot really discuss their properties (in a field that contained infinitesimals, however, it would be trivial to prove that there would be at least countable many). You still haven't given me your definition of infinitesimals or of the Archimedean property. You haven't shown how the Archimedean property implies the LUB property. It looks like you didn't really mean that "If the infinitesimal does not belong to the reals, then pi, e or any other number with similar properties does not belong to the reals also", though. There are no formal mathematical proofs in the archive. Feel free to make one here, however. WP:NOR just means that even if you should be able to produce a valid proof for 0.999... ≠ 1 and convince everybody here that it is correct, you do not get to move the article to Proof that 0.999... does not equal 1. You would have to produce a reputable source. It doesn't mean that we can't produce lots of talk-page material, that we can move to another archive, once we are done. Rasmus (talk) 19:43, 16 November 2005 (UTC)

Really? So how is it that you understand that sum |x| (i to n) < 1 where x is infinitesimal (and you don't know your ear from your nose either, never mind what an infinitesimal is or is not; you are also unable to define it in any rational way) and you use this in your faulty proof to show that 0.999... = 1 ? Look, the fact that you have a master's or a PHd in Mathematics does not mean anything. You are in many ways more ignorant than someone without any qualification at all. I have not given you a definition for infinitesimal because it is a concept that makes no sense to me. If you cannot define any concept rationally, it is in fact *illogical* (any surprise?) and consequently rubbish that cannot be used to prove anything. The Archimedean property is well known. Your webpage says you have a master's and this is something that is taught in real analysis. Were you not required to take this course? Just google it for crying out loud and you will know what it is.

As for formal mathematical proofs: There is a proof in the archive and I'll state it again:

 Sum (i to n as n approaches infinity) =  (ar - ar^n)/(1-r)      for |r| < 1

We cannot compute an infinite sum but we can investigate whether it has a limit or not. In the case of 9/10 + 9/100 + 9/1000 + ... it can be easily verified that this limit is 1. This states that even if we could sum this to infinity, its value would never reach, nor exceed 1. Thus it is *clearly* evident that 0.999... < 1.

All proofs that try to show it is equal are faulty but the one used most convincingly is a consequence of the Archimedean property:

The law of trichotomy applies *only* to finitely represented numbers, so you can't use an algebraic process that leads to 0.999... not less than 1 and 0.999... not greater than 1 implies 0.999... = 1. 0.999... is not a finitely represented number. Any arithmetic on such a number can only be an approximation (like pi, e, sqrt(2) etc). And yes, there should be a page called "Proof that 0.999... < 1" because contrary to what you think, it is not generally agreed that 0.999... = 1. Except perhaps in the case of the fools who run Wikipedia? 192.67.48.22

Hmmm. If you prefer to google, rather than reading a proper textbook on the subject, let us do that: The two top hits are our own Archimedean property, which uses the exact same definition as me, and Planet Math, which uses a slightly different, but equivalent formulation (Exercise: show how they are equivalent. Hint: use corollary 1, select y=0.5 and find the contrapositive). So now we have established that there are no non-zero real numbers x, for which , we should be able to agree on the validity of the proof!

As for your 'proof', let me see if we can clear up what you mean. First you type:

Sum (i to n as n approaches infinity) = (ar - ar^n)/(1-r) for |r| < 1

I assume you mean (you don't specify the left hand side).

We can certainly agree on that.

Then you assert that the limit of 9/10 + 9/100 + 9/1000 + ... is 1. I assume you mean

.

That takes some more work to prove, even using the above, but if you accept that , I won't disagree (the proof is not hard, but it takes some work to get all the epsilons and deltas right).

Then you say:

"This states that even if we could sum this to infinity, its value would never reach, nor exceed 1. Thus it is *clearly* evident that 0.999... < 1.".

When someone says *clearly* it is a clear sign that they are handwaving. Please prove this assertion! Hint: It might be a nice start to define exactly what you mean by 0.999... . Most people would mean

Ouch. You now claim that (R,<) isn't trichotomous. Most people define the real numbers so that (R,≤) is a total ordering. Care to prove your claim? Or just define your ordering. In any case, if you hold to this claim, you are talking about a different set of numbers than what the rest of us call the real numbers (and in that case anything might be true. In Z/2, 1=3).

By the way, you still haven't shown how the Archimedean property implies the LUB property.

Rasmus (talk) 14:32, 17 November 2005 (UTC)

My word but you do love yourself, don't you? And you sure know how to use this system. If I knew it half as well as you did, I would draw some nice sigmas, infinity symbols and why, of course beautiful epsilons and deltas to make every Phd green with envy. Now, there is no handwaving in anything I wrote. It is very clear that the sum on the lhs will never exceed 1:

If we split up the quotient as follows: a/(1-r) - ar^n/(1-r) the first term is independent of n and its value is 1. The second term becomes very small (and using Weierstrass's faulty logic - 'as small as you like' but always greater than *zero*). Thus we have 1 - s where s is some value greater than zero. This being the case, when we consider the difference, we always have a value that is *less than 1*. This is very *clear*. Got it? Hey, if you don't get it now, you must be thicker than I thought. Please don't tell me this is not mathematical or robust enough or else you are a disgrace to all the institutions of learning you have ever attended. Look, when I use words like *clear* and phrases like *by definition*, I do not use these in the same ignorant way as most Phds do. So relax. Don't build a brick wall around everyone else when you feel it necessary to do this for yourself. 192.67.48.22

You don't need to draw the nice LaTeX formulas to make yourself clear. You do however need to setup the formulas correctly (not being lazy and skipping part of the equations) and be rigorous in how you setup your definitions and how you apply them. You might also want to skip the ad hominem arguments, you can't prove anything in maths using those. That being the case, let us get back to your proof:

You say: . Obviously 1 − 10 ^{− n} < 1 for all . But for any reasonable definition of 0.999..., there is no natural number n, so that , so you can't prove anything from that. Now it comes down to how you define 0.999... (which you haven't done yet). Most of us define 0.999... as ; but I guess you are thinking of some other definition? Anyway, just claiming that it is "*clear*" that because something holds for any finite n, it somehow also applies in the limit, doesn't make it true. You will have to prove it. And you can't really prove anything about 0.999... if you don't first make it clear which number you are talking about. While you are at it, please show me where to find your alternate definition of the Archimedean property, how to show that the Archimedean property implies the LUB property and give me your definition of the real numbers that doesn't include a total ordering.

Rasmus (talk) 20:22, 17 November 2005 (UTC)

Firstly, I am not attacking you or anyone else and your psychoanalysis is deeply in error just as is your mathematics. Your above formula is incorrect: It's not (9/10 x 10^-n)/(1-.1) but rather (9/10 + 10^-n)/(1-.1). While you are enjoying Latex so much, you may as well do the job right. Okay, so you made a typo. I'll forgive you for this. Now let's move on. You say there is no natural number s.t 0.999... = sum (i to n) 9/10^i Well aside from stating the obvious, what are you trying to say? My proof considers what happens to the difference as n becomes infinitely large. There is nothing strange about this - it's used in limits and calculus and many other branches of mathematics. Regarding my proof: it is very *accurate* and *valid*. The problem is not with my proof but with your *understanding*. You are very confused. You have not answered my question:

You state that sum |x| (i to n) < 1 where x is infinitesimal (yet you are unable to define infinitesimal in any rational way) and you use this in your faulty proof to show that 0.999... = 1 ?

While you are trying to answer this, let me pose some more questions to you: If the real number system has 'holes' (as you claim it does), then how can you use epsilon-delta proofs at all? What does 'as small as you like' and 'as close as you like' mean? How small is small and how close is close?

This is true handwaving mathematics that has been taught the last 100 years. Real analysis is mostly a load of rubbish. Unfortunately you are the product of Weierstrass' ideas and logic that have some serious flaws. In answer to your question: I know the Archimedean principle the same way as it is published on planet math. Finally, please stop regurgitating what they taught you in your real analysis and honor's class and start thinking for yourself. This will be the best thing you can ever do. 192.67.48.22

Calling people ignorant and saying that if people do not agree with you is ad hominem arguments, and generally regarded as poor style. As for the formula, I won't claim there is no mistakes, but I can't really see what you are referring to. If it should be + rather than x, you would get for n=2: 0.9/(1-0.1) - (0.9+10^-2)/(1-0.1) = 1-0.91/0.9 != 0.99.

You wan't to prove something about what happens to the difference between 1 and as n grows to infinity. Great! How will you do that? Weiserstrass and I would do it by proving that since for any ε>0 we can find a n so that the difference is less than 0, the limit is 0 and thus . I would guess you are trying to argue that since the difference is >0 for all n, then also . That is not at all "*clear*". Actually, I claim that it is false, so if you want to convince me, you ought to provide a proof.

As for infinitesimals. I state that if for all x, then x is infinitesimal. That ought to be a clear definition. I also claim that the real numbers contain no such numbers. A different formulation would be that for all real numbers x!=0, there exists a natural number n, so that . That would work just as well for the proof, since we have shown that for all natural numbers n, thus either 1-0.999...<0 or 1-0.999...=0.

What are those "holes" you claim I postulate? "as small as you like" and "as close as you like" can be translated into clear logical statements: for a sequence {x_i} to "become as small as you like", it has to hold that . But you might take note of the fact, that the "Archimedean property"-proof did not use any limits or epsilons or deltas, so even if you reject Weierstrass methods, you still have to accept that proof. If you agree with PlanetMath, you should have no problem with the proof (unless you still claim that the real numbers aren't totally ordered).

You still need to show that the Archimedean property implies the LUB property and give me your definition of the real numbers that doesn't include a total ordering. Rasmus (talk) 14:32, 18 November 2005 (UTC)

Your formulas keep changing so I will just ignore these.

Unlike you and Weierstrass, I will not use epsilon-delta arguments because these imply the existence of infinitesimals. You are right about one thing: I am arguing that since the difference >0 for all n, then it is also true that the sum < 1. And yes, it is very *clear*. You claim it is false and I challenge you to find a counter example! So far you have only been able to regurtitate rubbish. I passed my real analysis course a long time ago. It was garbage then and it's still garbage now. My professors were ignorant then and today I find them to be even more ignorant. And saying that you are ignorant is not *ad hominem*. Once again my friend, you are trying to psychoanalyze me. I suggest you stick to proving the math and forget about everything else.

So here's your first exercise: find me an 'n' for which my claim is false and then I will believe you. No more BS (and MS and PHd) please. 192.67.48.22

So you claim this:

I do not challenge the truth of the left-hand side of this deduction. I challenge the "=>" part. You haven't given any arguments (not to mention proofs) as to why you should be able to conclude the right-hand side from the left-hand side. And no: "clearly" isn't an argument, not even if you surround it with asteriskes. You challenge me to find a 'n' for which the above it false. Please read Logical conditional for a discussion about how you prove or disprove a logical implication.

But let us play around with it a bit: If the above holds, then *clearly* also:

We have just substituted 0.999... for 1, and the left-hand side follows easily from the equations we studied earlier. So 0.999...<0.999...?

Ignoring my proofs won't make them go away. You haven't been able to show any incorrectness in any of the versions (beyond trying to redefine the real numbers). You haven't even given a proper definition of 0.999..., not to mention proving how the Archimedean property implies the LUB property and giving me your definition of the real numbers that doesn't include a total ordering. Rasmus (talk) 21:00, 18 November 2005 (UTC)

Nonsense. You made a rigid statement that what I *proved* is false. You then contradicted yourself by stating that you have no objection to the lhs and continue to talk screeds about *proofs*. You do not challenge the lhs of this equation? If you do not challenge the lhs, you cannot challenge the rhs and yes, you cannot challenge the implication either unless you able to provide a counterexample. So far, you have displayed exceptional ability with Latex but your arguments lack any sound proof or direction. You are *clearly* lost and are only deceiving yourself. You have *not* said anything *relevant* or *logical*, much less *proved* anything. Your diversion tactics show that you are trying to save face. Once again, the *implication* is *clearly* that 0.999... < 1. The nonsense you wrote about 0.999... < 0.999... does not deserve a response. I have defined 0.999... *clearly*: Once again: 0.999... = 9/10 + 9/100 + 9/1000 + .... It is an infinite sum. Now you do not know the difference between an *infinite sum* and the *limit of an infinite sum*. I am going to tell you these things are *not* the same: not by *definition* or *otherwise* - I leave this as an exercise for you. The Archimedean property states that if y is any real number, then a number p exists such that y < p. This implies that p is an *upper bound*. It does not take too much intelligence to figure out where this leads. The rest is an exercise for you to show how it leads to the definition of some least upper bound. If you have an MS in mathematics, this is something you ought to be able to prove on your own. Finally, forget about the Archimedes principle and stop trying to coerce me into giving you a definition of a real number system with a different ordering. Please stay with the subject. You will not earn any points for a dissertation on irrelevant topics. 192.67.48.22

You seem to lack some basic understanding about the concept of implications. Consider this. Let me claim:

(note the = on the rhs).

We clearly agree that the lhs is true. Is it then true that you can't challenge the rhs? Or can you provide a counterexample?

It looks like you believe that something like this holds: Obviously the "0.999... < 0.999..." example I gave is a counterexample to this (and actually, it can be shown that the lhs implies that b = c).

You haven't been able to properly articulate the difference between an infinite sum and the limit of a sequence of finite sums. You keep claiming that they are different, but you haven't explained what you believe an infinite sum is. You keep trying to make a proof using some property of the finite sums, but you haven't provided any link between the finite sums and the infinite.

You should try rereading Least upper bound. Apparently you have forgotten what it means. Obviously any single number has an upper bound (it is limited by itself - you do not need to invoke the Archimedean property to prove that). But to prove that an ordered set S has the LUB-property, you have to prove that any upper bounded subset of S has a least upper bound in S. So your proof-sketch isn't even close. (And since we have already seen that the rational numbers has the Archimedean property, but not the LUB-property, I am still really interested in seeing how you are going to actually prove your assertion).

In trying to find an error with the proof I gave earlier, you claimed (contrary to the common definition) that in the real numbers "The law of trichotomy applies *only* to finitely represented numbers". It is you that are trying to change the subject by not answering this. Do you only answer the easy questions? Rasmus (talk) 15:30, 21 November 2005 (UTC)

You are evidence that an Ms in mathematics is absolutely worthless. You cannot claim that the lhs implies 0.999... = 1 if the lhs specifically states 0.999... < 1. How do you arrive at this logic?! I don't need to challenge your rhs - it is irrelevant and gibberish that only you seem to understand. How can I challenge something that is incorrect? If you provided a correct statement and then drew an implication from this, I would be able to respond sensibly. But I can't respond to nonsense. You need to learn what 'implication' means. I am stating that you cannot find the result of an infinite sum - you can only investigate what its limiting value is (if it has one). This is a clear distinction between a infinite sum and the limit of an infinite sum. Your example that 0.999.. < 0.999... is *not a counterexample* but rather an example of the nonsense you are writing.

LUB: This is the smallest of all the upper bounds a number can have. In mathematics, if S is a set of upper bounds then there exists an element m of S such that m is less than or equal to all the other elements of S.

The law of trichotomy (LOT) applies only to finitely represented numbers: What this means is that if you have a representation of any number in any radix form that is infinite, LOT no longer applies. If this is untrue, *all* your arguments fail without any further support because 0.999... and 1 are two different numbers and yet you claim they are equal. Please don't tell me they are different representations of the same number. They are *not*. Radix systems are designed for *unique* representation. You were not paying attention in primary school when they taught you about 'tens and units'. Perhaps you should go back to primary school and relearn what you seem to have forgotten.

Oh, and if you are trying to put words in my mouth, then at least you ought to try and do it right. I would say: and not what you stated. To say that it implies b < c is nonsense. It is already given that b < c. Boy, you are a sorry MS! 192.67.48.22

The lhs doesn't prove that .999... < 1. It proves that for all finite n. That is very much not the same thing. You simply don't understand the concept of limits. The lhs impies the rhs in Rasmus's most recent version due to the concept of limits (and the fact that you can become arbitrarily close to 1 - that is, with any finite number of nines, you can come as close to 1 as you want. .999... and 1 are both the limit of that sum as n approaches infinity.) And your patronizing of Rasmus doesn't make your arguments look any more viable. --HeroicJay 17:36, 21 November 2005 (UTC)

And the law of trichotomy doesn't say anything whatsoever about only applying to numbers with a finite representation. It simply says that, for any two real numbers x and y, exactly one of these three things is true: x > y, x = y, or x < y. That's the whole law. --HeroicJay 17:46, 21 November 2005 (UTC)

For the new little bit that you added, might I point out that it makes no sense. If , then there exists no b for which . It's a contradiction in terms. Plus, your rhs pulls an i out of thin air (it's no longer part of the limit.) (EDIT: Scratched out because he replaced it with an n, even though that still doesn't make the assertion he thinks it does. The part about nonsense concerning limits still appplies.) --HeroicJay 17:50, 21 November 2005 (UTC)

HeroicJay: You should add lack of reading comprehension to your list of setbacks: *I* said that the law of trichotomy applies to numbers with a *finite representation*. Why don't you allow Rasmus to defend himself? I am sure he does not need your help yet. 192.67.48.22

I know *you* said it. *You* were either mistaken or lying. And, given that, I don't think *you* should be the one talking about reading comprehension. --HeroicJay 18:00, 21 November 2005 (UTC)

*I* did not add it - *Rasmus* added it. He was trying to prove his point. So in saying I am wrong, you are saying he is also wrong. However, from the way *you* (I will continue to do this because it annoys little boys like *you*) write, I can tell you are some immature young punk who has nothing better to do with his time. Have you not heard the saying?

******Speak when you are spoken to boy!******** 192.67.48.22 —The preceding unsigned comment was added by 158.35.225.229 (talk • contribs).

You must play by the rules here, which state that the talk page is only to be used to try to improve the article. In particular, you are to remain civil at all times, and no personal attacks are allowed. If you do not obey these rules, you will be blocked from editing Wikipedia. -- Jitse Niesen (talk) 21:38, 21 November 2005 (UTC)

So, you're going to deny what you just admitted that you admitted. Even though I just searched the whole conversation and the only person stating that the law of trichotomy only applied to finitely represented numbers was you, while Rasmus himself insisted that that was not the case after you said it was the first time. Okay, whatever. And I thought this was a public website that anyone could contribute to, so thank you for proving me wrong about that while not responding to any of the facts that I presented just because I wasn't the person you were initially speaking to. I'm sure Rasmus was extremely offended by my addition to this conversation, as I must thank you for pointing out, which is why he said absolutely nothing to or about me when responding to comments that you made after I made my first comments in this conversation. I'm sure his lack of response indicated how angry he was. And thank you for being so much higher and mightier than me, which you apparently deem yourself to be. Now, would you like to discuss facts, or would you like to lie about who said what and insult me some more? --HeroicJay 20:27, 21 November 2005 (UTC)

Anyway, more on limits. The definition of the limit of a sequence of real numbers is as follows: . You're saying that your e is c - b? Well, no matter what, there's an n such that, for all x > n, c - a(x) < c - b, which in turn states that a(x) > b (remember, you flip an inequality sign when multiplying by a negative number. Also, since all a(n) with finite n in this case are smaller than 1 anyway, the absolute value sign has been negated.) Don't even waste your time telling me that this isn't the case; that is the definition of limits. --HeroicJay 18:22, 21 November 2005 (UTC)

192.67.48.22: Obviously your manners are as bad as your math skills. This is not a private forum and it it quite ok for HeroicJay to join the fun. If this is the way you normally behave when confronted with people who argue against you, it might explain your conviction that your arguments are true: Everybody who have attempted to engage in dialogue with you have become disgusted by your lack of manners and ended the conversation, and you have mistakenly taken that to be an admissal of defeat.

HeroicJay: Welcome to the fun! I apologize in advance for repeating some of your arguments, I will probably just go through 192.67.48.22's posting and reply to his points one by one. (And everybody else: let us know if we should take this someplace else. I realize we are way off-topic here).

Back to 192.67.48.22: You claim that "the lhs specifically states 0.999... < 1". It does not. Your implication could be written in english like this :"(all finite representation numbers of the form 0.99...9 are strictly less than 1)=>(the infinite representation number 0.999... is strictly less than 1)". By any reasonable definition, 0.999... is not equal to any finite representation number.

You claim to use the rule: . Well, first of all: what you need is a rule that concludes that b < c (here , and c=1. Note that there is no n, so that b = a_n.) Secondly, even disregarding that, this form of implication would be what is called begging the question: your premises include the conclusion (namely that a_n < c). Note that the rule I wrote, did not have this problem, since I only assumed that not a_n < b < c (you might have misread that, judging from your comments).

As for your LUB-paragraph: you simply seem to invoke the greatest lower bound-rule on the set of upper bounds. Since the existence of LUB is equivalent to the existence of GLBs in a field, you are not getting closer to proving that the Archimedean-property implies the LUB-property. You are also completely ignoring the fact the the rational numbers have the Archimedean property but not the LUB property.

If you maintain that it is not true that for any real numbers x,y, either x<y, x=y or x>y; you are not talking about the same numbers as the rest of us. If that is the case, any discourse with you is pointless. (Though I wonder if you realize, that if you do not allow for the law of trichotomy for 0.999..., you cannot conclude that 0.999...<1 => 0.999 ≠ 1 ?)

Your handwaving about the purpose of radix-systems is pointless. Obviously radix-representations aren't unique since 0.1=0.10 . If we can accept these two representations of the same number, we should have no problem with accepting that 0.999...=1, if it is needed to make the system consistent with the underlying math. And as for your comment about primary school: perhaps you are trying to do primary school-math, the rest of us aren't. Once infinity is invoked, things gets a bit more complex and we can't rely on our primary school instincts any more. Rasmus (talk) 21:27, 21 November 2005 (UTC)

HeroicJay: Am sorry I hurt your feelings. However, you have not stated anything that I don't know and you ought to realize that simply commenting without relevance to the topic at hand does not help anyone. And, you were *riling* me with your provocative comments on my use of *asterisks*. I use these for emphasis because many people miss the main point of what is being written. If it annoys you so much, perhaps you should have stayed out of the fray, no? Rasmus has misrepresented so many facts, got so many things wrong and still adamantly insists he is correct. I have enough on my hands dealing with Rasmus without having to respond to you.

Back to Rasmus: Your entire previous paragraph is nonsense and all the accusations you make are baseless. I am taking you to task: find me one n for which the formula is untrue and I will believe you. BTW: Don't talk about LUB and Archimedes - you don't have the remotest idea what these are about and your writing shows that you lack understanding. So, let's deal with one thing at a time. Provide proof that the sum (i=1 to n) 9/10^i < 1 does not imply .999... < 1. The inequality you introduced is baseless and irrelevant. It demonstrates clearly how confused and deceptive you are. 192.67.48.22

You need to provide more context to your challenges. Which formula is it you want me to find an n for? Obviously I am not denying that . But you have not given the faintest argument for how you would use it to argue your claim that . Are you claiming that ?

Please remember that the discussion started out by you trying to refute the proof I gave. I have shown how your claims about the proof being incorrect were founded in an incorrect understanding of the real numbers, but you keep evading my challenges to either confirm that you work with another definition of the real numbers than the rest of us, or to admit that your claims were incorrect. For each step in "my" proof, I can show how it is founded in either some property of the real numbers (Archimedean property, total ordering etc.) or some self-evident property of 0.999... (0.99...9 < 0.999... <=1). You, however, when challenged to explain which property of the real numbers lead you to believe that "sum (i=1 to n) 9/10^i < 1 implies .999... < 1", evade the issue and challenge me to disprove it. Well, since you refuse to generalize it to another rule (such as the one I gave above), the only way to disprove it is by showing that the lhs is true, while the rhs is false. Well, the lhs can be shown to be true by using the sum for the geometric series, and the rhs is false, since by my earlier proof, 0.999...=1 (and since (R,<) are totally ordered, 0.999...=1 => (not) (0.999...<1)). I am sorry if you do not like this argument, but the only way to disprove the statement (A => B) is to prove A and disprove B. Rasmus (talk) 14:25, 22 November 2005 (UTC)

You have not provided any proof and the discussion did not start out with me disproving your 'proof'. You stated that and asked me if this is what I was claiming to which I responded in the affirmative. And yes, I am claiming that Show me *one* example where this is not true. A very simple proof is one by induction. Do you need me to help you with this? How did you obtain an MS in mathematics?

Simple Proof by induction:

 We have that k is true:  sum (i to k) a_i < 1

 Is k+1 true?   Yes since   a_k+1 + sum (i to k) a_i < 1  because no carry is possible.

 Thus it follows that we can choose any k and always find that k+1 is true. Q.E.D.

192.67.48.22

I already did. Let and . Then the lhs becomes , which is obviously true, and the rhs becomes , which is obviously false. Rasmus (talk) 14:50, 22 November 2005 (UTC)

You are in error. The rhs does not become what you state. Your reasoning is completely in error. How do you arrive at this? You cannot assume 1 = sum (i to infinity) 9/10^i and then use it in your argument! This proves nothing!! 192.67.48.22

I did not assume "1 = sum (i to infinity) 9/10^i". I chose a_i and x as mentioned above and substituted them in the implication. Surely you are able to substitute? Let us do it slowly: the lhs was:

.

We insert and :

That is the lhs. Now for the rhs:

again we insert a_i and x:

Putting both together we arrive at:

A true statement implying a false. We have made no assumptions beyond the original implication, yet we have an obvious contradiction.

As for your induction "proof", induction works in this way: If you have propositions P(k), and prove that P(1) is true and that P(k) => P(k+1), you have shown that P(k) is true for all natural numbers k. Since is not a natural number, you can't use this to show anything about infinite sums. Rasmus (talk) 15:26, 22 November 2005 (UTC)

No man, you got it wrong again. You *are* assuming that "1 = sum (i to infinity) 9/10^i" whichever way you look at it. You can't just *substitute* what you like either. As for my proof by induction - it is perfectly valid. Infinity does not have to be a *natural number*, nor does it have any relevance. Mathematical induction works regardless of what value n might assume. You are full of BS. Just eat humble pie and admit you wrong. Anyone with any sense reading this will be laughing at you. It's easy to tell you are a fake by some catch-phrases you use like "Invoke infinity" and by virtue of the fact that you waddle a lot about a lot of irrelevant side topics. Definition of Mathematical Induction: The truth of an *infinite* sequence of propositions P_i for i=1,...,infinity is established if (1) P_1 is true, and (2) P_k => P_k+1 for all k. This is the principle of mathematical induction. 192.67.48.22

Of course, I can substitute whatever I want. If you claim that , you must be able to choose a_i and x, however you want. I chose them to provide an obvious counterexample to your claim.

As for induction, you need to read up on Mathematical induction. You keep ignoring my reduction ad absurdum examples, but assume that what you claimed was true: Then let P(m)="m is a natural number". Obviously P(1) is true, and since the natural numbers are a group, if m is a natural number, then m+1 is a natural number, too. So P(m)=>P(m+1). But infinity is not a natural number. Rasmus (talk) 16:12, 22 November 2005 (UTC)

No. You cannot claim whatever you want and then use it to prove your point. Whatever you start off claiming must be *true*. So when you make a choice, it must be logical (and true - surprise?). You keep harping on the fact that infinity is not a natural number. So what? As I explained, the theory of mathematical induction deals with the truth of an *infinite* set of propositions. I am not concerned with *infinity* itself but rather about my propositions. I do not even try to deal with infinity. Please quit making loose statements such as "the natural numbers are group". Stay with the subject. HeroicJay makes some statements about the concept of infinity in higher math: even in *higher* math, we do not evaluate what happens at infinity but make assumptions regarding limiting values (if these exist and we know they do not always exist). Look Rasmus, you are wrong and don't have the balls to admit it. Male chauvinist pride? 192.67.48.22

I told you before to discuss the facts by themselves. It is never necessary to talk about the people you are having a discussion with. Stick to the maths. For your convenience, I repeat the relevant policies I mentioned before: Wikipedia:Civility. and Wikipedia:No personal attacks. Thank you. -- Jitse Niesen (talk) 18:57, 22 November 2005 (UTC)

Look cute boy, you may be the hottest mathematician but you don't appear to be the brightest. Please get off my case. I am not attacking anyone. Rasmus and others have been very disdainful to me, yet you have not reprimanded any of them. And I am the only one *sticking* to the math. Finally if anyone is going to take things too personally, he/she should refrain from the discussion. At the end of the day, nothing on this page is going to cause me to lose any sleep. You should be thankful that I am taking some of my valuable time to improve the content of your site. I believe that knowledge should not only be *free* but also *correct*. 192.67.48.22

I maintain that your comments were not civil. I would have said the same to Rasmus if I had seen him say something similar.

I don't think your comments are improving the contents. All mathematical textbooks say that 0.999... equals 1. It is the purpose of an encyclopaedia, like Wikipedia, to report what the experts say. So, Wikipedia will say that 0.999... equals 1. If you want to change that, then you have to publish a paper or a book proving that 0.999... does not equal 1, you have to convince some mathematicians so that they will also write that in their papers and books, and then it should be picked up by Wikipedia. This is one of the other policies here: our purpose is not to commit research ourselves, see no original research. -- Jitse Niesen (talk) 15:42, 23 November 2005 (UTC)

Why on earth do you have a talk page then? To see who can cite the most references even if idiots are the originators? How does publishing or writing a paper make knowledge more reliable? Thousands of papers have been written that contain junk worth less than the paper it's written on. There is no orginal research here, only logical rebuttal of past ideas that are incorrect and need to be changed. Given that there are many different points of view on this subject, you ought to represent both points of view equally well. To publish only what some academics think, is biased, morally wrong and does nothing to improve the quality of Wikipedia. It only propagates false knowledge and ideas. If this is your policy, then you seriously need to change it! 158.35.225.229 16:47, 23 November 2005 (UTC)

It is not only what some academics think, it is what all academics that I've come across, and certainly all mathematicians, think. If you want to try to change the policy, you can discuss this at Wikipedia talk:No original research or Wikipedia:Village pump (policy). I think the policy has served us very well. -- Jitse Niesen (talk) 17:40, 23 November 2005 (UTC)

You are wrong. You speak only your opinion when you say all mathematicians. I would certainly not call you a mathematician or any of Wiki's puppets. It is what you choose to believe. You are arrogant and foolish. I will not waste my time with you anymore. 67.10.167.95 00:06, 24 November 2005 (UTC)

Full circle. I again refer to my 4 < 4 reductio ad absurdum argument below, or, for an infinite summation argument, . I think Mr. 192's biggest problem here is that he's treating infinity as though it is a natural number - an arbitrarily big one, specifically. It's not a number at all. It's a concept, which is really only useful in some concepts in higher math, such as limits and integrals and so forth. --HeroicJay 16:54, 22 November 2005 (UTC)

You claimed that the implication was true. I provided a sequence a_i and real number x for which it wasn't true as a counterexample. Ergo, the implication can't be true.

Yes, using induction you can prove the truth of an infinite set of propositions. However, that does not mean you can ignore for which infinite set you prove your propositions. I probably repeat myself now, but using your proposition P(k)="", induction only proves the truth for . There is no so that P(k)="".

That (N,+) is a group has a clear meaning, and is sufficient to conclude that . It is also relevant, since it is a counterexample to your assumption that

Rasmus (talk) 18:58, 22 November 2005 (UTC)

I repeat myself: you have provided no counter-example at any time. You cannot start out by choosing a_i and x as you *please*. I explained this to you. Your choice has to be *true*. You are assuming that sum (i to n) 9/10^i < sum (i to infinity) 9/10^i. This is correct. However, you cannot proceed to argue that sum (i to infinity) 9/10^i < sum (i to infinity) 9/10^i because these are exactly the same thing! My, but you are slow. That (N,+) is a group is *irrelevant*. The last part of your statement is yet again nonsense: You write: For all k is an element of (N union infinity) .... - N is *infinite*. What nonsense are you writing? And what exactly does it mean to have (N union infinity) ? Rasmus, just admit you are wrong. Deal with it and move on. You are trying to save face but everyone who reads this knows you are wrong. You are only making a fool of yourself each time you respond the way you have. 192.67.48.22

N is the set of natural numbers. It is infinite in size, but it does not include infinity. Hence, means "The set of all natural numbers, and infinity." Now, it still seems you do not understand the point of reductio ad absurdum. The point is to take a logical construct and find a case leading to a contradiction if it's true, meaning that the original construct was false. As Rasmus was attempting to explain, your main argument was in the form . If this is true, it must be true for all a_i and x so long as the left side of the implication holds. He was substituting an a_i and x for which the left side held, but the right didn't (because, as you so keenly noticed, it makes a number less than itself.) Thus, cannot hold in all cases, and you have yet to provide a special reason why your a_i and x hold but others don't. And re: your comment earlier about assuming what happens at infinity, it's not really an assumption if you can prove it. And finally, can I assume that your "everyone" doesn't include me? --HeroicJay 21:13, 22 November 2005 (UTC)

Do you even know what a counterexample is? If I make some claim, say , a counterexample is a pair of numbers x,y so that x² > y but not x > y. In this case x=-1, y=0 would be a counterexample. I am not sure what you mean by restricting the choice to "true" choices. If you mean restricting to choices for which the implication holds, you wouldn't be able to give a counterexample to something so obviously false as the implication above.

I meant , not . Sorry. The point was that you try to conclude the truth of the statement from the induction proof. N is infinite, but does not include the element . You are not applying the Principle of Mathematical Induction correctly. Rasmus (talk) 21:25, 22 November 2005 (UTC) (I wrote this at the same time as HeroicJay, which is why we make the same points)

Actually it is you who are not applying the principle of induction correctly. Firstly, you are in grave error to write P(infinity) - there is no such thing and we never consider any such proposition. You need to learn what induction really means. I have stated it correctly and you have failed to understand it. I have *proved* conclusively using induction that 0.999... < 1 for any value of n. I do not need to consider P(infinity) - in fact this is impossible to consider in any scenario. As for counterexample - I know very well what it means - even before you were born. You have failed to provide any counterexample and it is evident you don't know what it is. If Newton said that 0.999... = 1, I would call him a fool. How much more I would call less intelligent men like you a fool? You are regurgitating (incoherently) what was brainwashed into your head at whatever institution you attended. So far all you have done is ramble on about irrelevant topics and have not provided a shred of logic to show that 0.999... = 1. And you have been unable to refute my proof in any way. Finally, if my induction is incorrect here, then all mathematical induction is *incorrect* and it cannot be used to prove the validity of any proposition. 192.67.48.22

(This has been detatched from what it replied to, so I'm removing the indentation.) Even if Mr. 192 isn't swayed by that, I now know what he meant by "Rasmus added that" when I went out of my way to show that his a < b < c assertion was meaningless with the limit (yes, it's true when you type . He apparently misunderstood the inequality sign.) I had thought he was still going on about the trichotomy thing. --HeroicJay 21:47, 21 November 2005 (UTC)

Heh, I was following the conversation and was about to make a similar point to the above by Rasmus until I saw it. In plainer English, while it is true that .9999...9 (with a finite number of nines) is less than one, no matter how large the finite number of nines, it also happens to be less than .999... (with infinite nines), so it doesn't really prove that .999... is unequal to 1. That argument is (loosely) similar to this: 1 < 4, 1 + 1 < 4, 1 + 1 + 1 < 4; thus, 1 + 1 + 1 + 1 < 4. Q.E.D. It's much more obvious that my "argument" is silly, but it follows the same principle: just because a bunch of numbers smaller than .999... are also smaller than 1 doesn't make .999... smaller than 1. (Technically, a better parallel would be , which is absurd to anyone who understands the terminology, but that uses infinite sums and infinity and Mr. 192 whines about those.) --HeroicJay 01:26, 20 November 2005 (UTC)

Really? Do you care to prove what you wrote about 0.999... < 0.999... ? Do you know what QED means? You have demonstrated nothing but your ignorance.192.67.48.22

I was being ironic. Of course I don't believe that 1 + 1 + 1 + 1 < 4; that's just ridiculous. But it's a consequence of the line of logic you're providing. And the fact that 0.999... < 0.999... is a logical consequence of the same, that is, that because .9 < 1, and .99 < 1, and .999 < 1, and so on for all finite numbers of digits, that .999... < 1. But .9 < .999..., and .99 < .999..., and .999 < .999..., and so on for all finite numbers of digits, so you didn't *prove* anything *clearly* at *all*. --HeroicJay 17:36, 21 November 2005 (UTC)

I have another way to 'sort-of' prove that 0.999... = 1. It doesn't exactly prove that 0.999... = 1 but the same principles work. Assume a graph f(x) = x². Then, by differentiating by first principles, . Then assume we want to find f'(2). We can set δx to become closer and closer to 0 in order to find the gradient at that point. Notice that , ,

... Then we can see that therefore . However, if f(x) = x², then we know that as then , and then f'(2) = 4. It must be then that as is defined to be zero. But the basis of my proof is that 4.000...1 (4, dot, an infinite number of 0s, 1) = 4. And then 5 - 4.000...1 = 0.999... = 5 - 4 = 1. Therefore 0.999... = 1. If you do not agree with this proof (except for very stupid errors I have made, such as not fixing my LaTeX properly), then, unfortunately, you must disprove all of calculus as it is all based upon it. x42bn6 Talk 07:10, 23 November 2005 (UTC)

Not so. You have not proved anything, nor said anything new. You make statements such as 10^(-infinity) is *defined to be zero*. Nothing could be farther from the truth. Zero is defined as *zero*, nothing else. I can see you are also a good product of the establishment. You need to start thinking for yourself - this is all I can say to you. See page on Mathematical analysis for an example of how you can find derivatives without even considering what an infinitesimal is. 192.

Then assume that as instead of putting . I just put that there instead of putting .000...1. x42bn6 Talk 01:52, 24 November 2005 (UTC)

intro

Passed by and changed the intro. No article on Wikipedia is a place of discussion, nor should any article strive to "prove" anything. Articles are here to present the facts in as unbiased a manner as possible. This article, if it continues to exist, should give information on known proofs of the concept. Nothing more, nothing less. Discussion is for talk pages. -- WikidSmaht (talk) 10:55, 20 November 2005 (UTC)

This article not only misrepresents 'facts' but is completely biased and is the 'opinion' of the Wiki clowns and authors. If Wiki allows this article to exist, it demonstrates clearly to any astute thinker the dependability of information from Wiki. You are a laughing stock amongst many readers. Any time someone desires a good laugh. all he/she needs to do is search for a Wiki article on a controversial topic. 0.999... has never been equal to 1, is not equal to 1 and will never be equal to 1. This is the fact. What you write here is absolute rubbish. Many of your articles are non-factual and contain serious errors because your editors are idiots. —The preceding unsigned comment was added by 158.35.225.229 (talk • contribs) 14:09, 2005 November 21.

This article is one of many in mathematics, where the consensus is that proofs are as important as facts. Most proofs have been refactored into their own pages. It is necessary to discuss context for the statement, its meaning, and its proofs, as is routine in mathematics articles. (This is by no means the kind of "discussion" found on a talk page.) Each of the four proofs presented is completely different. This well-established route to mathematical understanding is found in a number of Wikipedia "Proof" articles. Here, it makes the material useful and interesting for a wide audience.

It is my impression that many outside of mathematics think that if a theorem has been proved once, that we can thereafter forget the proof and just use the fact. Among mathematicians, the view is dramatically different. For example, the great mathematician Carl Friedrich Gauß proved the law of quadratic reciprocity in eight ways, and the Wikipedia page of proofs links to a web site listing over 200 variations. If the only interest were in establishing the validity of the theorem, this would be serious overkill!

Wikipedia has a vigorous community of mathematical contributors. Come talk at WikiProject Mathematics if you would like to learn more about the role of proofs, and their importance for Wikipedia. --KSmrq^T 14:42, 20 November 2005 (UTC)

I understand what proofs are and their importance, but asserting that "The purpose of [an] article is to discuss and prove" something is contrary to the intended function of Wikipedia. The purpose of a talk page may be a discussion, and the purpose of the proof(s) is certainly to prove the statemenr, but the purpose of a Wikipedia article is to present facts about the subject based on accepted data and consensus from said talk pages. -- WikidSmaht (talk) 07:25, 21 November 2005 (UTC)

Most likely we broadly agree on the purpose of a Wikipedia article; I can't be sure. If you wish to assert that Wikipedia has no place for proofs, I again suggest that you raise the issue at WikiProject Mathematics. The implications of a decision to ban proofs would go far beyond this one article, so this talk page is not an adequate venue for the discussion. I might also point out that at the Village Pump one regularly sees requests for more discussion in mathematical articles, never less. And again, this is discussion in the sense of presenting facts with their prerequisites, context, and implications, not in the sense of the discussion we're having on this talk page. I think we can agree that a Wikipedia article is not a chat room. :-) --KSmrq^T 13:51, 21 November 2005 (UTC)

I definitely wasn't saying that proofs shouldn't have articles. My point is that I objected to the original intro wording because the purpose of the proof is to prove the concept, whereas the purpose of the article about the proof is only to inform, NOT to "discuss" or "prove" anything. -- WikidSmaht (talk) 19:31, 24 November 2005 (UTC)

About the above discussion

I've been reading the above discussion for a while, and it seems to me that our anonymous friend thinks that:

Now I've been thinking about this. Suppose we calculate the intersection of closed intervals from 0 to the inverses of natural numbers. (A closed interval means one that includes its endpoints.) Specifically, we investigate the intersection of all where . Now it's clear that if n is a natural number, its inverse is going to be greater than 0. Thus, for all n \in N, the interval is going to include numbers greater than 0. Any smaller such interval is contained in any bigger such interval, so:

Now let's investigate the supremum (least upper bound) of an intersection of such closed intervals. It shouldn't be too difficult to figure out that:

This is because no matter how big n gets, its inverse is always going to be included in that intersection, and is such its supremum. Now according to our anonymous friend, this means that:

Call this supremum x. Clearly x is greater than 0. Now let m be a natural number whose inverse is smaller than x. (Because there's infinitely many natural numbers, we can always find such a natural number.) This would then mean that:

• 
• 

Now we have a case where the supremum of a subset is greater than the supremum of its superset. According to my understanding of suprema, this can't be possible.

So obviously the assumption is invalid, and the correct way should be:

Here we can't "let m be a natural number whose inverse is smaller than" 0, because then its inverse would have to be negative, and that would mean that the natural number itself would have to be negative.

But wait! Didn't we say that no matter how big n is, in the intersection of all intervals from 1 to n, its supremum is always going to be greater than 0? Now we say that in the intersection of all intervals from 1 to infinity, its supremum is going to be 0! Surely this would mean that an infinitely large collection of finitely large things is different from an infinitely large thing!

Well, that's exactly how it is. We all know that, and our anonymous friend should learn that too. — JIP | Talk 16:50, 23 November 2005 (UTC)

JIP: I think you have made an error in assuming that the supremum is 'x'.

You stated that:

"Call this supremum x. Clearly x is greater than 0. Now let m be a natural number whose inverse is smaller than x."

Way I see it: if x were the supremum, you would not be able to find an 'm' such that your conclusion is true. I see no problems with her logic. Her method of induction is valid and in my opinion proves conclusively that 0.999... < 1. Don't know why I didn't ever think of a proof by induction. —The preceding unsigned comment was added by 71.248.130.218 (talk • contribs) 15:52, 2005 November 25.

If x is the supremum, and it's greater than 0, then we take its inverse, . This inverse is either a natural number or a positive real number that is between two natural numbers. We simply need to find a natural number that is larger than and call it m. We can always do it, because otherwise there would be a finitely many natural numbers, as natural numbers are evenly spaced and do not converge on anything. — JIP | Talk 14:13, 26 November 2005 (UTC)

Just thinking, can the set you described above have a supremum? If the upper bounds keep getting closer to zero, then zero will be the intersection and the only member of the set. In this case it would be a supremum. I fail to see the analogy with her induction proof and how this proves 0.999... = 1? In her proof you are looking at a sum of things, not a collection of finitely large things. Very confusing. 71.248.136.206 01:01, 26 November 2005 (UTC)

The thing that 0 would be the only member of the intersection is the whole point of my analogy. Even though 0 is not the only member of any of the finite intersections (of which there are infinitely many), it is the only member of the infinite set. The point was to illustrate reductio ad absurdum, i.e. if a premise leads to a contradictory conclusion, the premise has to be false. The analogy is that I started from the same premise this "she" of yours used: If P is true for infinitely many finite sets, it's true for an infinite set.

(However, if you don't understand reductio ad absurdum, which I'm not sure you do, then this whole thing will have been in vain.) — JIP | Talk 14:13, 26 November 2005 (UTC)

I think I can see what you have done but I still cannot see the analogy and I don't see how reductio ad absurdum in this problem proves anything. If as you say P is true for infinitely many finite sets, it's true for an infinite set, I think you may have contradicted yourself for you stated earlier: Surely this would mean that an infinitely large collection of finitely large things is different from an infinitely large thing! If an infinitely large collection of finitely large things does not imply an infinitely large thing, how do you reach thi conclusion? Sorry, I am wondering if it's me who needs to learn something and not you? 71.248.130.225 19:43, 26 November 2005 (UTC)

Did you read my messages fully? Yes, the statements "If P is true for infinitely many finite sets, it's true for an infinite set" and "Surely this would mean that an infinitely large collection of finitely large things is different from an infinitely large thing!" are contradictory. That was my whole point! The first statement (true for infinitely many finite sets => true for an infinite set) was the premise your "she" was using. I used reductio ad absurdum to show this premise was false. Ergo, the second statement (infinitely large collection of finitely large things != infinitely large thing) is true. (Not P) is false. Ergo, (P) is true. You understand? — JIP | Talk 19:52, 26 November 2005 (UTC)

Your whole point whatever it is, indicates your argument is confusing and at best blurry? She (192.67.48.22) uses mathematical induction in much the same way as I have seen anywhere else. Reductio de absurdum is Latin for reducing an argument to the absurd and then concluding it is false. You have not done this for your argument is complete with contradictions. I must be frank - I do not have a clue of what you are trying to accomplish and I do not see any connection between your example and her proof. 70.110.85.150 13:18, 27 November 2005 (UTC)

Again, did you read my messages fully? I know what reductio ad absurdum is. I reduced an argument, i.e. that if P is true for infinitely many finite sets, it's true for an infinite set, to the absurd, i.e. a contradiction that resulted from a subset whose supremum was greater than that of its subset. Now, according to reductio ad absurdum, the argument must be false. The anonymous person's whole "induction proof" was based on this assumption. The anonymous user was able to prove by induction, correctly, that no matter how large a finite set is, P is true for it. What he failed to prove was anything related to P being true for an infinite set. Back to the original discussion at hand, the anonymous user proved by induction, correctly, that none of 0.9, 0.99, 0.999, 0.9999, 0.99999 and so on, are equal to 1. What proof by induction does NOT do is make this mean that 0.999... is not equal to 1. — JIP | Talk 13:38, 27 November 2005 (UTC)

Well, I guess I still don't get your point. You showed that 0 is the intersection of an infinite number of finite sets. She (I think it's a she because one of the comments is about male chauvinism?) is discussing an infinite sum that is not the same as a set. How do you draw conclusions relating your example to her's? It seems (imo) that these are two different things. In mathematical induction we do not try to prove P(infinity) ever, do we? If P(k) implies P(k+1), it seems logical and natural that P(n) is true regardless of the value of n. Can we at any time prove any induction result for P(infinity)? I don't think so because we cannot make any assumptions about P(infinity). If what you are stating is true (you are saying that P(k+1) does not imply P(infinity), aren't you?), then no mathematical induction can be correct because it clearly states that the results hold true for any value of n. It seems spurious to think at all about P(infinity) since there is nothing we can deduce from it. If 0.9, 0.99, 0.999, 0.9999 and so on are less than 1, then what makes you think that there will ever be a term that will make it equal to 1? The terms get closer and closer to zero, so the overall sum remains less than 1. If this is not true, then the Cauchy theorems do not make sense: are you saying that each x_i term does not get smaller as the i_s get larger? Does not m>n imply x_m < x_n ? Does infinity wrap round at some point?70.110.85.150 15:59, 27 November 2005 (UTC)

You seem to be contradicting yourself here a little. I am saying that even if P(k) implies P(k+1), neither P(k) or P(k+1) implies P(infinity). That's what this has been about all along. Then you proceed to say that if P(k+1) doesn't imply P(infinity) then "no mathematical induction can be correct". This is false. You need to understand the difference between "as large as you like" and "infinite". You really do. In the sequence 0.9, 0.99, 0.999, 0.9999, etc, there will never be a term equal to 1. That is true. But 0.999... is not in that sequence. It is what the sequence is asymptotically approaching. You could write a googolplex to the power of googolplex of nines after the decimal point, and the number would still be less than 1, because a googolplex to the power of googolplex is a natural number. Infinity is not a natural number. I don't understand how you can simultaneously say "Can we at any time prove any indication result for P(infinity)? I don't think so" and "if P(k+1) does not imply P(infinity) then no mathematical induction can be correct". If P(k+1) implies P(infinity) then we are proving P(infinity).

Yes, the x_i terms get smaller as i gets larger. Yes, m>n implies x_m < x_n. Note that each x_i is a natural number, and i, m and n are also natural numbers. I don't see how you could conclude from my messages that the x_i terms would not get smaller.

The point is, I read the original anonymous user's posts, and saw that he/she seemed to be saying that because in the sequence 0.9, 0.99, 0.999, 0.9999, etc., there will never be a term equal to 1 (which is correct), then 0.999... would not be equal to 1 either. How could this be proven? The only way I see would be: , i.e. if proof by induction works for all natural numbers (no matter how large), it also works for infinity. This is false. I demonstrated this with a counterexample: a sequence of intersections over a set of intervals. As the sequence progresses, the intersections over the set of intervals get smaller and smaller, but they never become equal to {0}, i.e. they never contain only 0, they always contain other numbers. Then I demonstrated that an intersection over an infinite set of intervals is equal to {0}, i.e. it contains only 0. We have is true, and is false. Premise is true, conclusion is false. Therefore the implication is invalid. Simple basic logic. Thus, the only way I see the anonymous user would be able to prove 0.999... != 1 from 0.9 != 1, 0.99 != 1, 0.999 != 1, 0.9999 != 1, etc, is invalid and the conclusion is false.

Finally, about the use of gender pronouns: Some anonymous user (either you or someone else) originally started calling the first anonymous user "she". I didn't see any mention of his/her gender. Therefore I hesitated to assume the gender of a person I don't know anything about. — JIP | Talk 17:37, 27 November 2005 (UTC)

I don't think you read the original poster's comments properly. You may want to reread these? She/he/it(whatever), did not state there will never be a term equal to 1. Why don't you reread her post? As far as is concerned, there is no room for this in Mathematical induction. Have you perhaps missed this point? As for examples, you have not been able to convince me of anything. Are you a student of Rasmus's? He appeared to think that his counterexamples were valid too. You first need to present a counterexample that is related to what is being discussed. What you started talking about is material that is commonly found in beginner's courses (0 the intersection of the sets...). This has no connection to the induction proof. It is completely unrelated and anything you deduce from it whether it be by reductio ad absurdum or otherwise is not true. Once you find a counterexample that is connected, then you need to show how it nullifies her proof. Really, you have not been able to prove anything imo. 70.110.85.150 20:36, 27 November 2005 (UTC)

Mathematicians who think 0.999... and 1 are different numbers

I was reading Jitse Niesen's comment that he knew of no mathematician who might think otherwise. Well, here is one mathematician who questions whether 0.999.. and 1 are actually equal. His name is Fred Richman and the URL is:

http://www.math.fau.edu/Richman/HTML/999.htm

Title of this page is: Is 0.999... = 1?

Another mathematician/computer scientist is Frode Fjeld:

http://www.cs.uit.no/~frodef/frodef.html

This is from the first google page returned for "proof 0.999 less than 1" —The preceding unsigned comment was added by 71.248.136.206 (talk • contribs).

Anon, how about signing your posts? Use four tildas for that, like this: ~~~~

By the way, I am tired chasing after you as you change IP address each time. What if you make an account? Oleg Alexandrov (talk) 22:06, 25 November 2005 (UTC)

Okay, granted that there are some experts that seem to have a problem with limits just like all the anonnies in this conversation. Can you prove that most (>50%; simple majority here) mathematicians disagree that .999... = 1? I'd be willing to bet that you can't. --HeroicJay 17:48, 27 November 2005 (UTC)

PS. Where on Fjeld's page is .999... mentioned at all? --HeroicJay 18:00, 27 November 2005 (UTC)

Two points. First, Richman questions whether 0.999... = 1. This is not the same thing at all as saying "0.999... and 1 are different numbers". Just because a proof of P is incorrect does not magically prove not(P) right away. Second, I couldn't find any mention of 0.999... != 1 on Fjeld's page either. Did 71.248.136.206 just hear it from him personally? — JIP | Talk 18:15, 27 November 2005 (UTC)

Good grief. Richman is merely having a little constructivist fun looking at an alternative to reals. He states clearly (towards the bottom) that the numbers he constructs are not traditional reals, and that they have serious problems. His numbers are no more a challenge than, say, p-adic numbers or GF(5). --KSmrq^T 21:06, 27 November 2005 (UTC)

I don't think so. The reals also have serious problems. This is but one of those problems. I think whoever came up with this idea (that 0.999... = 1) did so because it would close the door on those who argue that the reals are but an approximation to actual values where these cannot be expressed as rational numbers. 70.110.85.150 23:17, 27 November 2005 (UTC)

I am willing to bet that you will be surprised how many mathematicians think this is nonsense because it cannot be proved in any way. Out of curiosity, does anyone reading this page know who first came up with the idea that these two numbers are equal? Was it Frechet? Weierstrass? Cantor? Someone else? Would be interesting to know who. 70.110.85.150 20:40, 27 November 2005 (UTC)

Can you prove 0.999... does not equal 1? If it doesn't, then what does it equal? Keep in mind that 0.999... is a real number. It's either smaller than, equal to, or greater than, 1. I think we can discount "greater than 1" right away. So if it's not equal to 1, it's smaller than 1, but smaller by how much? — JIP | Talk 21:22, 27 November 2005 (UTC)

I don't have to prove it. I think 192.67.48.22 proved it conclusively by mathematical induction in the above posts. I don't think anyone can refute this proof. It is 100% solid imo. Go ahead and reread it. Yeah, I would think it's safe to say 0.999... is not greater or equal, therefore it must be smaller than 1. By how much is not really important. Is it a real number? Yes, I think it is, but I also think that it is not a rational number. Can it be expressed as a/b or a finite polynomial sum that reduces to a/b s.t b <> 0 and both a and b are integers? I don't think so. 70.110.85.150 23:17, 27 November 2005 (UTC)

Okay, 70.110.85.150, let's say that 0.999... is smaller than 1. Let's also stick with the theme of rational numbers. Everyone likes rational numbers. Are there any rational numbers in between? Can you find a fraction a/b such that

0.999... < a/b < 1? :How about it? Melchoir 11:40, 28 November 2005 (UTC)

I don't think he can do it. Leaving aside the issue of rationality, we're all agreed that both 1 and 0.999... are real numbers. The reals are closed under substraction, so 1-0.999... is also a real number. Which real number is it? Keep in mind that 0.999... is equal to . Now because this is an infinite series, then no matter how small the difference between 1 and 0.999... is, we can always find a point in the series where the difference between the sum and 1 is smaller than that. For example, assume the difference is 1 divided by a googolplex. Then, from i=googolplex+1 onwards, the sum is closer to 1 than that. Either the values are the same, or there are infitesimals in the reals. — JIP | Talk 11:57, 28 November 2005 (UTC)

Yes, how about it. This is exactly what I wrote in my previous paragraph. 0.999... is not a rational number. Can you find a number between 3.141... and the actual value of pi? 68.238.98.71 11:50, 28 November 2005 (UTC)

If by 3.141... you mean the limit of the series that gets closer and closer to pi by decimal places, then no, of course I can't. But what you are trying to prove with that? Seems to me like the same argument also says why 0.999... is equal to 1, because we can't find any real number between them. — JIP | Talk 11:57, 28 November 2005 (UTC)

No, we're not going to do this merry dance.

The subsection in which Melchoir attempts to impose order:

What we're going to do is discuss one point at a time. 70.110.85.150, or 68.238.98.71, or wherever you are: I am addressing you. Is there a rational a/b such that 0.999... < a/b < 1? There are two acceptable answers: yes or no. Melchoir 12:05, 28 November 2005 (UTC)

I have answered this: no. 68.238.98.71 12:11, 28 November 2005 (UTC)

Thank you for your conclusive reply. Now: is it your belief that two Dedekind cuts can partition the rational numbers in the same way, yet still be different? Melchoir 12:16, 28 November 2005 (UTC)

I would like to continue discussing this with you but I have to rush off to work. So I will answer later. 68.238.98.71 12:19, 28 November 2005 (UTC)

Fair enough. Melchoir 12:20, 28 November 2005 (UTC)

I am not sure that your question is that clear: Do you mean a number represented by Dedekind cuts as in x < a and x >= a implies a ? I have always thought Dedekind cuts to be problematic because a cut can represent the number a and also the number immediately preceding a. If so, then do you mean are the Dedekind cuts different or are the numbers different? If you mean numbers, then the Dedekind definition is ambiguous. If you mean cuts, then obviously the cuts are different. I read your comment further on about 'definition'. Sorry, I do not agree with you that you can prove anything by definition. 68.238.103.187 21:58, 28 November 2005 (UTC)

68.238.103.187, are you a different person from 70.110.85.150 and/or 70.247.52.236? Please respond, and additionally, please create an account to avoid future confusion. Seriously, you're killing us. I'm not letting you off the hook: is it your belief that two Dedekind cuts can partition the rational numbers in the same way, yet still be different? The absence of the word "real" in this question was, and still is, intentional. Also note that I have never used the word "definition" on this page. Melchoir 22:26, 28 November 2005 (UTC)

My ip is 71.248.136.206. I keep signing my posts with 68.238.103.187 22:33, 28 November 2005 (UTC) as per Oleg A's request.

I have responded to your question. Unless you clarify what you mean, you will not be able to obtain any further help from me. I have no intention of creating an account. If you want, I can use a nickname. You choose it. 68.238.103.187 22:33, 28 November 2005 (UTC)

Why would you avoid creating an account? I'm honestly curious. You may have responded to my question, but you haven't answered it. The definition of a Dedekind cut as a partition of the rationals is unambiguous. I will rephrase my question very slightly: can two Dedekind cuts consist of the same partition of the rationals, yet still be different cuts? There is now a second question that you also have not answered: are you a different person from 70.110.85.150 and/or 70.247.52.236? Melchoir 22:45, 28 November 2005 (UTC)

If they consist of the same partition of rationals, how can they be different cuts? Your question remains unclear and confusing. Now time for you to answer my question and until you do, no more responses from me. Show me how to represent pi using Dedekind cuts.68.238.103.187 23:02, 28 November 2005 (UTC)

That sounds like a no. I must also assume that you are 70.110.85.150 and 70.247.52.236, since you refuse to say otherwise. You do ask an interesting question: how to represent pi using Dedekind cuts. I am willing to show you, but you must point me to a definition of pi that you agree not to argue with. If I do this, will you agree that all real numbers can be represented by Dedekind cuts? Melchoir 23:29, 28 November 2005 (UTC)

It does not matter who I am. There are over 153 pi formulas/definitions. The definition of pi I know states that it is the ratio of the circumference to the diameter. If you can do this, I will concede that all real numbers can be represented by Dedekind cuts. 68.238.103.187 01:57, 29 November 2005 (UTC)

Good, let's use circles. Let A be the set of positive rational numbers x such that a circle of circumference x has diameter strictly less than 1. Let B be the union of A with the set of rationals less than or equal to zero. Then B is easily seen to be closed downward, and it should also be clear that B has no greatest element. Let C be the complement of B in the rationals; it is closed upwards. Let p = (B, C); then p is a Dedekind cut.

Take care to note that the definition of p does not require, assume, mention, or beg for the word pi, the Greek letter pi; the number pi; or any hint of foreknowledge of the existence, in a mathematical, physical, metaphysical, sociological, or anthropological sense, of the concept known as pi.

Now, consider the question: what does it mean for a Dedekind cut to represent a real number? We can answer this question without deciding on a particular definition of "real number"; we need only agree that the set of real numbers is an ordered set which extends the rationals. One says that a cut (X, Y) represents the real number z if X consists exactly of those rationals which are less than z. This paragraph is non-negotiable.

Back to our problem. We all agree that pi is real. Every element of the set B is less than pi, and every rational number less than pi is in B. Therefore the cut p represents pi. Melchoir 06:55, 29 November 2005 (UTC)

You have made a few mistakes but this is how I expected you might respond. This Dedekind cut does not describe pi. 68.238.102.162 12:05, 29 November 2005 (UTC)

Which part do you disagree with? Melchoir 18:49, 29 November 2005 (UTC)

For all intensive purposes you have said the same as the poster below who stated:

pi = { (-inf,pi),[pi,inf) }. This says nothing about the value of pi - only that it lies in this cut. Problem with Dedekind cuts is that they can represent more than one number and now I see what it is you were trying to get me to commit to earlier and which I would not because your question was unclear. You assumed I said 'no' but I remained uncomitted. In the form: { (-inf,x), [x,inf) } we are talking about x or the number immediately preceding it. In the form { (-inf,x], (x,inf) } we are talking about x or the number immediately succeeding it.

In fact Dedekind cuts prove exactly the opposite of what you are trying to prove, i.e. 0.999... is less than 1. If you compare the partial sums term by term, you would not say that 0.999... could fall into the cut (1,inf) or [1,inf) ever. 70.110.81.253 22:16, 29 November 2005 (UTC)

Please do not refer to yourself as "the poster below ". I think I finally understand what you are trying to say about Dedekind cuts, and we can talk about that later. For now, however, I ask you to focus on the question at hand: pi. My argument above proceeds in steps:

1. You have defined a number pi.
2. I have defined a Dedekind cut p.
3. p represents pi.

Obviously you disagree with at least one of the above. Which is the first point with which you disagree? Melchoir 23:02, 29 November 2005 (UTC)

Well, the last step is problematic because although you have defined the cut, it still says nothing about the value of pi. Again, you would like to say that 0.999... and 1 are equal because they can be represented by the same cut. Unlike Hardy, I do not believe that between any two real numbers, there are some rational numbers. If you deny this, then the real number system has gaps in it and all your theory is nonsense. Dedekind cuts assume the real number system has no gaps. 0.999... is not a rational number. If Hardy understood this, he would not comment the way he does without thinking. 70.110.81.253 23:16, 29 November 2005 (UTC)

I have a great deal to say about the "gaps" to which you allude, and I will do so later. However, I would venture that Michael Hardy understands more than both of us put together, which is saying something considering our... differences.

So, you agree with (1) and (2) but not with (3). This is encouraging. Tell me, what does it mean to you to say that a Dedekind cut (X, Y) of rational numbers represents a number z? Melchoir 23:29, 29 November 2005 (UTC)

I very much doubt that Hardy understands even a tenth of what I understand. He may know more than you - but I also have my doubts here. He sounds like a fool. Let me ask you: what does a Dedekind cut mean to you? No, wait I am not going to play games with you. Present proof that 0.999... = 1 or otherwise you are wasting both my time and yours. 70.110.81.253 23:43, 29 November 2005 (UTC)

Please refrain from personal attacks. You claim that we are wasting our time, but I am learning more from you than you realize; you could make an attempt to learn from me. If I read you correctly, you are threatening to leave this discussion. Make no mistake: most folks here at Wikipedia would be happy to see you go, but I would not.

By "Dedekind cut" I mean the definition in that article, and I have already told you what I mean by "represent". I quote:

• • Now, consider the question: what does it mean for a Dedekind cut to represent a real number? We can answer this question without deciding on a particular definition of "real number"; we need only agree that the set of real numbers is an ordered set which extends the rationals. One says that a cut (X, Y) represents the real number z if X consists exactly of those rationals which are less than z. This paragraph is non-negotiable.

Since B consists exactly of those rationals which are less than pi, p represents pi. Now, please notice that as far as this argument takes us, p might also represent numbers that aren't pi. Conceivably p could represent lots of numbers very close to pi. Infinitely many, even. But p still represents pi. Do you disagree with this simple statement? Melchoir 00:33, 30 November 2005 (UTC)

I disagree that there are infinitely many numbers described by your cut p. There are essentially two numbers your cut describes: pi and the number that immediately precedes it. 70.110.81.253 01:10, 30 November 2005 (UTC)

That's fine; it's not going to matter how many numbers p, or any other Dedekind cut, represents. Just to be clear, you agree that p represents pi? Melchoir 01:21, 30 November 2005 (UTC)

As I stated, cut p represents pi and its predecessor. Why don't you just finish your proof? 68.238.103.215 21:53, 30 November 2005 (UTC)

You once claimed about finding a cut to represent pi: "If you can do this, I will concede that all real numbers can be represented by Dedekind cuts." I cannot proceed until you do exactly that. Melchoir 00:18, 1 December 2005 (UTC)

Yes, and you have not found a cut that contains only pi. Also you have not stated anything different from the contributor who described pi in a equivalent way using Dedekind cuts. No Dedekind cut contains only one real number. Well, if your proof relies on acceptance of this fact, it confirms what I already know - you have no proof. 68.238.103.215 01:06, 1 December 2005 (UTC)

I have been extremely careful not to claim that p represents only pi. You asked me, "Show me how to represent pi using Dedekind cuts." You never asked for a Dedekind cut that represents only pi, and I never claimed to give you one. I now ask you to acknowledge that, among the numbers that p happens to represent, one of them is pi. Melchoir 01:43, 1 December 2005 (UTC)

I have acknowledged this several times in what I've written. However, this is not about pi, but about whether 0.999... = 1. You have not been able to show this. Please do not find the limit and then claim this is the same as the sum. This is a common misconception amongst Phds. Many numbers have one as their limit but are in fact not equal to one. Let's see, do you agree that infinite sum and limit of infinite sum are two different things? 70.110.85.3 11:22, 1 December 2005 (UTC)

You asked me to find a cut representing pi; presumably you thought that I couldn't, and you could be a little more humble and upfront in admitting that I did. And, please do not refer to "common misconception"s among PhDs to try and strengthen your point. I have all the authority in the world behind me, but I have not tried to cram it down your throat. If you want a sociological holy war, start one on your user page. This is about math.

Now, let me make a definition. Let a Melchoir number be a set of "numbers whose limits are the same", whatever that phrase means to you. We can add, subtract, and multiply Melchoir numbers; if we're careful, we can even divide them; and we can compare two Melchoir numbers by comparing their elements. The set of Melchoir numbers is an ordered field. Do you object? Melchoir 18:31, 1 December 2005 (UTC)

You did not find a cut that contains pi alone and this is what I was objecting to. One of the posters beat you to it. Your cut contains pi and its predecessor - just like the example of the poster. I am familiar with the kind of logic you have been trying to use with me. I am not inferior to you in any way. I understand the theory of fields and know what ordering is. You are not the only one to have studied number theory. Now I will not answer any more of your questions seeing you do not answer mine. A sociological war? Wow, that one is really far out. Just when I was beginning to think you made more sense than anyone else in Wiki's ranks... 70.110.91.41 22:42, 1 December 2005 (UTC)

I speak of a "sociological war" because you speak of a "common misconception amongst Phds". I do not hold a PhD, and I have never studied number theory. If you're looking for a fight against the Establishment, I can't help you. I never called you inferior. Perhaps you think "my logic" is patronizing because I am bending over backwards to agree with your terminology, along with that of logamath1 at the bottom of the page. Would you really prefer that I cry out every time you defy a universally accepted definition?

I agree that there is such a mathematical object as a formal infinite series; in a topological group, there is also such a mathematical concept as the sum of a formal series. Sure, these are different things. There is a formal series of rationals that perhaps best represents the idea of the string of symbols "0.999..."; let's call it M. There is another formal series of rationals that perhaps best represents the idea of the string of symbols "1.000..."; let's call it N. So M is different from N, but who cares? Both of these formal series are Cauchy, and as luck would have it, they are co-Cauchy; their partials sums can be made arbitrarily close to each other, can they not? Melchoir 23:31, 1 December 2005 (UTC)

I don't care about logamath1. My discussion is with you. You have not answered my question. Let me repeat it: Do you agree that there is a difference between an infinite sum and the limit of an infinite sum? btw: if you think patronization is not an indication you feel superior to others, then you have another think coming your way. 70.110.87.58 12:03, 2 December 2005 (UTC)

Please define the phrases "infinite sum" and "limit of an infinite sum", so that I will be able to decide whether or not they are the same. Meanwhile, by speaking of formal series I have bypassed the whole issue. Do you agree with M and N are co-Cauchy? Melchoir 18:36, 2 December 2005 (UTC)

Melchoir cordons the following off so that 70.110.81.253 is not tempted to respond:

No, a Dedekind cut cannot represent more than one real number, because between any two real numbers there are some rational numbers. And people would be more confident in what you write if you could understand the phrase "for all intents and purposes". Michael Hardy 22:47, 29 November 2005 (UTC)

Okay Hardy. You are correct, it is "for all intents and purposes". However, I have heard very educated people use it this way and since they were not corrected or challenged, I assumed it was acceptable to use. I speak/read/write 6 languages. And I am proud of the way I learned English seeing it it not my mother tongue. You are nothing but an arrogant fool and a miserable man who thinks he has a high IQ. "For all intents and purposes", most people who are not arrogant farts like you, won't even notice I used this cliched phrase incorrectly. So unless you can prove me wrong by focusing on the math, I suggest you save your grammar talents for a more appropriate occasion. Do you know more than one language Hardy? 70.110.81.253 01:05, 30 November 2005 (UTC)

This is true, but in the absence of a definition of "real number", your argument is circular. I haven't tried to force a definition upon 70.110.81.253, since it will not accept the reasonableness of any definition until its existing misconceptions are cleared up. I have infinite respect for you, but I don't think you're helping. Melchoir 23:10, 29 November 2005 (UTC)

You can't be serious? That's exactly what a Dedekind cut is saying or has it not dawned on you yet? 70.110.81.253 23:16, 29 November 2005 (UTC)

I seem to have failed. Everyone else, please do not repond to the above comment. Melchoir 23:23, 29 November 2005 (UTC)

While we're waiting:

If there is no rational number between 0.999... and 1, and rationals are dense in the reals, seems to me like 0.999... is equal to 1. — JIP | Talk 12:21, 28 November 2005 (UTC)

Of course you're right, but my whole point is that we don't need to call on any theorems from analysis or topology. The simplest link between the reals and the rationals is a definition. Melchoir 17:15, 28 November 2005 (UTC)

If you believe that Dedekind cuts can be used to represent any real number, then please show me how to represent pi and e. 68.238.103.187 22:15, 28 November 2005 (UTC)

Surely you don't believe that there exist rational numbers which are neither less than pi, nor greater than pi. Melchoir 23:05, 28 November 2005 (UTC)

To which yet another IP says:

You may not need to call on any theorems because they won't help you.

I know where this is leading: If 68.238.98.71 says no, then you will try to say that {(-inf,1);[1,inf)} = 1 and {(-inf,1);[1,inf)} = 0.999... therefore they must be the same. However, by saying this you would concede that 0.999... is less than 1 since (-inf,1) does not contain 1 and I don't think you are quite ready to say that 0.999... is a member of [1,inf), are you? 70.247.52.236 18:13, 28 November 2005 (UTC)

I am unfamiliar with the { ; } notation, but I'll address your other points. No, of course (-inf,1) doesn't contain 1. But I find your second claim very strange. Where do you base your claim that we are "not quite ready to say that" 0.999... is a member of [1,inf)? We've been saying 0.999...=1 all along, therefore it is of course a member of [1,inf). Do you want us to say "no, 0.999... is not a member of [1, inf)"? If we say that, then it's less than 1, and that would prove your argument. So essentially you're asking us to say that we are "not quite ready to say that" 0.999...=1. — JIP | Talk 18:59, 28 November 2005 (UTC)

I was afraid you would respond this way. You cannot use Dedekind cuts to support your argument because then you are assuming that 0.999... = 1 without any proof. 1 is the upper bound of the infinite sum 0.999... therefore it cannot be a member of [1,inf). Using your argument, one can equally well assume that 0.999... is a member of (-inf,1) for there is no number between (-inf,1) and [1,inf). Dedekind cuts cannot be used to represent all real numbers. How would you represent pi? Please don't tell me: {(-inf,pi),[pi,inf)} because you do not know what pi is. You will have to come up with a far more convincing proof if you are to defeat 68.238.98.71. 70.247.52.236 22:13, 28 November 2005 (UTC)

70.247.52.236, please create an account. I cannot respond to you because, for all I know, you are 68.238.98.71. Melchoir 22:31, 28 November 2005 (UTC)

The two IPs do come from the same area... - Fredrik | tc 23:12, 28 November 2005 (UTC)

Enough.

Please create an account. Melchoir 19:30, 28 November 2005 (UTC)

Algebra Proof

I'm not really comfortable with this section, how do you guys here feel about changing it?, maybe something like:

Another kind of proof adapts to any repeating decimal. Let the number 0.999… be called c, then 10×c = 10×0.999… = 9.999…; If we substract c, we have 10c - c = 9c and 9.999… - 0.999… = 9, thus 9c = 9. Dividing both sides by 9 completes the proof: c = 0.999… = 1

Please edit this entry, so we can have a 'definite' version of this section. Or am i the only one who has trouble with the current wording? Thank you. Jesushaces 05:05, 30 November 2005 (UTC)

I strongly oppose such a change. Your rewording is less meticulous than that currently in the article, and therefore liable to attacks that have plagued this topic both here and elsewhere. --KSmrq^T 06:16, 30 November 2005 (UTC)

I support the vague idea of a compromise between the current wording and Jesushaces'. The current wording suffers from a pretention to precision that isn't really justified, and the fact that it's more detailed than the previous proof (using 1/3) seems to suggest that the latter proof stands on firmer ground, which it doesn't. Now, I don't think there is a danger of attack if we switch to a less meticulous wording, since the article already has disclaimers about elementary proofs. However, even if there were an increased danger of attack, one must always attempt a balance between readability and precision, even in a textbook or a journal article, to say nothing of an encyclopedia. Right now it's unreadable. Melchoir 07:04, 30 November 2005 (UTC)

I will not agree to any wording that does not include the present details. Previously this article was constantly being revised. It degenerated into a horrible hash. Everything that the proposed revision omits has been shown to lead to attack and confusion. The current version of the article is by far the most stable ever. The algebra proof pretends nothing; it honestly points out subtle issues. I guarantee you that if you strip out the safeguards the clamor will begin anew. Such a "compromise" will bring not a risk of attack, but a certainty; and the consequences will be ugly. Maybe the current wording is a little awkward, but that's nothing compared to the mess it replaced, and that you want to "risk" again. I think that would be a damned foolish choice. --KSmrq^T 09:03, 30 November 2005 (UTC)

If you knew me better, you wouldn't have provoked me like that...

Ksmrq, I have reviewed the edit history of this article. It has racked up a hundred edits in six months of existence, which is a lot for such a low-visibility page but not completely out of line. It was indeed a "horrible hash" until you rewrote it on 26 October 2005, and in some sense it is a testament to your skill that its content has not changed since then. Although the article's stasis reflects well upon you, it is not necessarily a good thing, and it certainly should not be defended as an end in itself.

Since your 26 October 2005 rewrite was so extreme, we must treat the article as if it were completely new. Indeed, its content was written by one person and has since been revised only cosmetically. Most relevantly: we cannot predict the effect of modifying the new article by extrapolating from the old article. You speak of a "certainty" of "attack". I don't know how you can be so certain.

On to the math. You claim "The algebra proof pretends nothing". In fact, the algebra proof relies on the properties of the real numbers while pretending that it doesn't.

Take a look at this webpage, which was presented by an anon as an example of a mathematician who supposedly thought that 0.999... and 1 are different. The anon was wrong, and you correctly pointed out: "Good grief. Richman is merely having a little constructivist fun looking at an alternative to reals." Richman is sort of playing the iconoclast by referring to "traditional real numbers", an extremely unfortunate phrase given that it is far too late to redefine the reals, even if it were a good idea, and it isn't. However, his mathematics looks sound to me, and he does two important things. First, he names the decimal numbers and takes them seriously. This is important because many people seem to think that the reals numbers are the decimal numbers, and the Elementary section of our article doesn't even suggest otherwise. Unfortunately, Richman does not provide a symbolic representation for decimal numbers (he reuses the notation everyone else reserves for real numbers, another poor choice), so I will invent one: let's agree to write decimal numbers as we would real numbers, except that we replace the decimal point by the letter d. For example, 0d999... and 1d000... are different decimal numbers, essentially by fiat.

Second, he defines addition and multiplication operations on the decimal numbers. This allows him to observe, for example, that 3x 0d333... is not 1d000... but 0d999. Indeed, it is impossible to divide 1d000... by three.

Why am I writing so much about the decimal numbers? Because they provide a counterexample to the reasoning in the "Elementary proofs" section. For the decimal numbers, "manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition". Therefore the reader should expect the two elementary proofs given to work for the decimal numbers. But they don't. In fact, we can pinpoint where they break down. In the first proof:

• The standard method used to convert the fraction 1⁄3 to decimal form is long division, and the well-known result is 0.3333…

Long division doesn't work on decimal numbers, and 0d3333... is not one third of 1d0000... This proof relies on the numbers discussed being real. In the second proof:

• Subtracting the smaller number from the larger can proceed digit by digit

There is no unique way to subtract two decimal numbers, but your argument can be rearranged to avoid subtraction. If you made this rearrangement, you would still need to say at some point

• Then 10c − c = 9. Then 10c = 9 + c. This is the same as 9c = 9.

If c is a decimal number, then the second equation does not follow from the first. As Richman might say, 0d999... is not cancellable. This proof also relies on the numbers discussed being real.

The arguments listed under "Elementary proofs" only work if they are applied to real numbers. However, the article currently does not contain the word "real" until the "Advanced" section. This is bad logic and possibly even dishonest. It's not just that the elementary proofs lack "sophistication and rigor"; as stated, they are simply wrong. I see three possible fixes to this problem:

1. Explicitly warn that the elementary proofs rely on unstated properties of the real numbers.
2. Modify the disclaimer: "Elementary proofs assume that manipulations at the digit level are well-defined, meaningful, and work the way we intuitively expect them to, even in the presence of infinite repetition'"
3. Do not refer to the elementary arguments as proofs. Call them plausibility arguments.

The first two options have the same intent: in a more or less subtle way, tell the readers that something essential is being hidden from them. Sadly, we live in a world filled with disclaimers, and many readers will fail to heed the warning. They will interpret the elementary arguments as purported self-contained proofs and think that we are wrong. This is, in fact, what is currently going on and what you should fear the most. I think my cynical prediction of the future is much more likely than your cynical prediction of the future.

The third option seems like the only honest one to me, and it hides nothing. Just admit that they aren't proofs at all but sleight-of-hand manipulations that are too pursuasive for their own good. In that spirit, if I may quote myself, "The current wording suffers from a pretention to precision that isn't really justified." I stand by it. Melchoir 11:51, 30 November 2005 (UTC)

And now for something different:

I am the originator of the 'Induction Proof' that 0.999... < 1. I have read what you have written in the above paragraphs and I think you are only fooling yourselves. The 10x proof does not rely on the unstated properties of real numbers or any established facts and it is entirely false. It seems to me that Dedekind cuts can also be used to prove that 0.999... < 1 as suggested by one reader. The induction proof is solid. You do not need to rewrite the nonsense you have in this article - you need to delete it because it misleads those who do not have enough sense to see it is false. I said I would not post but I hope you will do what is right, not what you think is right, but what I am telling you is right. I know what I am talking about. Am I smarter than most of you? The arrogant answer is yes. Most of you are sincerely deceived. You love ignorance and it shows. 158.35.225.231 12:57, 30 November 2005 (UTC)

158.35.225.231, I've never read your earlier posts, since I am relatively new to this discussion page. However, if you dispute the properties of the real numbers, then you cannot help us improve the article. Please do not intrude on an effort you do not wish to help. Melchoir 13:25, 30 November 2005 (UTC)

Who said anything about disputing the properties of real numbers? It's your misinterpretation of the number 0.999... that makes it appear there is conflict in the real numbers unless it equals 1. 158.35.225.231 14:50, 30 November 2005 (UTC)

You are claiming that 0.999... and 1 are different real numbers, which is not the position taken by the article. Surely you understand how that prevents you from contributing to the article. As for which of us is wrong, I am currently attempting to converse with one anonymous user already. You might be the same person. I cannot engage you unless you create an account. Melchoir 15:19, 30 November 2005 (UTC)

I don't want to contribute to your article because it's false. Your main concern is how to remove the contradiction of there being no real numbers between 0.999... and 1. Here's how you can do it: There are infinitely many numbers between 9/10 + 9/100 + 9/1000 + ... and 1 as the following demonstrates.

 Let  X = 9

      X/10  +  X/10^2  +  X/10^3 + ....                        [Radix 10]

   A= .9    +  .09     + .009    + ....       = .999...

 Let  X = 99

      X/10^2  +  X/10^4  +  X/10^6  + ....                     [Radix 100]

   B= .99     +  .0099   +  .000099 + ...     = .999999...

 Let  X = 999

      X/10^3  +  X/10^6  +  X/10^9     + ....                  [Radix 1000]

   C= .999    +  .000999 +  .000000999 + ...  = .999999999...

   A < B < C < 1  

The terms of C decrease much faster than B and those of B much faster than A so that no matter how large any of these numbers become, the difference between C and 1 will always be the smallest. Let's call this difference Dc. Likewise let's call Db and Da the differences of b and a with 1 respectively. It is easy to see then that Dc < Db < Da. Therefore no matter how small these differences become, they will always be greater than zero and the ordering will always be preserved, i.e. Dc will always be less than Db which will always be less than Da. Would you be able to represent the number C in Radix 100? Would you be able to represent the number B in Radix 10? Answer is no. Part of this controversy has to do with the way numbers are represented in a radix system. You cannot say (A) 9/10 + 9/100 + 9/1000 +... = 1 for then you must have (B) 99/100 + 99/10000 + 99/1000000 + ... = 1 and we know that A is less than B. Remember we are not comparing limits for this tells us nothing about the size of the number. We can only compare the sum of a finite number of terms, term by term. 158.35.225.231 17:56, 30 November 2005 (UTC)

Are you the same person as 70.110.81.253 or not? Melchoir 19:16, 30 November 2005 (UTC)

No. 158.35.225.231 19:33, 30 November 2005 (UTC)

And yet you share 70.110.81.253's reluctance to create an account and save us from guessing? Should we not be suspicious that you have similar views, and when he disappears, you reappear? Melchoir 19:40, 30 November 2005 (UTC)

Here you go. Hate to break it to you but I hardly ever use your website. And now you have one more account... Logamath1 20:02, 30 November 2005 (UTC)

Wow, thanks for logging in. None of the other anons on this page ever agreed to do so, and I admit that I was losing hope that anyone ever would. I apologize if I insulted you unnecessarily.

As for there being one more account, I'm sure nobody will mind the burden, even if you never use it again. Accounts are probably cheaper than dirt.

Let me understand what you have done above. You have defined three formal series, A, B, and C, and you have defined an ordering between them based on their rates of growth, or their terms' rates of decay. You explicitly state that you do not mean to compare the "limits" of A, B, and C; if I may paraphrase, you want to avoid actually summing A, B, and C, because you wish to retain the term-by-term information contained in a formal series. This viewpoint allows you to observe that, for example, "the difference between C and 1 will always be the smallest". Formally, you might say that for any positive integer n, the first n terms of C are always closer to 1 than the first n terms of A or B. You observe that A, B, and C cannot be distinguished by decimal expansions, since a decimal expansion loses the information of how quickly it grows. On the other hand, using multiple radices (10, 100, 1000) allows you to faithfully represent the different growth rates of A, B, and C. This is all fine. In fact, since you can add and subtract formal series such as A, B, and C, among other algebraic manipulations, you might even be justified in calling them "numbers".

In order to most easily compare A, B, and C to 1, perhaps we should define 1 itself to be the formal series 1 + 0 + 0 + 0 + ... in this context. (I have used the symbol 1 in two different ways in the last sentence, but only for convenience.) In that case, I cannot disagree with you that A < B < C < 1, and there are infinitely many more such "numbers" in between.

Is that a fair representation? I swear I am not trying to trick you. Melchoir 22:23, 30 November 2005 (UTC)

Am I missing something here? Yes, 0.9 < 0.99 < 0.999 < 1, and 0.99 < 0.9999 < 0.999999 < 1, 0.999 < 0.999999 < 0.999999999 < 1, and so on. In fact, for any natural number n, even a googolplex to the power of googolplex, the sum of the first n numbers in the first series is smaller than that of the first n numbers of the second, which is smaller than that of the first n numbers of the third, which is smaller than 1. But that doesn't mean the numbers A, B and C are smaller than each other, or smaller than 1, because they are the sums of infinitely long series. In fact, in my opinion, A=B=C=1. It seems to me that you've still demonstrated a lack of understanding between the sum of a finite series and the sum of an infinite series, which I've been pointing out for several weeks now. Perhaps you should read about Hilbert's hotel, where the manager was able to get shoes for every room, even though the shoes for half of the rooms were missing? — JIP | Talk 07:56, 1 December 2005 (UTC)

Well, JIP, you're kind of forcing my hand, but my point was going to be that even if we leave A, B, and C unsummed and agree that they are different numbers, for some definition of "number", they all still correspond to the same real number, which is of course 1. Someone made a good point on Real number about the meaning of R being "complete"; apparently Hilbert used the term not to suggest that we must start with a small set of numbers (the rationals) and build them up, but that we can also start with a large set of numbers (such as whatever Logamath1 is talking about) and collapse away all the infinitesimals. I suspect that for novices in mathematics, it might be easier to first imagine a whole jungle of "numbers" and then cut them down until we get what we want, rather than the usual method of bootstrapping up from the skeleton that is Q and making sure that we don't overshoot and introduce infinitesimals in the process. Of course, we know how not to overshoot, but I don't want to have to convince the newbies that we do. Melchoir 09:58, 1 December 2005 (UTC)

Actually that's the problem - they do not all correspond to the same real number 1. The limit of those numbers is 1. Their sum is not 1. Their sum is undefined. And no, you cannot say that at infinity their sum will be 1. If you consider the differences Dc, Db, Da - at infinity these will still not be zero even if these are close to zero. In fact, if you believe in infinitesimals, Dc, Db and Da would be the closest thing to what these might resemble. Of course there is no such thing as infinitesimal (not in non-standard analysis, surreal numbers or any other kind of number system which is a load of rubbish) This is the point and yes JIP, you have missed it. The definition you have for infinitesimal in Wikipedia is really amusing. But then so many of your articles are amusing. logamath1

The problem is that you're not understanding the terminology. The number system as we know it was devised by humans in order to be logically consistent with both the real world and itself. And, in order to remain logically consistent, .999... has to be exactly equal to 1, because the definition of an infinitely repeating decimal is defined as the limit of the sum of the digits (which is 1 in this case), and it would require infinitesimals (which do not exist in the real number system) otherwise, since between any two unequal reals exist infinitely many other reals. You keep speaking as though the number system has some inherent property such that inifitely repeating decimals are undefined or not properly defined (or that they don't really have an infinite number of digits), but that runs contrary to the truth. I'm sure you could probably devise a number system where infinitely repeating decimals don't exist, but it wouldn't be analogous to the reals as we know them. --HeroicJay 17:21, 1 December 2005 (UTC)

I think I understand the terminology fairly well. 0.999... does not have to be 1 for the real number system to be logically consistent. There is an obsession that things might fall apart if these two numbers are not equal. As I have shown above, you can have numbers between 0.999... and 1 - this does not violate the Archimedean corollary. In fact the Archimedean property holds for all numbers whether they be real (or not) provided these are treated as approximations. Whether you like it or not, this is the way we have always dealt with real numbers - as approximations. The problem occurs with the interpretation of expressions such as 'finite sum' and 'limit of infinite sum'. These are not the same and should not be treated equivalently. You talk about infinitesimals without having a well-defined meaning. Infinitesimals are a figment of the imagination. Wiki's article on this is a joke (as are several others). In fact having 0.999... = 1 causes the real number system to fall apart. Why? Well, the real number system is ordered. There is an ordering amongst decimals too. This ordering exists so that we can make meaningful comparisons. If you compare a number that differs in the 15 zillionth digit of pi, how would you compare it? You would start from the most significant digit which is to the left of the radix point and proceed until you found the digit that differs. In the same way, our decimal system fails if we break this rule by setting 0.999... = 1 because we compare these in exactly the same way as we would compare pi. I put it to you that whoever came up with the idea that 0.999.. = 1 was a fool. logamath1

Logamath1, it will be enlightening for you to see an explicit example of an ordered field containing infinitesimals. Let F be the field of all rational functions on the rational numbers. So, a typical element of F is a function f defined by, say,

It is obvious how to add,subtract, multiply, and divide elements of F. Now for the ordering: let the statement f < g be defined to mean that there exists an M such that for all x > M, f(x) < g(x). Essentially, one function is less than another if its values "eventually" are lesser. Let h be the function h(x) = 1/x. Then for any positive integer n, we have 0 < h < 1/n. This is what we mean by calling h infinitesimal.

Please do not tell me that the function 1/x is "a figment of the imagination"!

This is all relevant to the real numbers because all of the equivalent formulations of the real numbers eliminate any infinitesimals essentially by definition. Any two "numbers" that are infinitesimally close must correspond to the same real number. To pick a random example, in the real numbers, 0.999... equals 1. Melchoir 18:46, 1 December 2005 (UTC)

You have stated a corollary of the Archimedean property. h is not an infinitesimal but a very small number. An infinitesimal does not exist. logamath1

And you say "Infinitesimals are a figment of the imagination." I'm not disagreeing with that point. But, if .999... doesn't equal 1, then what is 1 - .999... but an infinitesimal? (That's a reductio ad absurdum, in case you missed it; if 1 = .999..., then 1 - .999... = 0, but the question is what value you obtain if they're not equal.) Can you provide the solution to 1 - .999...? And just note that saying it's .000...1 doesn't cut it; if the number of zeroes is finite, then you're wrong, and if the number of zeroes is infinite, then there can't be a 1 at the end of them. --HeroicJay 19:51, 1 December 2005 (UTC)

And, oh yes, let's not add conditions to mathematical theorems that don't require them. That's already come up in this conversation before. --HeroicJay 19:56, 1 December 2005 (UTC)

HeroicJay: It's quite incredible how you agree that infinitesimals are a figment of the imagination and then you call the difference 1 - 0.999... an infinitesimal - isn't it? You are very confused I would say. Logamath1

1-0.999... is undefined HeroicJay. It's as simple as this. We can only approximate its value just like all the other repeating decimal numbers in the decimal system. What is so hard about this to you? Best we can do in this case is to approximate it by 0. However, this does not mean it is exactly equal to 0 but is good enough. logamath1

Logamath1, HeroicJay is making a very simple argument. Roughly speaking:

1. If 0.999... were a real number less than 1 then 1- 0.999... would be an infinitesimal real number.
2. But there are no infinitesimal real numbers.
3. Therefore the real number 0.999... is not less than 1.

Do you follow? Melchoir 20:09, 1 December 2005 (UTC)

Not so Melchoir: 1-0.999... would be a very small real number, not an infinitesimal because an infinitesimal does not exist! Therefore 0.999... is less than 1. Do you follow? logamath1

No. We do not. It would be a very small real number that you refuse to describe, which isn't an infinitesimal because you say so. I don't follow that at all. --HeroicJay 20:14, 1 December 2005 (UTC)

Fine, you refuse to use our definition of "infinitesimal". Let's avoid it. Roughly speaking:

1. If 0.999... were a real number less than 1 then 1- 0.999... would be an very small real number.
2. But there are no very small real numbers.
3. Therefore the real number 0.999... is not less than 1.

Do you follow? Melchoir 20:09, 1 December 2005 (UTC)

Wrong. There are very small real numbers for if there were not:

1. 1. Weierstrass's theory is all garbage
2. 2. Calculus is a load of hogwash
3. 3. Therefore the real number 0.999... is less than 1

logamath1

Let's see if I can say what I was trying to say earlier without getting Edit Conflicted again. Repeating decimals are a very real and clearly defined part of mathematics that do not require approximations in pure math (you do require approximations in science, due to the way science works, but science is not pure math.) --HeroicJay 20:17, 1 December 2005 (UTC)

They do require approximations because you cannot do arithmetic on any numbers that are not finitely represented. This has nothing to do with science but everything with pure math. logamath1

Back up, logamath1. I say there are no very small real numbers, and you say there are. We can settle this by going to the definition. Please visit the article Construction of real numbers. Pick out any construction you like. I will prove to you that it has no very small numbers. Melchoir 20:23, 1 December 2005 (UTC)

I am only going to deal with Melchoir. It's too difficult because of edit conflicts. Sorry HeroicJay. Try Decimal construction. Otherwise you choose. logamath1

I'm dealing with Edit Conflicts from the both of you. Sounds like a lame excuse so avoid answering my question to me. --HeroicJay 20:28, 1 December 2005 (UTC)

It's not a lame excuse. Don't take anything personally. Besides, it's one against the whole of Wikipedia and most of the Western World's Phds. Cut me some slack please. logamath1

Doesn't that say something to you? As I said before, the real number system is defined by humans (in such a manner that it remains consistent with the real world and itself), so if the people who are experts in the field say that .999... = 1, who are you to argue? --HeroicJay 20:35, 1 December 2005 (UTC)

Who says you can't do arithmetic on numbers that aren't finitely represented? Besides you, I mean. --HeroicJay 20:24, 1 December 2005 (UTC)

Have fun. I will check in tomorrow or later tonight if I get a chance. Sorry I gotta leave you. logamath1

C'mon, the decimal construction is the easiest one. Let x be a positive infinite decimal expansion. If x is greater than, say, 0.2, then it isn't very small. So x has the form 0.abcdefg..., where a through g and beyond are digits. If all the digits of x are zero, then x is zero by definition. Otherwise, x has a nonzero digit in the nth decimal place for some n. So x is greater than 10^-n. In summary, any positive infinite decimal expansion is not very small. Melchoir 20:42, 1 December 2005 (UTC)

You could have fooled anyone else but you don't fool me. How small is very small? Please Melchoir, this is about the worst argument you have presented thus far. It's total nonsense! By the way, where is that boy Jitse Niessen? Do you see how HeroicJay is provoking me? He needs to be reprimanded!! Who the hell is he to ask me why I argue?! Why, I would take this up with God if I had to until the very last breath!! I am very tempted now to tell this creep off. Let's see if Niessen will reprimand him or not... logamath1

A very small number is a positive number that is less than all positive rational numbers. Melchoir 20:55, 1 December 2005 (UTC)

By the way, you should calm down. HeroicJay is asking a question that is only half rhetorical. Melchoir 20:58, 1 December 2005 (UTC)

It seems "very small real number" is a confusing term as well - let me rewrite the whole thing fully to avoid and argument about whether it's reasonable to use that term:

1. If 0.999... were a real number less than 1 then 1- 0.999... would be a positive real number less than all positive rational numbers.
2. But there are no positive real numbers less than all positive rational numbers.
3. Therefore the real number 0.999... is not less than 1.

JPD 10:58, 2 December 2005 (UTC)

It's still wrong because:

1) You are referring to a positive real number less than all positive real numbers. There is no such number. How do you know such a number exists? 2) Having assumed that there is such a number, you then state that there is no such number.

Have you thought at all about this? Are you just another brainless PHd? Melchoir's argument even though it is flawed, is far superior to yours. Come on, who do you think you are fooling? logamath1

I give up. My argument is exactly the same as Melchoir's, only trying to write it all out in language you accept. The whole point of the reductio ad absurdam argument is that such a number doesn't exist. The point is that such a number needs to exist in order for 0.999... to be a real number less than 1. I never said that such a number existed, I said "If 0.999... were a real number less than 1, then 1-0.999... would be such a number".

If you think Melchoir is saying anything different, you obviously don't understand his point, which would explain why you think it is flawed. JPD (talk) 13:14, 2 December 2005 (UTC)

I am saying that 0.999... is a real number less than 1 such that at least one real number 1-0.999... exists. Although I can't find this number because I believe 0.999... is irrational, does not mean it does not exist. You talk about a reductio ad absurdum proof but there is no such proof. Let 1-0.999... = x and let me show you how this reduces to the absurd if x is equal to 0. Here is a reductio ad absurdum proof:

  Let   1  =  0.999...
  Now add .1 to both sides:

         1.1  =  1.0999....

  which is clearly false, therefore 0.999... cannot equal 1 and thus it must be less than 1.
  Don't try to tell me that .1 = 0.0999... - it is not. When we compare numbers in the decimal
  system, we begin the comparison with the most significant digit.

logamath1

Clearly false?? It looks very true to me. Comparing number in the decimal system works the way you describe, as long as neither of them ends in a 999... . You say that the real number 1-0.999... exists, and is not zero. Lets assume you're right and call it x. Then x must be less than 10^-i for any natural number i. Otherwise, 0.999.... would be less than 0.999...99 with i 9s. Such an x is not possible. JPD (talk) 14:21, 2 December 2005 (UTC)

Yes, it is clearly false. Such an x is possible always. Give me any 10^-i and I will simply give you 10^-(i+1). Of course it is possible for any natural number i. And comparing numbers in the decimal system works the same for all numbers including numbers that end in 0.999.... logamath1

You have not given a reason why comparing numbers should work in the way you say it is, or why it should be false. You have also misinterpreted what I said. If what you say is true, then x=1-0.999... should be a single positive real number that is less than 10^-i for every i. This is quite different to saying that for every i there is an x < 10^-i, and you yourself said earlier that such a positive real number does not exist. JPD (talk) 16:09, 2 December 2005 (UTC)

I do not have to provide a reason for why decimal numbers are compared this way - this is how the decimal system was designed. You are making a mistake by assuming that x=1-0.999... should be a single positive real number less than 10^-i for any i. This is faulty logic. I am making no assumptions about the value of x except that it exists. I do not call it a very small real number and I do not call it an infinitesimal. I only call it a real number that is greater than 0. Finally I did not say at any time that such a positive real number does not exist. Look JDP, the fact that you cannot find the exact value of pi in any radix system does not mean that pi is not a finite value. Yet do you ask: Is there an x such that x-pi = 0? Of course not. Please don't tell me x=pi (in fact it is pi but not by your reasoning. We reach this conclusion because we know pi exists but do not know its exact value just as we don't know the exact value of 0.999... or the value of 1-0.999...) because we don't know the exact value of pi. What you have with 0.999... is very similar: x-0.999... = 0 means x=0.999... and not that x=1. logamath1

Logamath1, you make a mistake when you say "this is how the decimal system was designed". In fact, any careful definition of the real numbers based on decimal expansions SETS 0.999... equal to 1. This is why I believe you are instead talking about Richman's decimal numbers. Nobody is saying the real numbers and the decimal numbers are the same, except perhaps you. Melchoir 18:32, 2 December 2005 (UTC)

Melchoir is completely right. Your comments about pi are a sidetrack, so I will only answer your comments about what you call my faulty logic. The only assumptions about x that I am making are parts of the definition of the real numbers. If x=1-0.999... it must be a single real number, since the reals are a field. If x>10^-i for any natural number i, then 0.999... = 1-x < 1-10^-i = 0.99..99 with i 9s, which is a contradiction. This is not an assumption, it is simple algebra. Therefore x<10^-i for all natural numbers i. I did all this without saying anything about the value of x that didn't follow from it's definition. I didn't call it infinitesimal, or very small. I did, however, show that it is smaller than any positive rational number. Because there are no positive real numbers that are smaller than every positive rational number, x must be less than or equal to 0. If you are dealing with a different number system, like the decimal numbers mentioned above, then you get different results, but it must be 0 in the reals. JPD (talk) 18:45, 2 December 2005 (UTC)

(Backing up a little bit, and I hit an edit conflict) Half-rhetorical nothing, that was a serious question. What theorem, what postulate, what what says you can't do arithmetic on infinitely repeating decimals? Now, admittedly, you can't use any grade-school techniques that require you to go digit-by-digit the whole way through, especially if they require you to start at the least significant digit, but arithmetic is not defined by the techniques you use to solve it.

For instance, take the number .23454545... and .43212121... These are equivalent (whether you accept it or not) to the fractions 129/550 and 713/1650.

Now, let's add the decimals. You can't start at the least significant digit, as there is none. So let's start at the most significant digit and work backwards! Yes, that is totally legal, so long as you remember that any carry digits still get added to the previous digit. That doesn't even matter in this case; there will be no carry digits (yes, I chose those decimals with that intent). We also won't be looking at every digit, but since both fractions repeat forever, all we have to do is add far enough to find the pattern. Which, in this case, is really easy: .66666666... Which is equivalent to 2/3.

Now let's try adding the finitely-represented-so-you-shouldn't-scream-at-me-about-them fractions. We get the exact same result.

Let's try another one. This one will use carry digits. .1717... and .1616... equivalent to 17/99 and 16/99. Adding the most significant digits produces .2; adding the next most significant produces 13, and the carry digit is added to the already-existing 2, so we get .33. And everything else is simply repetition, so we end up with .333... or 1/3. And, lo and behold, .

In fact, you will get the exact same result with any two rational numbers of your choosing, whether in fraction form or decimal form. The only place where we'll disagree is the .99999... case; for instance, adding 1/3 (.333...) and 2/3 (.666...) But, as pretty much everyone has been trying to tell you, this case works too since .999... and 1 are the same number (even if you don't accept that, you have to admit that the number of cases where adding rational numbers does work is uncanny.) So addition, at the very least, works just fine with infinitely repeating decimals, thankyouverymuch. And subtraction is simply the inverse of addition, so there's no reason why 1 - .999... would be undefined.

And, one final note. Do you really believe that your use of the term "PhD" is an insult? --HeroicJay 19:04, 2 December 2005 (UTC)

Melchoir has cordoned off the following from the above:

You are either sincerely mistaken or you are lying. You challenged logamath1 to suggest a construct for the reals. He suggested the Decimal construct. You then tried to show this is true and were not able to do so. 70.110.92.137 19:57, 2 December 2005 (UTC)

Forget about Melchoir and let's see what you are saying: Your logic is strange. You first say that x must be a real number. I agree with you on this. Then you say, "If x > 10^-i for any natural i, then 0.999... = 1 - x < 1-10^-i ..." This is rubbish so I'll stop here for no one will start off an argument by saying that x > 10^-i. You must be a newbie. 70.110.92.137 20:05, 2 December 2005 (UTC)

Please show me how you can do any accurate arithmetic with pi, e or sqrt(2). It is all approximate whether real or decimal and we are talking about decimals here. 70.110.92.137 20:05, 2 December 2005 (UTC)

HeroicJay: As long as you are adding fractions you can obtain accurate answers because these are rational (provided you leave your answers in fraction form). The minute you begin dealing with decimals, all bets are off. As for the part about PhD, I leave this to logamath1 - he can probably answer this better than I. 70.110.92.137 20:05, 2 December 2005 (UTC)

70.110.92.137, I am beginning to wonder again whether or not you are logamath1, despite logamath1's statement otherwise. I will assume you are different people for now. You have insulted me for the last time. I will not respond to anything further you say, and all of your future edits will be cordoned off from discussions with other users. Melchoir 20:14, 2 December 2005 (UTC)

There will be no more edits. I have wasted my time. 71.248.138.198 02:56, 3 December 2005 (UTC)

You didn't understand my argument. You obtain the same results whether you add decimals or fractions in both cases and, indeed, in any cases you can imagine. --HeroicJay 20:33, 2 December 2005 (UTC)

As for your other point, you can do perfectly accurate arithmetic with pi, e, and the square root of 2. In these cases, using the decimals are useless for exact answers because there's no pattern in them, but you can (in fact, many people do) work with those numbers in their symbolic form. --HeroicJay 20:37, 2 December 2005 (UTC)

Heroicjay: I have never met a more obstinate fool than you. Indeed you are duller than most and I see no point in carrying on any exchanges with you.

Wikipedia: In addition to the nonsense you write in this article, there is more nonsense in other math articles. Actually your math articles are referred to as the WikiMath joke articles. Your infinitesimal article states: "...an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number.." - This is such incoherent and illogical garbage for the first part implies it is a positive real number and the second part that it is zero. How can it be both zero and a small positive real number? I leave you to break your stupid heads over this. 71.248.138.198 02:56, 3 December 2005 (UTC)

How pleasant of you to insult me rather than make any comments regarding my points. The infinitesimal article states that it's larger than zero but smaller than any positive real number - this is true, because an infinitesimal is not a real number. (Infinity is larger than zero without being a real number as well.) --HeroicJay 03:35, 3 December 2005 (UTC)

"There will be no more edits. I have wasted my time. 71.248.138.198 02:56, 3 December 2005 (UTC)" You said you'd stop. Please do, you make no attempt to reach consensus, merely to push your viewpoint time and again on people who aren't going to accept it. You've now moved on to personal attacks, which really have no place on wikipedia. 81.86.207.192 13:04, 3 December 2005 (UTC)

To whatever credit he deserves, he posted his insult to me at the same time (in the same edit) as his "no more posts" comment. Though that doesn't really justify insults and adding more arguments immediately below if he really does intend not to defend them. --HeroicJay 16:54, 3 December 2005 (UTC)

Here we go again!

Melchoir writes: "...any careful definition of the real numbers based on decimal expansions SETS 0.999... equal to 1." - Melchoir

Not only is this an outright lie but it is clearly demonstrated to be false:

1) The Indians did not use a decimal point. This was an invention of Scotland.

The notation we use today first appeared in a book called "Descriptio" by the Edinburgh 
mathematician, John Napier, Laird of Merchiston, in the 1616. He used a decimal point 
to separate the whole number part from the decimal number part. Known as 
'Marvellous Merchiston", he published many other treatises including 
"Mirifici logarithmorum" (1614) and Rabdologia (1615) on systems of arithmetic using 
calculation aids known as Napiers Bones. 

2) Fractions were invented long before the decimal point.

In the 10th cent. A.D. Arab mathematicians extended the decimal numeral system to include fractions.

Melchoir will try to justify himself by using terminology such as any careful definition of the real numbers - this is entirely subjective and is open to debate.

I asked who came up with the idea that 0.999... = 1. No one on Wiki's board of idiots was able to answer this. Why? The answer is simple: they are all products of today's institutions that are run by a bunch of baboons masquerading as Phds. My apologies to the few Phds who really earned their diplomas but the majority are to be regarded as ignoramuses who are spineless and incapable of thinking on their own.

It's not a concept that is unique to the Wiki in any sort, and it's a simple application of Calculus. If 1 != .999..., then several concepts in Calculus are completely false — and I've yet to see evidence that anything in Calculus is false. And your talking about number systems that don't use decimals says nothing about number systems that do. Melchior talked about "any careful definition of the real numbers based on decimal expansions." And insulting my intelligence, or implying that I'm just "going along with the crowd" (a laugh and a half from anyone that knows me in the real world) is not appreciated. Your complaints against the institution that defined the decimal number system is ludicrous at best. Sure, there are other logically consistent number systems, but they're not the one used by most of the world today. And, in the decimal system, an infinitely recurring decimal is defined as the limit of the sum of its digits. And, as it just so happens, the limit of the sum of the digits in .999... is 1.0. --HeroicJay 16:41, 5 December 2005 (UTC)

The definition of the real numbers is not open to debate. Mathematics is useless as a language unless we agree on the meaning of its jargon, and "real number" is too entrenched a concept to change, even if it were a good idea, and it really isn't.

81.86.207.192 158.35.225.228, I wonder if you think the rigor of modern mathematics is somehow a challenge to your creativity. It's a fine intellectual exercise to dream up number systems in which the analogues of true statements for the real numbers turn out to be false. One might even make a living doing this. It doesn't make me a liar.

As the self-appointed dictator of this talk page, I declare that it is closed to personal attacks. Comments insulting other users or the Establishment will be erased. The purpose of the talk page is to discuss improvements to the article. If you have a suggestion, let's hear it. If you're confused but willing to learn, ask a question. If you're here to whine about PhDs, go away. Melchoir 19:12, 5 December 2005 (UTC)

You have not provided one proof or argument that shows why the real number system must have 0.999... = 1. It is easy to strike out what you don't agree with but much harder to admit you are wrong.

We have. Multiple times. Most of them clearly seen above. We've shown that the limit of the series .9, .99, .999, etc. (which is how a repeating decimal such as .999... is defined) is 1. We've shown that there must be a real number between any two unequal real numbers, but there's nothing between .999... and 1. We've shown that simple algrebra or arithmetic lead to .999... = 1, although these aren't as convincing as the Calculus stuff. You just simply refuse to acknowledge any of it. --HeroicJay 22:44, 5 December 2005 (UTC)

...The limit of the series is not the sum of the series and a definition does not make it true. There are real numbers between 0.999... and 1 as has been demonstrated above... There has been a solid proof by induction that has also been rejected... --anon

I'm not re-adding it, but I'm going to reply to the "points" made by Mr. IP. (While, yes, he is being abusive, I don't think deleting any his posts other than the insults is really accomplishing anything.) The limit of a a sum of an infinite number of terms is an infinite summation, so .999... is the limit is of the sequence .9, .99, .999... and the sum of the series .9 + .09 + .009 + ... (the result is the same either way.) To say that something can be the definition of a term and not be a property of that term is just mind-bogglingly silly. And if there is a number between .999... and 1, could you tell us what it is or describe it? The answer is no, you can't, because it would require that, somewhere along the line, one of the digits would have to be larger than 9. For instance, for something between the numbers .333... and .4, a number in between would have a digit larger than 3 in some place other than the first. (such as .34, .33501, .333333338, etc.) However, there is no digit larger than 9, and the nines repeat forever, so there isn't any other digit to increase (without resulting in 1 or something larger.) --HeroicJay 02:19, 6 December 2005 (UTC)

Okay, I've restored the non-abusive bits of anon's latest post. I hope that my deletions do accomplish something; I hope that they convince would-be insult throwers to reconsider their methods. Melchoir 02:35, 6 December 2005 (UTC)

No complaints with deleting the insults. None whatsoever. But deleting his arguments, shaky as they are, kinda gives the vibe that we don't want to (or can't) respond to him, or are simply trying to silence him for whatever goals, which is kind of exactly what he implies with his conspiracy theories and derision at the word "PhD" and insults. Last thing I want to do is make him think his silly conspiracy theories are true. :p --HeroicJay 03:36, 6 December 2005 (UTC)

Indeed. Melchoir 05:30, 6 December 2005 (UTC)

"The limit of a a sum of an infinite number of terms is an infinite summation, so .999... is the limit is of the sequence .9, .99, .999... and the sum of the series .9 + .09 + .009 + ... (the result is the same either way.)" ...I will let the readers decide on this. The limit of an infinite number of terms IS NOT AN infinite summation. An infinite sum is impossible. And no, you cannot define it this way because it does not make sense... You saying it one million times or claiming that it is true does not make it true. A Definition is useless unless it makes sense AND it is TRUE. —The preceding unsigned comment was added by 70.110.81.224 (talk • contribs) 2005 December 6.

Oh please!!! If you are going to restore points, make sure you restore everything... YOu have only restored what makes you look good. The fact that you have final editing say displays clearly that this is not a peoples encyclopedia... Wiki is a view of the world according to YOU... —The preceding unsigned comment was added by 70.110.81.224 (talk • contribs) 2005 December 6.

...You cannot show me one reputable source that states 0.999... = 1... Show me an encyclopedia Brittanica or something that has been around for a long time that states this. I put it to you that ... 0.999... = 1 has not been around that long. It was probably conceived in the early or late twentieth century... ...This article should be deleted. It is not encyclopedic... —The preceding unsigned comment was added by 70.110.81.224 (talk • contribs) 2005 December 6.

I don't know whether I would call the article encyclopaedic, but it's quite correct. You say an infinite sum is impossible, and you might have a bit of a point in that it isn't quite the same beast as a nice finite sum. However, what most of the world means when they write 0.999... (or 0.333... or similar things, for that matter) is the limit of the truncated decimals. It doesn't really matter whether you call this an infinite sum or not, but it's what we mean when we write 0.999... . I can't tell what you mean when you write it, but if that's not what you mean, then we're talking about different things, so it's not surprising we think they're different! JPD (talk) 16:58, 6 December 2005 (UTC)

This is exactly the problem: most of the world do not think of 0.999... as the limit of 9/10 + 9/100 + 9/1000 + ... They think of it as an infinite sum or a number. If you talk of the limit of the partial sums of (9/10;9/100;9/1000;...), then its limit is 1. The same principle applies to 1/3, 1/6 and any other fraction that cannot be represented exactly in base 10. 0.333... is not equal to 1/3. 0.333... is an approximation for 1/3 in base 10. Although it cannot be represented exactly in base 10, it can be represented exactly in many other bases, e.g. 1/3 = 0.1 (base 3) 1/3 = 0.2 (base 6) and so on. Now a 1/4 in base 3 is approximated by 0.020202... It is not equal to 0.020202... even though the limit of 0.020202... is 1/4. I do not have a bit of a point, I have a major point. Terminology is extremely important and infinite sum and limit of an infinite sum are not the same thing! Lastly, 0.999... was never defined as a limit, neither were the real numbers defined as a limit until Weierstrass, Cauchy and others decided to 'rigourize' mathematics. In many respects they inhibited the progress in understanding the real numbers. Real analysis is filled with inaccuracies and contradictions as a result. —The preceding unsigned comment was added by 158.35.225.228 (talk • contribs) 18:09, 2005 December 6.

158.35.225.228, I'd like to respond to you, but I don't know where I ought to start. What would you say is your main point? (Oh, and by the way, if I recall correctly you're Logamath1. You might want to sign in so that I don't accidentally confuse you with 70.110.81.224. Ironically, remembering numbers isn't my strong point!) Melchoir 19:21, 6 December 2005 (UTC)

An infinitely repeating decimal represents a limit, but it doesn't have a limit in and of itself - it's a single number. (I'm reminded of T-shirts that say that 2 = 3 for sufficiently high values of 2.) There is no contradiction with saying that .999... = 1.0; it just seems like there is because of their different appearances. No, the contradictions come into play when you say that they're unequal, for all of the various reasons we've said that they're equal. --HeroicJay 20:09, 6 December 2005 (UTC)

You say that most people think of 0.999... as an infinite sum or a number. I agree. My point is, that when we think of it as a real number, the real number that we think of is the limit of the truncated decimals, which is 1. This is what the notation usually means. I don't really understand how you and the other anons are interpreting it. If you prefer to consider it as an infinite sum and say that infinite sums are impossible (meaning that they can't be evaluated as a single real number), then what you mean by 0.999... is not a real number and cannot be compared to 1. If, instead, this infinite sum is to be considered a real number, then the only reasonable way to do so is to take it to be the limit of the truncated sums. It is contradictary to treat the infinite sum as a real number that is less than 1. If you think that a number representing this infinite sum exists and is less than one, then you are dealing with a different set of numbers. You might think that this set is better because you don't like definitions with limits and so on, but if so, that is a matter of philosophy, not mathematics, because the simple fact is that it is not the same "real numbers" that the rest of us are talking about. JPD (talk) 10:57, 7 December 2005 (UTC)

0.999... is less than 1 for all the reasons you say it is equal to 1.

That doesn't make any sense whatsoever, and I assume without even checking the history that you're Mr. 70 and not Logmath1. --HeroicJay 20:18, 6 December 2005 (UTC)

Whoa! I was wrong! --HeroicJay 20:19, 6 December 2005 (UTC)

It makes perfect sense. You are the one who is having difficulty understanding any of this.

If I may speak to the article itself...

I have just found the most wonderful paper:

Conflicts in the Learning of Real Numbers and Limits by D. O. Tall & R. L. E. Schwarzenberger, University of Warwick Published in Mathematics Teaching, 82, 44–49 (1978). pdf webpage

It's 13 pages long. Whoever finds this sentence in the first place should go skim it; it's got some gems.

I propose to rewrite the article with Tall as our motivation and primary reference. In fact, I would have proposed this before, but without Tall it really would have been Original Research. Additional references would be great, including a more modern one, but it's a good start.

The article currently does not address many common misconceptions; I think we should mention them all and explain why they are wrong, although preferably after the proofs. I want to make it clear that I'm not backing down under fire and "teaching the controversy". There is no controversy, and the article must make that plain. There is, however, a whole lot of confusion that can be productively cleared up in the article itself. In some sense, this effort would be similar to a page on an urban myth.

My questions are:

1. Is this a good idea?
2. How should it be done? (gradually, subpage here, subpage of Melchoir, new article, blank-and-rewrite, or not at all?)
3. Does anyone have another reference on popular difficulties with 0.999...?

Melchoir 21:16, 6 December 2005 (UTC)

Also, this earlier paper by the same author is mostly generalized nonsense, but it gets interesting and relevant on page 10. Melchoir 22:12, 6 December 2005 (UTC)

The paper you mention, "Conflicts in the Learning of Real Numbers and Limits" (PDF), strikes me as more psychological speculation than mathematics, but I believe it could be helpful to teachers and others. I would support adding a citation to it in the article.

As for your question 3, my web searches have turned up little of substance. For the mathematician and game enthusiast, there is a fun article on "Hackenstrings" [1]; I hesitate to cite it, even though it explicitly states a non-conflict: "It should not be thought that the Hackenstrings approach in any way shows that conventional mathematics is `wrong'. Rather, the axioms of Hackenstrings arithmetic are different from those of the real numbers…". Perhaps it would help some folks to see explicit alternatives to the reals, but I'm afraid it would confuse many more. The "Ask Dr. Math" [2] citation currently listed in the article is the best of those cited, but still leaves a lot to be desired. If you find more material, please do bring it up here.

Question 1, your proposal to rewrite based on Tall, is a temptation I think we should resist. The proofs already avoid every source of confusion Tall discusses, and actually do so better than Tall. Which leaves the proposal to mention common misconceptions and explain why they are wrong. First suppose we can do it to well; we will give students a head full of misconceptions, and maybe they'll understand the mistakes. But chances are they won't understand better; they'll be more confused than ever. Now be more realistic. As soon as we say "Here's what you're thinking and here's where you went wrong", we set ourselves up for failure. Neither you nor I nor anyone can speak for others, to assert we know what they're thinking. And when we try to do so, they will feel obliged to say that our explanation does not work for them. This talk page alone attests to that! In fact, Tall's paper on "Conflicts…" does a questionable job, mathematically and psychologically, of addressing the misconceptions he observes.

The article presently includes two solid advanced proofs. Far better to stick with that and not even try to second-guess (badly) and correct all the possible mistakes. "It ain't broke; don't fix it!"

May I suggest, however, that Wikipedia has many pages that are broken. For mathematics, in particular, see Wikipedia:WikiProject Mathematics#Things to do. --KSmrq^T 02:59, 7 December 2005 (UTC)

Yes, the papers I link to contain speculation, but they are published, and they contain quotes by students displaying some misconceptions. If we address those misconceptions, such a source is necessary to avoid speculation on our part!

That Hackenstrings bit is very interesting. I've seen nimbers before, but not those. As for the citations currently in the article, the first two don't address any misconceptions, and the third is so confused that if it's preserved it should only be as a bad example! I think we can do a better job.

As for rewriting, I must apologize for what was honestly just a poor choice of words. As a mathematician, I think your proofs are the strongest point of the article, and my proposal does not require that we alter them. For now, I simply want to tack on a section at the end exploring the misconceptions. Nor, of course, do I propose to reproduce Tall, who avoids the structure of the real numbers altogether. His article could be useful mainly in that

1. he offers advice on misleading phrases and concepts to avoid, and
2. he records concrete examples of misconceptions which we may address.

If he doesn't address them well enough, we can do better than him, too.

Of course we can't guess what the reader is thinking. The reader, in fact, might agree with us; surely teachers will read our article, not just students! But the encyclopedic voice isn't really appropriate to connect with readers on a personal level anyway. I don't propose that we attempt to open a dialogue with the reader, but that we describe what students are known to think. With references such as Tall, we don't have to guess on that.

As for there being better articles to work on, you're right. Melchoir 04:14, 7 December 2005 (UTC)

I'm assuming teachers will read Tall, and benefit from his survey of misconceptions. Also, should an encyclopedia merely present facts, or is it appropriate for it to air and debunk misconceptions? Since you've asked for comments from WikiProject Mathematics, I'm curious to see what opinions others may have.

I won't try to conceal my dissatisfaction with the current "External links", especially the "Repeating Nines" one. If I had found good replacements I would have tossed the inherited ones long ago. To my dismay, the discussions turned up by a web search were consistently poor. Even the sci.math FAQ entry is disappointing. It's no wonder skepticism persists in the face of arguments that are generally about at the level of "Trust me." Please, if you can find better links, do so! --KSmrq^T 05:13, 7 December 2005 (UTC)

Of course you already know my answer to your question, but I'll give it anyway: yes, it is appropriate to air and debunk misconceptions in an encyclopedia article. I would go so far as to say that it's not only appropriate but required, at least when the misconceptions are so pervasive and well-known. This view seems to be accepted elsewhere on Wikipedia; note the existence of Category:Urban legends, Category:Logical fallacies, and in fact all the material under Category:Communication of falsehoods. Melchoir 05:37, 7 December 2005 (UTC)

Mind the following:

The article is neither interesting, nor relative. It has nothing about proofs and talks of infinitesimals as if these were widely understood. There is no such thing as an infinitesimal.... If an infinitesimal is greater than zero but less than every positive number, there can only be one of it.... It would be a good idea to delete this article and if you really want to talk about comparing 0.999... with 1, then you should make it clear that 0.999... is less than 1. Tall refers to some questions posed to students and discusses their response. He tries to infer that students answered incorrectly because they were unsure of the denotation of recurring. ...this is taught in elementary school and those same students claimed that they knew the definition of a limit.... —The preceding unsigned comment was added by 68.238.99.174 (talk • contribs) 02:26, 2005 December 7.

In mathematics, any sentence that doesn't name a set and starts with "you can't" or "there is no such thing" is already wrong. Melchoir 02:53, 7 December 2005 (UTC)

In mathematics there is no such thing is correct if such a thing is not properly defined. And you can't dream up fancy words that mean nothing and develop theories out of these. --anon

If an infinitesimal is greater than zero but less than every positive number, there can only be one of it....

Surely there could be one blue, one red? - Fredrik | tc 05:49, 7 December 2005 (UTC)

Care to explain?... --anon

Show me where it is written that "infinitesimal" means "nothing". Melchoir 19:13, 7 December 2005 (UTC)