Talk:0.999...: Difference between revisions

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

			:Rasmus is going to say that
			:* the Archimedean property is a property for the entire set of numbers, and it applies.
			:* x is indeed a real number, but not one that is greater than 0.
			:After all, what we are trying to show is just x=0. If you are now willing to sacrifice the Archimedean property for your brand of "real" numbers, you will probably agree that yours are not what mathematicians usually call the real numbers.

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

Revision as of 17:15, 21 December 2005

Because of their length, the previous discussions on this page have been archived. If further archiving is needed, see Misplaced Pages:How to archive a talk page.

Previous discussions:

Merge proposal

I have proposed that this page be merged with positional notation, as it is only really interesting because it expresses the peculiar property of positional notation that there are multiple distinct sequences of digits representing identical real numbers. I also would consider making it briefer. Deco 02:15, 10 December 2005 (UTC)

Disagree. If anything, this article should be longer. The topic may have limited interest in real mathematics, but it's a fascinating educational issue. You're welcome to expand Positional notation, but this article couldn't be merged there without destroying it. Melchoir 02:59, 10 December 2005 (UTC)

I have to question the appropriateness of this page. I also find the question fascinating as an educational issue, and I think a coherent evaluation of the topic is a very useful thing for a lot of students. However, the article seems a bit too "instructive," and I wonder if it really belongs in an encyclopedia. --Monguin61 03:08, 10 December 2005 (UTC)

Monguin61, maybe the article gets "instructive" at times, but surely that can be fixed, leaving an informative article, without deleting it? Melchoir 03:26, 10 December 2005 (UTC)

That might be possible, but its the topic itself that bothers me more than the style. There aren't many other articles that go into such depth on a specific problem or example, are there? Removing the content entirely is unnecessary, but I do think that in light of the nature of the article, merging it with positional notation or a similar article might be in order. --Monguin61 03:31, 10 December 2005 (UTC)

Yes, I doubt that there is a comparable article of comparable length. Prompted by the merge suggestion, I've been looking for a better possible target for the merge. Positional notation is a poor choice, since it doesn't address infinite tails, and it doesn't specialize to the decimal system. I've found Recurring decimal, which is a much more natural home -- and similar content is already there! The arguments at Recurring decimal are a mess, so if you want to merge this article there, I won't complain. However, someday I'll take a crack at writing an encyclopedia article on 0.999... myself. Melchoir at 85.195.123.22 04:18, 10 December 2005 (UTC)

Scratch that, Monty Hall problem is longer, and has a section on "aids to understanding." Considering that it was a featured article, I guess this one is alright. --Monguin61 10:16, 10 December 2005 (UTC)

Oppose. So you find this topic boring and trivial. I find Madonna boring and trivial. Each still demands a dedicated Misplaced Pages article because they draw notable attention. This topic is raised repeatedly: on the sci.math newsgroup, in schools around the world, and all over the net. It is, in fact, a Frequently Asked Question for sci.math; but nowhere, that I have found, is the topic treated nearly as well as here.

And why is that? Because people with a course in real analysis under their belts dismiss it with a handwave, and elementary school teachers who know the psychological obstacles their students face do not have the expertise in foundations to handle it securely.

Or look at the endless discussions on this very talk page. (First, notice that half the posts ignore the most basic rules for all talk pages and lack signatures, indenting, and civility. Second, notice that, contrary to the stated purpose of talk pages, the discussions ignore the article itself.) Most telling, neither side in the debate argues well; the advanced proofs in the article are noticeably more careful than those seen on the talk page.

It takes a certain amount of mathematical sophistication to appreciate the difference, to get beyond the handwave "it's a limit, deal with it" argument repeated over and over to no avail. Also, as with many sensitive topics (abortion, for example), we must be careful in more than mere facts; we must choose our words carefully, because psychology and emotional impact matter. Which is also why it would be insensitive and misguided to shorten the article. Mathematically, a proof page is needed for no theorem, certainly not this one, because Misplaced Pages includes no original research. Nevertheless, we find proof articles valuable. On such pages we do not limit ourselves to a single brief proof; consider Proofs of Fermat's little theorem, which has five proofs.

Both the merge proposal and the shortening proposal are ill-considered, and should be rejected. --KSmrq 06:12, 10 December 2005 (UTC)

I appreciate your sentiment above, but as you mention, the topic is a Frequently Asked Question, and it seems that the article is there to attempt to resolve that question for anyone who might be looking for an answer. Is that really what Misplaced Pages is for? On the other hand, you do have a good point that the topic is treated better here than elsewhere, and to me that is reason enough for the content to remain. --Monguin61 06:36, 10 December 2005 (UTC)

Oppose. This article is far from complete. Among other things, it fails to point out that the z-transform of a p-adic expansion of a real number has an image that is a Cantor space. What does this mean? Consider the 10-adic (aka "decimal") expansion of a real number, with digits $d_{n}$ . The z-transform is then

\sum _{n}d_{n}z^{n}\,

Then, for z=1/10, one gets exactly the decimal expansion of a real number, but for other z-values, even z-values infinitessimally close to 1/10, one gets Cantor sets of various shapes. This is a non-trivial observation, and seems to underly the properties of fractals in particular, and chaos in general. The resulting topologies are non-trivial in a variety of ways. Note, in particular, the z-transform of 0.9999... is not at all equal to the z-transform of 1.00000... except when z=1/10 (precisely and exactly). (An example of a high-falutin' version of this is the Alexander horned sphere.) linas 07:45, 10 December 2005 (UTC)

Oh, and why should one mention this? I think this will make a lot of the people who don't believe that 0.999..=1.000... a lot happier, and give them something to think about. Although it is "technically" true that 0.999..=1.000.., the z-transform makes it clear that decimal expansions are actually very pathological. Its a mathematical slight-of-hand that holds true only for the very special value of z=1/10, and makes something look smooth when its really not. So I'm hoping that this extra tit-bit can make everyone go home happy. linas 07:55, 10 December 2005 (UTC)

Are you really? And I'm hoping for world peace. ;-) --KSmrq 08:49, 10 December 2005 (UTC)

In particular, one can construct maps from the Cantor set onto the unit interval, in such a way that 0.9999... and 1.000 correspond to two distinct points of the cantor set. (note the word "onto": the cantor set really does have the cardinality of the real number line, plus, you could say, "a little bit more", an extra countable infinity of points).linas 08:06, 10 December 2005 (UTC)

That's a pretty interesting topic in itself. As far as satisfying those who don't believe that 0.999...=1, though, I think it would work more just to confuse. Correct me if I'm wrong, but this article only covers z=1/10, right? Using any other value for z is simply asking a different question, and only tangentially related. --Monguin61 08:08, 10 December 2005 (UTC)

Removing tongue from cheek, you do make a valid point, linas, about fun diversions. The article mentions that the construction of reals from rationals depends on ordering, so there is a hook there for the valuation that leads to p-adic numbers. It could also be instructive to delve into non-standard analysis, or the impact of replacing the usual Set topos with something more interesting. For example, Dedekind cuts and Cauchy sequences can give different "reals" in a non-Set topos. But I don't think I'm clever enough to write such an article without losing either the audience or my composure. And given the fun the forces of chaos have without encouragement, I'm not inclined to hand them more toys. --KSmrq 09:16, 10 December 2005 (UTC)

To respond to my various protestors: I don't find this topic boring at all, I have a Bachelors degree in mathematics. I proposed the merge because I thought it was a reasonable compromise between those who wanted the page removed altogether and its supporters — I'd rather see the content preserved than lost. At the very least I think positional notation should include a brief summary of this article and a link to it. Deco 20:05, 10 December 2005 (UTC)

Another response: You say "Positional notation is a poor choice, since it doesn't address infinite tails, and it doesn't specialize to the decimal system." It certainly should discuss infinite tails, and the 0.999... = 1 issue is not at all specific to the decimal system. For example, in octal, 0.777... = 1, and in binary, 0.111... = 1. Moreover, it extends to any real number ending in an infinite sequence of (largest digit), such as 0.263999... = 0.264. The example discussed by this page is just a very specific case. Deco 19:08, 13 December 2005 (UTC)

I agree about expanding Positional notation to discuss the issue of infinite tails, and in all bases, with a link here. I think I meant to say simply that Positional notation would be a poor choice for a merge, especially since there are better homes. But it's been awhile, and after reading Monguin61's, linas', and KSmrq's replies, I would no longer support a merge to any article, so my original comment is kind of moot. Melchoir 00:09, 14 December 2005 (UTC)

In light of the views stated here, I'm removing the "proposed merge" tag. If Deco or anyone else thinks I'm acting too hastily, feel free to revert. --KSmrq 00:31, 14 December 2005 (UTC)

Adding the z-transform

KSmrq is worried, but I say go for it, linas. Rather than encourage the forces of chaos, I think such examples might help convince our doubters that, yes, mathematicians have seriously considered the possibility of setting 0.999... not equal to one, and the consequences thereof. It shouldn't cause trouble as long as you write it after the discussion of the decimal series, and you don't reuse notation. Melchoir 18:04, 10 December 2005 (UTC)

Delete: ... 0.999... is equal to 0.999... - this is its sum. It is not equal to 1. In fact, simple induction shows it is less than 1 in the decimal system which is how most people interpret 0.999... People do not think of 0.999... as the limit of its partial sums any less than they think of 4 as the limit of its partial sums (4 + 0/10+0/100+...) This article ... should be deleted or rewritten to show that in a positional system, 0.999... is less than 1. ... 0.999... = 1 is an ambiguous and chancy statement. Either qualify it or discard it completely. Do you want people to be attracted to Mathematics ...? Limit are fairly easy to understand. If you want to talk about limits, then you should make it clear you are talking about limits. In every other sense, 0.999... is treated as a number just like 0.333... just like 4, just like any other number in the decimal system. Do the right thing - DELETE! Let KSmrq write another article... — Preceding unsigned comment added by 68.238.101.29 (talk • contribs) 2005 December 13 (UTC)

Anon, if you honestly want to discuss this article, you're going to have to be a lot more polite about it. Melchoir 18:20, 13 December 2005 (UTC)

No, he has a good point. There should be a section at the top giving a precise definition of the terms and notations used. With that, no more argument would be possible. --Monguin61 23:28, 13 December 2005 (UTC)

He may have a good point, and we should be more precise in the article. But have you seen anon's above post before I got to it? I won't encourage that sort of behavior. Melchoir 00:03, 14 December 2005 (UTC)

In what way should we be more precise? Each of the advanced proofs defines reals, equality of reals, and the interpretation of a recurring decimal in terms of reals. However, I would vigorously disagree with putting such material at the top. Quoting from Misplaced Pages:Manual of Style (mathematics):

A general approach is to start simple, then move toward more abstract and general statements as the article proceeds.
It is a good idea to also have an informal introduction to the topic, without rigor, …

The article follows the guidelines appropriately, down to the concluding "Generalizations" and "See also" sections. --KSmrq 02:33, 14 December 2005 (UTC)

Anon's post wasn't appropriate, obviously. The thing about the general guidelines is that they're general, and they don't take into account the nature of this particular article/debate. This debate is going to continue on the internet as long as there are people that do not understand the entire question being asked. I think, for the sake of driving away the impolite anons that have some insatiable desire to argue here, an exact definition of terms and notation should be introduced in the article, very clearly. I think if this article exists at all, it should do everything possible to quell the discussion. Unlike some of the other subjects of debate on this site, the specifics of the decimal system do not depend on your politics, or your opinions. There is one, only one, right answer. It is unfortunate that so much must be written to convince some people of this, but it is true. --Monguin61 19:15, 14 December 2005 (UTC)

...you have stated that you "love to mercilessly edit and correct these"... 158.35.225.231 13:36, 14 December 2005 (UTC)

Isn't it ironic that after you maliciously misquote me, I delete everything else you say? Gosh, the world's a funny place. Melchoir 18:42, 14 December 2005 (UTC)

EEK

I'd just like to take a timeout and say...ouch. This talk page makes my head hurt. Since when is mathematics so...political. I love math because there are right and wrong answers, and I seriously can't believe this has progressed over 2 archives. Oh, and psst.. .999...99 = 1. :D Arkon 05:14, 15 December 2005 (UTC)

If you think this is crazy you should see Misplaced Pages: Lamest edit wars ever. :-) Deco 05:28, 15 December 2005 (UTC)

Thanks for the reminder; we were due for a new archive!

I won't comment on the sociology or politics. Mathematics does have right and wrong answers; it also has choices. Over the millennia some of the choices troubled people. Should we write numbers using powers of 60, as in Babylon? Should units, tens, and hundreds have their own symbols, as in Rome? Are negative numbers really numbers; can they be correct solutions to equations? What are we to make of incommensurable numbers like √2, which caused consternation in Greek religion? And yes, the solution of some real equations seems to pass through "imaginary numbers" to get real answers, but are those "imaginaries" really numbers? Following the quaternions, can we make up a number system following crazy rules like "multiplication doesn't commute"? Does it make sense for a proof to be able to conclude anything from a falsity, to use the law of the excluded middle; or should proofs deny existence without construction? If a statement is true, will we always be able to prove it?

Are these old, settled questions? Our clocks and angles still use 60. Accountants use double-entry bookkeeping and parentheses so they never touch negative numbers. We still call √(−1) an imaginary number. Is it really that surprising that repeating decimals trouble some people?

I don't think that's what generated the volume of discussion on this page, but it has done so elsewhere.

Mathematics is a human endeavor. So is Misplaced Pages. So is life. Ain't it grand? :-) --KSmrq 07:20, 15 December 2005 (UTC)

I say the whole topic is simultaneously grand and lame. Melchoir 07:24, 15 December 2005 (UTC)

...Who the hell do you think you are Melchoir? .... AS for me, I do have better things to do. --anon

I am neither an admin nor a sysop; I practice soft power and I get away with it because I am right. Please, don't let me hold you up on your business elsewhere. Melchoir 20:06, 15 December 2005 (UTC)

The anon is being rude, but I personally disagree with this practice of editing others' talk page comments solely to remove mildly offensive rantings, even if they are off-topic. The beauty of free speech is that when everybody speaks their mind, ridiculous things like trolling and personal attacks just get ignored. Don't worry about it. Deco 20:40, 15 December 2005 (UTC)

Well, I started removing the off-topic stuff because I thought it would discourage the addition of more. It seems I have failed in that aim. Melchoir 21:52, 15 December 2005 (UTC)

You seem to have failed at a lot of things and now you are trying to forcibly silence your opponents? This is what a dictator does - so you are not wrong by calling yourself one.

Are you done? Melchoir 23:35, 15 December 2005 (UTC)

What's it to you? Don't read the posts if they are too harsh for your wimpish character.

Now are you done? Melchoir 00:16, 16 December 2005 (UTC)

Again, what's it to you?

I am interested in discussing the article. Melchoir 00:18, 16 December 2005 (UTC)

Well, now I am done.

New content

I've removed the following recently added "Counter Argument" from the article:

"Consider this: 0 is nothing, 1 - 0.9_ is not nothing, but 0.0_1 . 0 is not equal to something infinitely larger than 0, it's smaller. Therefore 0<0.0_1, 1>0.9_ . 1 is larger than the indication of a number approaching 1, no matter how close to 1 it is, because of this, it is never equal, always greater. The problem with the argument of whether .9_ = 1 is the definition/workings of the decimal, non-rational form of expressing rational numbers. The only situation where you could say .9_ = 1 is if you assume that by definition, that numbers follows a repetitive series infinitely, but infinity has a limit, which is where the number is equal to it's limit. This is a flawed interpretation of the notation; Really what the notation would mean is that the number continues as stated by the repeater indicator, but continues INDEFINITELY, NEVER reaching it's limit, which is why this notation is inaccurate to a rational notation.

If you say you have 1 of something, you have 1. you do not have an infinitely repeating quantity of numbers which approach 1. If you wanted to say you have an infinitely close quantity to 1, then you would use .9 +repeater symbol (underscore in this case (0.9_) and NOT 1."

I'm not sure what that passage is trying to say, but it contradicts the rest of the article. Whoever boldly added it didn't do anything wrong, but if you want to keep it in the article, you'll have to defend it here first. Melchoir 20:21, 16 December 2005 (UTC)

I concur. An article should never contradict itself. It also doesn't really make much sense to me. The phrase "infinitely close to" is meaningless, at least in this setting, many sequences never reach their limit, and the rest of it seems to contradict the usual definition of the intended meaning of infinite expansions in positional notation. Deco 20:34, 16 December 2005 (UTC)

Actually, they did do something wrong. Misplaced Pages has a strict policy of No Original Research. The proposed material violates that policy, just as surely as a statement that the Earth is flat or denial of the Nazi Holocaust. Attempts to insert these kinds of things in any article are routinely reverted, and repeated attempts will result in a block on the perpetrator. --KSmrq 23:37, 16 December 2005 (UTC)

Whoever boldly added it has more common sense than the lot of you combined. He does not need to defend it - it is self-explanatory. KSmrq needs to defend the article. Every one of his so-called 'proofs' has been refuted by me. Again, I ask: "what does infinity working behind the scenes mean"? Melchoir says the phrase "infinitely close to" is meaningless - what non-sense. It's as meaningless as when you talk about "as close as you like" or "as small as you want". This article is about as useful as me stating that there are not 360 degrees in a circle. Of course there are not 360 degrees in a circle - we have defined it this way. However, there is circular measure that can't be disputed (radians). 1/3 is not equal to 0.333... so Ksmrq's first proof fails. The limit of 0.333... is 1/3 but its sum is 0.333... Under Advanced proofs there is really nothiing but a whole lot of handwaving. As for the order proof or Archimedean proof - this too is false for you can find infinitely many numbers between 0.999... and 1 as I have demonstrated clearly. The limit proof is nonsense because the actual sum (0.999...) is not equal to its limit. KSmrq gets horribly confused under generalizations by misunderstanding that this happens in every radix system simply because many numbers cannot be represented finitely in all radix systems. This is a limitation of using radix systems. This article says nothing: it starts off with a vague claim without any definitions. It is very badly written and assumes that the reader thinks along the same lines as Ksmrq - wrong! Unfortunately this will probably also be edited because it's true and Misplaced Pages only publishes inaccurate information. Until just a few years ago, anyone who would have said there were more than 9 planets in the solar system would have been condemned as a fool. Today he would have been correct. Once again, this article is Original research. Show me one reliable source - an encyclopedia or a textbook wrwitten by more than one authority that proves what this article claims. You have not been able to do this because this article is false. — Preceding unsigned comment added by 71.248.136.72 (talk • contribs) 2005 December 17 (UTC)

First of all, I made the statement that "infinitely close to" is meaningless - or at least, very informal. The inserted text isn't exactly wrong, but it's so informal and vague that it makes no verifiable or meaningful assertion. However, describing our assumptions — in particular, describing the definition for the meaning of an infinite decimal in positional notation — is a great idea and I'll do this. Deco 02:19, 17 December 2005 (UTC)

I inserted a bunch of new text. The idea behind everything I wrote is that "it's natural to define the value of an infinite decimal using infinite sums". I think this is the point of contention here. I also tried to make it clear that the point of the proofs is not so much to prove this fact, but to justify the choice of definition; with a different definition, these proofs would fail, and we'd have to throw all sorts of familiar proof techniques out the window. I hope this helps to find some consensus. Deco 03:13, 17 December 2005 (UTC)

I've moved the new text to the end, and restored the previous intro. Here's why.

My first major objection is that it is not suitable for kids. In its first paragraph it uses exponential notation, 10, including a negative and a variable. In the second paragraph, two sentences later, it talks about countable and uncountable sets. The third paragraph introduces infinite summation, with sigma notation, and limits. Such material at the top ignores the guidance found in Misplaced Pages:Manual of Style (mathematics):

Probably the hardest part of writing a mathematical article … is the difficulty of addressing the level of mathematical knowledge on the part of the reader.
A general approach is to start simple, then move toward more abstract and general statements as the article proceeds.
It is a good idea to also have an informal introduction to the topic, without rigor, …

This article will be seen by students in elementary school. The notation and level of mathematics used in the original intro (now restored) try to respect that; this attempt doesn't even come close.

My second objection is that it clashes with the careful definitions and proofs already given in the body. It is fine to give yet another set of definitions and proofs in the advanced section, if that will help some readers. But in the intro, this kind of text gives the sense that we must use infinite series and limits, which is neither true nor helpful. It's especially troubling that it does not even lay the groundwork for a naive reader to understand what these mean. Most readers do not have a background in analysis, and Melchoir dug up some informal surveys of college mathematics students, discussed in an archive, that shows widespread confusion.

My third objection is that, despite the length, the new material does not leave me with a sense of greater confidence and clarity. The language is not simple, the concepts are not simple, and both pile on. One minor, but perhaps telling, example of overly complicated language is the following sentence:

This non-uniqueness of nonterminating representations is often counterintuitive for people more familiar with terminating representations.

We're looking at three negations in a row (non-uniqueness, nonterminating, counterintuitive); that's not reader-friendly. It's especially unfriendly for the elementary school students who are likely to be exactly the ones "more familiar with terminating representations". Compare this to the simple, everyday language of the restored intro.

My fourth objection is to this statement:

Note that the point behind these proofs is not to actually show that some distinct nonterminating decimal sequences represent the same value, but rather to justify our definitions above.

That is not my view, and I find it confusing. I agree that it may be helpful and appropriate to explain why we define reals and the meaning of decimal notation as we do, but that is not what the proofs are about. The proofs take standard definitions and show how they lead to the asserted fact, 0.999… = 1.

I believe the new material needs to be simpler, shorter, and clearer, even in its new location. If the point is really to justify the standard definitions, then state that at the beginning of the new section, and tackle it directly.

And even if the justifications are wonderful, some folks will always object. --KSmrq 07:42, 17 December 2005 (UTC)

Thanks for the in-depth analysis, but it seems like you expect to be attacked on this! I for one am fine with it. Melchoir 07:58, 17 December 2005 (UTC)

Er, by "it" I mean your move. Melchoir 07:59, 17 December 2005 (UTC)

Trying to be courteous and constructive. When someone has gone to the trouble to write in good faith, however well they've succeeded, they would probably appreciate some explanation for a dissent, especially a revert. If there is no objection, unless the new material is substantially revised I'm inclined to proceed to actual removal.

Doing mathematics requires different knowledge and skills than writing about it in an encyclopedia. My analysis may turn out to be overkill, but (before it joins the archives) it may also be instructive. Suppose I say: (1) know your audience, (2) start simple, and (3) be clear and direct. Which is more helpful: the generic advice, or seeing it applied?

Besides, isn't it nice for a change to see a talk page used for its intended purpose? ;-) --KSmrq 08:27, 17 December 2005 (UTC)

It's quite the revolutionary concept, yes. Melchoir 09:28, 17 December 2005 (UTC)

Wheeee!

Of course the whole article is counter-intuitive. We can do almost anything we like in mathematics (as long as it is logical): we can have circles of 240 degrees. We can even define bases where transcendental numbers are no longer transcendental. Not only children are confused but many adults with Phds in mathematics. I have communicated with accomplished Phds in Mathematical Statistics, Applied mathematics and Pure Mathematics. Some of them agree that 0.999... is not equal to 1. Fine you say, and then use an Archimedean corollary to claim that a number can be found between the two if they are not equal. But of course you can find numbers between 0.999.. and 1 - infinitely many of them. The Archimedean property is one that supporters of the argument in favour of 0.999... = 1 like to use to show that 0.999... must be equal to 1. I would hardly call the limit argument a proof at all. It is simply irrelevant. However, it is misunderstood and incorrectly used. For example, any arithmetical operation with recurring decimals is not possible (except as an approximation) just as any exact operation with pi, e, etc is not possible either. To show that 1.999.../2 = 1 or 0.999..., you first have to demonstrate that the quotient is possible. There are problems with any radix system so that not all laws are satisfied and certain anomalies always exist. To make statements such as "infinity behind the scenes" is not only vague and ethereal but it implies a certain offensive arrogance that states the author understands infinity and we all know that the human mind can only speculate about infinity. Any profound results regarding infinity are arrived at through experimentation and speculation involving limits. The sum of 0.999... (since you are treating it as a sum) is 0.999..., not 1, not approximately 1 but certainly less than 1 is all we can say. This is how radix systems were constructed. — Preceding unsigned comment added by 68.238.103.100 (talk • contribs) 13:12, 17 December 2005 (UTC)

That we consider so many things in mathematics makes it all the more important that we stick to our definitions. Melchoir 18:37, 17 December 2005 (UTC)

Concerning the Archimedean property cited above: It is, iirc, not used to show that there are no numbers between 0.999... and 1, but to show that the difference between 0.999... and 1 cannot be greater than 0. A more detailed proof along these lines can be found in Archive 2. A short summary:

The difference is either 0 or an infinitesimal. 
By the Archimedean property, there are no infinitesimals in the set of real numbers. 
Thus, the difference must be 0.

If one intends to claim that 0.999... is not 1, one would have to doubt one of these statements. Due to its shortness here, the first one is most open to criticism, but I refer to the more detailed proof cited above.--Huon 21:36, 17 December 2005 (UTC)

The difference is not zero and you can't show that it is zero. You are making this assumption - erroneously. There is no such thing as an infinitesimal, not in the real numbers and not in any other number system. Wiki's definition of hyperreals, surreals, etc is all a load of crap. Robinsohn's non-standard analysis is unsound. And the Archimedean corollary is used to show that there are numbers between any two real numbers that are not equal. I have shown that there are infinitely many numbers between 0.999... and 1 in one of the archives so these two numbers need not be the same. It's quite incredible that you continue to argue with a concept (infinitesimal) that is neither true nor provable in any fashion because you know that every single one of the arguments that you have used is completely false and you refuse to admit it. — Preceding unsigned comment added by 71.248.129.104 (talk • contribs) 00:16, 18 December 2005 (UTC)

I don't know why I'm still trying, but let me ask you a question:

Is −0 equal to 0? Melchoir 00:42, 18 December 2005 (UTC)

Yes, because -0 means -1*0 = 0. Let me ask you a question: are you a teacher? Male/female? Okay, so you don't have to answer but I am infinitesimally curious... Common melchoir, what's so hard for you to understand? Dedekind cuts show that for every real number, there must be a real number immediately preceding it and succeeding it for otherwise the real number system would have holes. Now you promised to teach me something about this in an earlier archive and since I respect you the most, I expect you to keep your promises. See, I too am learning a lot from you. Really. And I am not trying to trick you or be sarcastic either. 71.248.130.143 13:37, 18 December 2005 (UTC)

I'm male and not a teacher, although the latter might change at some point. Now hold on, you claim that -1*0 = 0. What if I tell you this:

<<<No, obviously -1*0 = -0, which is diffferent from 0. Everyone knows how to multiply with negatives: a negative times a positive is a negative. In fact, -0 < 0. This is just part of a general trend that all negative numbers are less than all positive numbers, no matter what the numbers are. -3 < 4, -8 < 5, and -0 < 0. Clearly -0 is not equal to 0; in fact, -0 is the immediate predecessor of 0.>>>

Well? Melchoir 20:37, 18 December 2005 (UTC)

Yes, you can say: -1*0 = -0 but it is not true that -0 is different from 0 because -0 means -1*0 and we know that -1*0 = 0. We also know that the product of any number with 0 is 0. The expression -0 is different from 0 but when -0 is evaluated, it results in 0. There is no analogy here between 0.999... and 1 whatsoever. Next, to say that -0 is less than 0 is also false since -0 is a product whose result is 0. Finally, to state that -0 is the immediate predecessor of 0 is evidently very faulty. Melchoir, I am sure you can do a lot better than this? Now, what does 0.999... evaluate to? It evaluates to 0.999..., not 1. I can not see any analogy here that might support your argument for 0.999... = 1. Oh, and I am still wanting to know what you were going to say about the holey reals?70.110.93.158 22:47, 18 December 2005 (UTC)

-1 was to the invention of negative numbers as sqrt(-1) is to the invention of complex numbers. Just as the imaginary part of every nonzero complex number is a product of some real number and i, so every nonzero negative real number is a product of -1 and the same real number. 70.110.93.158 23:01, 18 December 2005 (UTC)

<<<You just keep saying -0 equals 0 but you cannot prove it because it isn't true. Who says -0 evaluates to 0? Obviously -0 evaluates to -0.>>> Melchoir 23:07, 18 December 2005 (UTC)

Okay, let me prove it: (i) A negative number is defined as the product of any real number (equal or greater than zero) and -1. (ii) The product of any number by zero is zero. This is well-defined. -0 means -1 * 0. (iii) Since -1 is a negative number and the by (ii) the product of any number by zero is zero, it follows that -1 * 0 = 0. But -1 * 0 = -0, therefore -0 = 0. Melchoir, my logic is flawless and I know very well what you are trying to do. You might as well give up here because there is no analogy between this proof and 0.999... = 1. There is simply no way you can start off with 0.999... and end up with 1 using transitivity of equality. Why? Because 0.999... is not equal to 1. It is less than 1. 158.35.225.231 13:19, 19 December 2005 (UTC)

Zero is the only point in the number line that is neither positive nor negative. It's how negative and positive have meaning: How far is this value from zero? It is |value| distant from zero. Which direction? Aha. -0 has no meaning in the sense of valuation. I'm not sure on the symbolism, but it may have meaning in the sense of limits (approaching zero from the negative). Arguing whether 0 = -0 is fruitless until you establish what both of those symbols mean. Although, I think you'll find any approach to the question (0 = -0?) a monumental waste of time.

Who are you? Melchoir 06:48, 19 December 2005 (UTC)

monumental waste of time -- that's certainly coming from the right person. Anyway, there are some contexts where it's useful to distinguish between -0 and 0, for example in floating-point arithmetic. Fredrik | t c 09:34, 19 December 2005 (UTC)

No, he has a valid point and I have proved it very easily. See above proof. 158.35.225.231 13:32, 19 December 2005 (UTC)

<<Your "proof" is wrong. You say the product of any number by zero is zero, but obviously the product of a negative number by zero is negative zero. So really all you've done is assume that negative zero and positive zero are the same, and then offer a "proof" based on that assumption! C'mon, I thought you could do better.>>>

You say you know very well what I am trying to do? Are you sure? Melchoir 19:35, 19 December 2005 (UTC)

Oh yeah? Read the proof again. It's okay for the product to be negative zero! I am not saying this is incorrect. I am making no assumptions: I showed that -1 * 0 = 0 and then I showed that -1 * 0 = -0 and by transitivity of equality (what this means is that the left hand side which is the same equals the rhs of 0 and -0 respectively, therefore 0 and -0 must be the same) these are equal. No rocket science required here Melchoir. 158.35.225.231 19:40, 19 December 2005 (UTC)

<<<You certainly did not "show" that -1 * 0 = 0, you simply assumed it, hoping I wouldn't notice. You say, "The product of any number by zero is zero". This is wrong. The product of a negative number with zero is negative zero. You are overextending your familiarity with positive numbers.>>> Melchoir 19:43, 19 December 2005 (UTC)

Am sorry. I defined this in my proof. I stated that the product of any number by zero is zero. This is well-defined. You ought to reread the proof. It has been stated correctly. I did not need to prove that -1 * 0 = 0, nor did I have to prove that -0 = -1 * 0. Since both the LHS of each statement is the same, this is implies that 0 must be equal to -0. Transitivity of equality. 158.35.225.231 20:30, 19 December 2005 (UTC)

<<<"I stated that the product of any number by zero is zero. This is well-defined." Yes, it is a well-defined statement, and it is wrong. Your argument hinges on that assumption, and it fails. By all means, if you can come up with a revised proof that doesn't include such false assumptions, please give it.>>> Melchoir 20:39, 19 December 2005 (UTC)

It is not an assumption and it is not incorrect. How do you define the product of any number by zero? 158.35.225.231 20:58, 19 December 2005 (UTC)

<<< I have already told you. The product of a positive number with zero is zero. The product of a negative number with zero is negative zero.>>> Melchoir 21:21, 19 December 2005 (UTC)

Okay, let it be so:

-1 * 0 = -0

-0 = -(0-0) = -0+0

Now since -0 and 0 are additive inverses -0+0 = 0. Therefore -0 = 0. Very easy eh? So just cut to the chase, what are you trying to pull here? 71.248.130.208 23:17, 19 December 2005 (UTC)

<<<Your second line makes no sense. In your text, who says -0 is the additive inverse of 0? In fact, -0 + 0 is a number between 0 and -0 and it cannot be represented by decimals. Don't try to fool me with your advanced math; I have taken a course in abstract algebra, and it was a load of rubbish. If abstract algebra depends on -0 equaling 0 then it is wrong. All this "additive inverse" nonsense is a recent invention of PhD algebraists who don't know what they're doing and can't admit they're wrong. In fact, the original, everyday, correct understanding of negative numbers is that they are negative versions of positive numbers and they are not the same. Would I try to prove to you that -1 = 1? Of course not, that's silly.>>> Melchoir 00:38, 20 December 2005 (UTC)

Immitation is the sincerest form of flattery. Do you love me Melchoir? 71.248.130.208 02:49, 20 December 2005 (UTC)

Is that a capitulation? Melchoir 03:36, 20 December 2005 (UTC)

>>Re: "Who are you? Just an passer-by...I was pointed to this discussion by a friend in another forum. I am merely attempting to point out that the notion of "negative zero" is absurd. The value of zero is neither positive nor negative. There must be such a reference point, or else the concepts of "negative" and "positive" have no distinctive properties. — Preceding unsigned comment added by 131.30.121.23 (talk • contribs) 11:01, 2005 December 20 (UTC)

Moi capitulate? No my dear Melchoir - evidently you have learned nothing about me through all these discussions. If you are trying to immitate me, you ought to do it properly, i.e. when I say a load of rubbish - I back it up with real facts, not myth. You went and erased some posts that you thought were maliciously critisizing you but had you read them correctly, you would have realized that I was attacking the arrogant fool called Kmsrq (Mr. Good Grief!) As for abstract algebra, I also passed this course too - it's was worth far more than basic analysis. I would not call this a load of rubbish but I would call real analysis a load of rubbish. It is full of errors and contradictions. I pity poor students who have to listen to ignorant professors try to explain that 0.999... = 1. Look at how you failed miserably: you started off with Dedekind cuts. You incorrectly assumed that a Dedekind cut represents only one number (you swallowed this willingly as a student without much thought about it - even Hardy (the pompous fart) did not know this). I bet Hardy can talk screeds about the central limit theorem and not impart any understanding to his students. Truth is that he is not way above them. The problem is not with students, it is with fools the likes of Hardy and company. You cannot teach what you don't understand! I look on people like Hardy with utmost contempt. He is a bitter and twisted individual who probably takes pleasure out of making his students seem incompetent. I would place Ksmrq in this category too. Birds of a feather...

Passer-by:: Melchoir knows it's absurd. You need not worry. He is playing a game with me. 158.35.225.231 13:24, 20 December 2005 (UTC)

My point in using an irrational alter-ego was in demonstrating that if you're willing to be sufficiently absurd, you can defend any false statement. Hopefully you now understand the way you sound to everybody else. The Dedekind cuts ARE the real numbers. If you refuse to accept my definitions, I can't help you and I feel no further obligation to try. I hereby declare victory over you and therefore over all IPs who can't be bothered to sign in. That's all. Melchoir 14:45, 20 December 2005 (UTC)

Actually you have no point at all. The only absurdity is what you have been using in your arguments defending the false statements you have been making. You might want to accurately represent what I stated and not maliciously twist it: I did not say Dedekind cuts cannot be used to represent real numbers, what I said is that any Dedekind cut represents more than one number. Why should I accept your definitions? You claim to speak for everybody else - you ought to be more specific: you speak for Misplaced Pages and others who are in agreement with you. You forget that there are those who are not in agreement with you. <> - this statement just goes to show how immature and unreasonable you are. You have in fact capitulated, not me. Finally, Melchoir, what gives you the impression I need your help? Do you think that I know less than you? Wait, of course you do because you are so blatantly disdainful and patronizing. You are so pathetic. 158.35.225.231 15:20, 20 December 2005 (UTC)

Rules of engagement
Dear all,
I assume all of us are well-meaning people, wanting a good result for this article. That makes me ask the following favor to anonymous contributors:
Please make an account. It is not productive for us to deal with a person who always uses a different IP address. It is impossible for us to keep in touch with you this way. Making an account will take you five secods. Just choose an imaginary username which has nothing to do with your real name, and a password. No more. But so much gained.
Please sign your posts. Use four tildas for that, like this: ~~~~. I would really, really ask you to give it a try. You leave a lot of unsigned comments and nobody can tell which is what you wrote, and which is somebody else, and which is another anonymous user.
I truly appreciate you taking your time to read all this, and do us a the small favors I asked which will take you a very small amount of time (not infinitesimal, but close :) and will make it so much more pleasant for us to have constructive discussions. Sincerely, Oleg Alexandrov (talk) 06:04, 18 December 2005 (UTC)
As an aside, a discussion about this very topic has been hotly debated by an on-line society for people with a high IQ. Keep it going SM! — Preceding unsigned comment added by 68.148.229.166 (talk • contribs) 06:30, 2005 December 18 (UTC)
What?! You mean Hardy is discussing this without me?! How can this be? I too have a high IQ (over 140) -not that it means anything. The IQ concept was developed by a human who I believe had to wipe his arse every time he had a BM. Do you think Hardy wipes or washes? I wash only. Toilet paper is for those who have bad hygienne and stinky butts. Although I don't care to0 much for Islam, this is something we could learn from them. Wait, I think the ancient Greeks invented the bidet if I am not mistaken? Oops, I think Melchoir is going to censor this when he wakes up. Hopefuly a few people will get to read it before he does and have a good laugh if nothing else. Not that it's any of my business but I am infinitesimally (singular ONLY) curious, do you wash or wipe Oleg? 71.248.130.143 14:14, 18 December 2005 (UTC)

Infinitesimals
I have not yet given up on the "infinitesimal" proof, especially since anon agrees there are no infinitesimals in the field of real numbers. (Compare the 0:16 post of 18 December 2005. By the way, meaningful concepts of infinitesimals in larger number sets can be found in Winnig Ways for your Mathematical Plays, part 1, which is written on a rather basic level.)
0.999... and 1 are real numbers. Let x = 1-0.999... be their difference. Note that I do not care whether I can give a decimal representation for x. Note also that I do not make claims about the existence of numbers between 0.999... and 1. All I claim is that I can subtract, and that the difference of two real numbers is again a real number. And that is due to the fact that the real numbers are a field. By definition, a number y ≠ 0 is an infinitesimal if every sum |y|+...+|y| of finitely many terms is less than 1, no matter how large the finite number of terms. We agreed such a thing does not exist in the reals. Now form any sum |x|+...+|x| of finitely many terms, say n terms. Obviously, |x| = 1-0.999... < 1/n. Thus, |x|+...+|x|<1. So if x were greater than 0, it would satisfy the definition given above. That can't be, since we agreed that no real number is an infinitesimal. Our only way out is x=0. Thus, 0.999...=1.
If there are problems with this proof, please be precise in denoting them.--Huon 00:00, 20 December 2005 (UTC)
The proof would be flawless if your definition of infinitesimal is true. Only problem is it is not true because it does matter how large the finite number of terms become. If you feel comfortable that the sum of these terms will always be less than 1, how is it that you do not feel the same way about the sum of 9/10+9/100+9/1000+... ? I can make the same statement here, i.e. for finitely many terms, this sum will always be less than 1. So what?! 71.248.130.208 02:54, 20 December 2005 (UTC)
Hmmm, I had sort of given up on this discussion, but one last try: Let x = 1-0.999... as above. Consider the set S = {x, 2x, 3x, 4x, ...} = {nx|n in N}. (If you don't agree with me setting x = 1-0.999... , just consider the set S = { (1-0.999...), 2(1-0.999...), 3(1-0.999...), ... } = { n(1-0.999... ) | n in N} instead ). Could you answer these questions?
Does S have an upper bound?

Does S have a least upper bound?

If S has a least upper bound, what is it? If you can't give an exact answer, can you give an interval (ie. 0.5 < sup S < 1)?

If 1-0.999... != 0, then 1/(1-0.999...) must be a real number. Can you describe the properties of 1/(1-0.999...)? For instance is there any natural number n, so that 1/(1-0.999...) < n ?

Rasmus (talk) 07:23, 20 December 2005 (UTC)
Answers:
1. S does not have an upper bound therefore it cannot have a least upper bound. So questions 2 and 3 are not relevant.
4. 1/(1-0.999...) is a real number. Properties: all we can say is that it is a very large indeterminate number comparable with infinity. There is no natural number n, so that 1/(1-0.999...) < n.
So now you are going to conclude that since 4 is true that 1/(1-0.999...) is not a real number - yes? What about 1/(3.15-pi)? Is there a natural number n so that 1/(3.15-pi) < n ? 158.35.225.231 13:09, 20 December 2005 (UTC)
All members of S are of the form nx. Consider one such member, nx. As we saw a month ago, we can choose a natural number $m>\log _{10}(n)$ , and use the fact that $0.999...>\sum _{i=1}^{m}{\frac {9}{10^{i}}}$ to show that $x=1-0.999...<1-\sum _{i=1}^{m}{\frac {9}{10^{i}}}={\frac {1}{10^{m}}}<{\frac {1}{10^{\log _{10}(n)}}}={\frac {1}{n}}$ , and thus $nx<n{\frac {1}{n}}=1$ . So all members of S are less than 1, yet you claim it has no upper bound?

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

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

Rasmus (talk) 15:28, 20 December 2005 (UTC)
You have just proved that 0.999... < 1: nx < n*1/n = 1 => x < 1. If x is a real number greater than 0, there exists a natural n such that 0 < 1/n < x or nx > 1. So if x = 0 (which is what you would require for having 0.999... = 1) then no n exists such that nx > 1. Hence x must be greater than 0 and if x is greater than 0, then 0.999... must be less than 1. Now do the right thing and delete this garbage article. ] 17:35, 20 December 2005 (UTC)
Could you clarify that argument please? In the above I showed how for all natural numbers n, nx < 1. If you accept that "If x is a real number greater than 0, there exists a natural n such that 0 < 1/n < x or nx > 1", the conclusion must be that x is not a real number greater than 0.

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

Rasmus (talk) 22:28, 20 December 2005 (UTC)
You showed nx < 1. The Archimedean property says there exists an n s.t. nx > 1. There is no x that satisfies nx < 1 and nx > 1. So how do you reach the conclusion that x = 0? You are looking only at nx < 1 and thus drawing the conclusion that x must be zero? Okay, let me try to understand what you are saying:
The Archimedean property shows the relationship between a natural number n and a number x greater than 0 such that nx > 1. This means that x and n must be greater than zero. Your proof demonstrates that a number x and some natural number n have the property that nx < 1. The only n that satifies this is n=0 for otherwise x must be zero. How do you associate the Archimedean property with your proof? They both state different facts. So what I am trying to say is this: if you are to draw any conclusion that is backed by the Archimedean property, then your proof must result in a form that resembles it, i.e. nx > 1 and not nx < 1. You cannot arrive at the conclusion that nx < 1 and then state by the Archimedean property that x is not a real number greater than 0. By demonstrating that x (1-0.999...) < 1/n for any n, you have proved conclusively that x is greater than 0 because for whatever 1/n you give me, I can always find an x that is smaller. This x is greater than zero and sounds very real to me. 00:26, 21 December 2005 (UTC)
What I showed before were that for x=1-0.999... and all natural numbers n: nx<1. (I actually only wanted to use it for showing that S had an upper bound, since you had earlier rejected the application of the Archimedean property). Since the Archimedean property state that for all real x>0, there exists a natural number n, so that nx>1, we have a contradiction unless x is not a real number greater than 0.

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

Rasmus (talk) 07:32, 21 December 2005 (UTC)
Fine. I see your argument now. It's always been confusing because for any 1/n, I can always find an x that is smaller but not zero. In an earlier discussion, you maintained that the induction proof was incorrect because it does not show P(infinity). Do you realize that one can say the same to you regarding this argument? You may say that the lowest x one can find is zero but then you are assuming P(infinity) is true. So although your argument is valid, you have not shown P(infinity). It seems to me that one can show equally well by induction that 0.999... < 1 and using your method that 0.999... = 1. How can 0.999... be both less than and equal to 1? This is strange... 158.35.225.229 13:14, 21 December 2005 (UTC)
Well, the difference is that I don't need to go to the limit. To use the Archimedean property, I only need to show that for all finite natural numbers n: nx<1. I do not need to show that " $\infty x<1$ " (whatever meaning one would assign to that statement). Rasmus (talk) 14:07, 21 December 2005 (UTC)
One can say exactly the same for the induction proof, i.e. only need to show that 0.9999xn < 1. Same thing. Let me get one thing straight: you are also saying that if the Archimedean property does not apply, then x cannot be a real number, right? If your answer is 'yes', then the Archimedean property only applies to finitely represented reals in any radix system. 0.999... is not finitely respresented. 158.35.225.229 14:20, 21 December 2005 (UTC)
Rasmus is going to say that
the Archimedean property is a property for the entire set of numbers, and it applies.

x is indeed a real number, but not one that is greater than 0.

After all, what we are trying to show is just x=0. If you are now willing to sacrifice the Archimedean property for your brand of "real" numbers, you will probably agree that yours are not what mathematicians usually call the real numbers.
Concerning the definition of infinitesimals I gave above: That was the Misplaced Pages definition; I just copied it. If you don't believe that definition to be correct, look it up in, say, Winnig Ways.

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