Talk:0.999.../Archive 8

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DO NOT EDIT OR POST REPLIES TO THIS PAGE. THIS PAGE IS AN ARCHIVE.

This archive page covers approximately the dates between 2006-02-23 and 2006-07-10.

Post replies to the main talk page, copying or summarizing the section you are replying to if necessary.

Please add new archivals to Talk:Proof that 0.999... equals 1/Archive09. (See Wikipedia:How to archive a talk page.)

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The real conclusion

I think this is a good time to close this discussion. Anon, I don't know what your mathematical background is, but it seems like you have a difficulty grasping several mathematical ideas, and in that case, this talk page is not the right place to advance your mathematical education. I'll just add a few final notes: Math is "what you define is what you get", so a sentence like "This (even if it were defined this way) is untrue" is an oximoron - if it is defined this way, then it is true. You haven't given one clear argument against such a definition, nor an alternative definition - Just vague, non-mathematical references to some mysterious comparisons between numbers. It can be shown that two numbers are equal iff they have the same decimal representations, the "s" is because some numbers have 2 decimal representations. You also fail to present a fundumental flaw in the axioms of the reals, further illustrating the fact that your argument is perhaps a philosophical, rather than a mathematical one.

If you wish to add a closing comment that would be fine, but I have no inclination to continue this discussion. If anyone else does wish to keep debating with you, I wish you all good luck. -- Meni Rosenfeld (talk) 19:54, 23 February 2006 (UTC)

If I may, I'm going to borrow a tune from Talk:Jyllands-Posten Muhammad cartoons controversy and segregate mathematical arguments into a subpage of this talk page. I'm sorry, ConMan, if this steps on your own idea, but I just learned about this mechanism, and I think it'll work. Melchoir 21:41, 23 February 2006 (UTC)

You're right, we got carried away. But I feel this is going to be a long-term debate, and we really should rewrite the article in a way that makes the point crystal clear - The way it stands now is not really that clear. We could try to refactor all the arguments and counter-arguments from the archives and present them in a coherent way in the article. Seems more cost-effective than making silly arguments with anyone who wishes to challenge fundumental mathematical notions. -- Meni Rosenfeld (talk) 07:07, 24 February 2006 (UTC)

I didn't mean to rebuke you, Meni! We've all been there. As for the article itself, my views haven't changed much since Talk:Proof_that_0.999..._equals_1/Archive02#If_I_may_speak_to_the_article_itself...:

• The article should ultimately expand in scope to fully address common misconceptions.
• Such an expansion must not be original research, but it should draw on published sources to determine which misconceptions are common and which are unique to the IPs on this talk page.

With the cynicism of experience, I must also add:

• Although a more complete article might be more helpful to some people, some of the IP addresses will never accept this or any other Wikipedia article that challenges them, or indeed any source at all.
• There are more important projects on Wikipedia, including other math articles needing expansion.

Melchoir 07:41, 24 February 2006 (UTC)

No offense taken. I agree, and will try to read the reference you gave carefully, and then see what we can do. I think changing the article bit by bit would be best. But perhaps refactoring all the archives first would be desirable... We'll see. -- Meni Rosenfeld (talk) 08:00, 24 February 2006 (UTC)

"Refactoring" the archives? No, you should scrap the article. But wait, no one takes Wikipedia seriously, so go ahead - refactor the archives whatever that means! 70.110.84.97 13:24, 25 February 2006 (UTC)

Though it is to some degree irrelavent to this conversation, I merely wish to mention, that in fact all numbers have at least 2 decimal representation, as opposed to "some" as said by Meni. If one replaces the last digit x with x-1, and then concetanate an infinite string of nines to the end, the new sequence will always represent the same number as the original sequence. He Who Is 18:45, 16 June 2006 (UTC)

How about 0? Or π? Rasmus (talk) 20:24, 16 June 2006 (UTC)

Any rantional number with a finite decimal representation other than zero. He Who Is 21:12, 16 June 2006 (UTC)

Where we go wrong

It might be interesting if this article included a section going into more detail on the logical errors that lead people to erroneously conclude that point nine repeating must be less than one. This "intuitive" conclusion is so powerful that many people, even when convinced point nine repeating does indeed equal 1, point to it as a failure of the mathematical system. --Trystan 00:53, 11 April 2006 (UTC)

This is a temptation that attends many topics. Experience shows that it is an endless road, a quixotic quest. Pædagogically and psychologically it confuses, it fails to convince, and it invites dissent. ("No, no; you misunderstood my objection!") A proof is a proof; it ends debate. The article includes four completely different arguments, which ought to be enough for anyone who seriously wants to understand. Both elementary arguments use language and examples accessible to a schoolchild. Both advanced arguments carefully sidestep many of the pitfalls observed in reading endless online "debates". (Typical confusion centers around limits, which the order proof uses not at all, and which the limit proof uses with disarming transparency.) Finally, the cranks are always with us; their minds are already fixed, so nothing we do will satisfy them. --KSmrq^T 02:36, 11 April 2006 (UTC)

You just presented some of the common pitfalls right there in your reply. Might it not be a worthwhile section? While I agree there is no need to try to put refutations or psuedo-counter-proofs here, presumably this article would not even exist were not for the strong tendency towards disbelief based on a few misunderstood concepts (i.e. limits and infinitesimals). --The Yar 21:33, 19 June 2006 (UTC)

I've always been confused by the fact that when it was shown the axiom of choice meant a sphere could be made into two spheres of equal size, people assumed the axiom of choice must be right and simply lead to strange conclusions; but something as simple and trival as to numbers (That otherwise have an infinitessimal difference anyway) are the same, everybody's outreages the world. -- He Who Is^{[ Talk ]} 17:48, 23 June 2006 (UTC)

Everyone is a different set in the two cases. How many people who have heard of Banach-Tarski object to this? Septentrionalis 23:17, 27 June 2006 (UTC)

Regardless, were it not for the legions of loyal disbelievers, all of whom share the same basic, intuitive misconceptions, this article would never have been created. The article seems to be lacking some acknowledgement of that. Not for any POV reason, but in the interest of explanation. --The Yar 20:12, 3 July 2006 (UTC)

I also think it would be a good idea to incorporate some sort of common pitfalls into the article. As I see it, there is at least one serious problem in trying to do this:

Not being "preachy". As User:Pmanderson has recently pointed out in the article, there are other axiomatizations of the "real" numbers which emphasize alternative properties. Who is to say that we are "right" about this particular definition? The answer, invariably, is going to be something like: "We're right because we defined things in this way." Such answers are fine for mathematicians, but to non-mathematicians they must sound arrogant or at least question-begging.

One route around this particular difficulty is to be relative in our description of the pitfalls. So, rather than say that these pitfalls are conceptual errors, merely to point out how one stipulated property of the real numbers is inconsistent with another stipulated property. (See the Socratic dialog below: no conclusion is actually reached, but an inconsistency is at least pointed out.) In the span of history, people have employed real numbers since the dawn of mathematics, however partial their understanding of the logical underpinnings. On this time scale, it is not until quite recently that the number system was sufficiently codified to establish beyond all doubt that 0.999... = 1. Nevertheless, there is evidence that even the ancient Babylonians were aware of problems such as these -- cf. with Fowler, Dedekind's Proof..., AMM (1985?).

From this point of view, understanding the ostensive pitfalls is a large step to understanding why the real numbers have been defined in the way they are. Silly rabbit 20:49, 3 July 2006 (UTC)

Capitalization of "Doppelgänger"

This is a trivial point to the article, but I will address it here anyway. The word doppelgänger should generally not be capitalized, with obvious exceptions such as when it is used as the first word in a sentence, or as the title of an article or book, etc. It should not be capitalized in the context in which it is currently used in this article. Please see the rules of capitalization at dictionary.com and grammarbook.com. It is true that doppelgänger is not native to English; however, this does not imply that it should always be capitalized. Please see the list of foreign words and phrases at infoplease.com. Most of these generally should not be capitalized. The only notable exception is when a word is derived from a proper noun. By examining the etymology of doppelgänger, it can be seen that this is not the case. Rishodi 20:01, 3 May 2006 (UTC)

The article on doppelgänger is not a helpful guide; it uses both capitalizations seemingly at random! I will defer to Wikipedia convention on this, whatever that happens to be. But if lower-case is used, write the link as I have in this note. --KSmrq^T 20:57, 3 May 2006 (UTC)

Yes, I looked at the Wikipedia article on doppelgänger and it improperly capitalizes the word in many places. I'm going to change the reference in this article back to lowercase, and at some point in the future I may have to go through the doppelgänger article and correct the instances of the word that are incorrectly capitalizated. Rishodi 21:58, 3 May 2006 (UTC)

Regardless of what the article does, in general (aside from the cases of beginning a sentence, etc.) doppelgänger is left uncapitalized in English, while in German it is always capitalized. -- He Who Is^{[ Talk ]} 02:06, 23 June 2006 (UTC)

Does this matter?

This is good material for kids like myself to prove to other kids that they're smarter in the endless debate of who's smarter. Other than that, I don't see much use to this. Not very sure if I got dumber or smarter after reading this but it was a good article with a correct theory. Case Closed! —The preceding unsigned comment was added by 70.127.34.98 (talk • contribs) 07:39, 11 May 2006 (UTC)

Actually, I think it's more important than just showing off your smarts - it's showing that some things that are mathematically provable that at first seem counterintuitive: 0.999... initially looks like it's strictly less than 1, but the article proves that they're equal. It just goes to show that you can't just look at a result and say "that's right", you have to be able to prove or disprove it before you can make any use of it. Confusing Manifestation 11:43, 11 May 2006 (UTC)

And conversely, many facts that mathematicians take for granted have been disproved over the centuries, although at a glance they seem to be easily verifiable. For instance, the product of the squared roots of to numbers equals the squared root of the product of those numbers. This has been proven to be true only over the principal branch of the real axis, and otherwise led to the assumption that any number is its own opposite. The axiom of choice is another example. -- He Who Is^{[ Talk ]} 17:44, 23 June 2006 (UTC)

The axiom of choice was disproved?? Any example of a fact "mathematicians take for granted" over the last century that has since been disproven? (Cj67 05:07, 25 June 2006 (UTC))

From the article on AC: "However, there are schools of mathematical thought, primarily within set theory, that either reject the axiom of choice or investigate consequences of axioms inconsistent with AC." While it has never been conclusively proven or disproven, in many mathematical disciplines $\neg AC$ is taken to be true. -- He Who Is^{[ Talk ]} 22:01, 27 June 2006 (UTC)

And there's a proof that the axiom of choice cannot be proven (or disproven), which most of these schools accept. Septentrionalis 23:19, 27 June 2006 (UTC)

A far cry from "ha[s] been disproved." In fact, it has been proven, as mentioned above, that the axiom cannot be disproven (within ZF). Still waiting for examples of disproved things. You are simply wrong, and you should admit it. I think you should also be more reluctant to consider yourself to be an "advanced mathematician" (although maybe you are joking about that). (Cj67 18:10, 28 June 2006 (UTC))

Yes, it was in fact a joke. But as for the examples you wanted: In my oiginal post, I did provide one when I mentioned the square root of a product. It was also once thought that any polynomial equation with algebraic coefficients would have algebraic roots and thus could be solved via some algorithm. This is now known to be untrue, thanks to Galois groups. It was also once beleived by many mathematicians that all real numbers could be represented as a fraction of integers and were thus rational. This is now known to be untrue as well. Moreover, similar beleifs were once held pretaining to algebraic numbers. In fact, it might even be a good idea to include some of this in the article in order to clarify the significance of this. -- He Who Is^{[ Talk ]} 18:53, 28 June 2006 (UTC)

Huh? The roots of a polynomial over the algebraic numbers are algebraic. Septentrionalis 01:24, 29 June 2006 (UTC)

I had restricted to within the last century, because of course there is a difference between now and the pre-rigor days. Over-and-out. (Cj67 00:13, 29 June 2006 (UTC))

Ah. I hadn't noticed. But in my earlier reply, I said over 'the centuries,' so I daresay this doesn't actually make my point any less true, just not as important in the present context. But I do wish do know, do you think the results presented in the article really are of any consequence, and just disagreed with my assertions, or did you disagree with the point that I used them to reinforce as well? Septentrionalis, see quintic. The roots of algebraic polynomialss degrees 0-4 are all all agebraic, but fo instance, the roots of the quintic polynomial x⁴ − x + 1 = 0 are not. For this reason, it has been proven that no {Quintic Formula" analogous to the quadratic, cubic, and quartic formulas can exist, nor can exist for degree higher than five. That is not to say no quintic polynomials have algebraic roots, though. It depends on a given polynomials Galois group. -- He Who Is^{[ Talk ]} 02:51, 30 June 2006 (UTC)

You seem to be using a different definition of algebraic. An algebraic number is defined to be any root of a polynomial with integer coefficients, so the roots of x⁴ − x + 1 = 0 are per definition algebraic. I think you mean "solvable by radicals". -- Jitse Niesen (talk) 03:13, 30 June 2006 (UTC)

Oh, I'd almost forgotten about this section! So, yeah, I've started an Applications section with one example: ternary representations of points in the Cantor set. Any other ideas? Melchoir 02:55, 30 June 2006 (UTC)

I'm not sure if this would really fit, but it certainly disproves the basis representation theorem in number theory, since it shows that more than one unique set of digits in the same radix can represent the same real number. -- He Who Is^{[ Talk ]} 03:07, 30 June 2006 (UTC)

Can you find a place to work in a mention of Zeno's paradox somewhere? It's not exactly a mathematical application, but is certainly in the same spirit. Silly rabbit 04:00, 30 June 2006 (UTC)

Jitse, I didn't notice your reply and I apologize. You're precisely correct. So inherently what I meant to say was that all polynomials with rational root are solvable by radicals. -- He Who Is^{[ Talk ]} 03:24, 30 June 2006 (UTC)

Is that true? I am not aware of a result which says that if the roots are rational, then it is solvable by radical, though the roots can be found by exhaustion using Viète's formulas. -lethe ^{talk +} 03:45, 30 June 2006 (UTC)

I think you misunderstood. I meant the opposite. At once it was believed that rational polynomials are solvable by radicals, but Galois disproved it. Although I must admit, as Cj67 said, this was centuries ago, before rigor in mathematics. -- He Who Is^{[ Talk ]} 13:51, 30 June 2006 (UTC)

If you meant to say "not all rational polynomials are solvable by radical", then you are of course correct. But that's not what you said, even on your second try. -lethe ^{talk +} 14:19, 30 June 2006 (UTC)

Bad idea

The Anome unilaterally inserted a new section purporting to discuss objections, despite a lack of consensus to do so. This is a Very Bad Idea, and will not stand. We have enough problems fighting off the "free thinkers" on the talk page, and spun off a separate /Arguments page to deal with the noise. The last thing we need to do is invite opposition on the article page itself. It will not convince the opponents. It will not make a stronger article. If you say "Don't think about a pink elephant", guess what everyone will think about. The right thing to do is what the article does now: present clear, compelling, and correct proofs. Leave readers to compare their mistaken impressions to the truth and sort out for themselves where they went wrong. Or they can bring it up on the arguments talk page.

Again, and again, and again, and again, and again (, and …) on the talk page we have seen attempts to second-guess the mistakes in a reader's world-view; the success-rate of that approach has measure zero. Other proof pages don't stoop to this level of nonsense, and neither should this one. Resist the temptation. State the facts and move on.

What I might support is the addition at the end of the article of a culture section, discussing how commonly this topic arises in various forums, and with what passion. But it would need to be handled delicately, so as not to start arguments. This would be an appropriate place to cite the fellow who looked into misconceptions and their roots in teaching.

We had a recent poorly-conceived vote for deletion, and there was no massive outcry saying, "What a great article, except that it should discuss objections." There is no consensus to include objections. The provocation was a bad idea, the vote was a bad idea, and this new section is a bad idea. Enough already! --KSmrq^T 09:48, 16 May 2006 (UTC)

I don't agree at all. I thought the The Anome's additions were pretty reasonable. I really don't understand why you are so dead set against acknowledging the difficulty people tend to have with the concept in the article. You say it will not convince opponents, fair enough, I don't suppose it will, but ignoring it will not convice opponents either. I think it will make for a better article however as it will address an issue that is of interest to a great many who view the article. The pink elephant is obvious, it's not going to go away just because we leave it out of the article. You are the one asking people to ignore it by refusing to include it in the first place.

The proofs are very good, but please understand that most people who visit this article will not be able to follow the advanced proofs and the elementary ones are not watertight. If this were a textbook I'd absolutely agree with your argument but as this is an encyclopedia article we should write for a broader audience. I think your criteria as to consensus may be somewhat stringent, as far as I can tell there is no consensus to exclude misconceptions from the article either.

Your compromise of including a "culture" section sounds like it may be workable to me. If you don't object I'll reinstate The Anome's edits under such a heading shortly. --Dv82matt 10:40, 16 May 2006 (UTC)

I do object to The Anome's content, as it is not about culture and it does invite trouble. Culture statements would be things like "This topic has generated over one thousand posts in the sci.math newsgroup, and is listed in its FAQ." The academic discipline of psychology long ago decided it would describe only observable behavior, not make up stories about what goes on in people's heads. Yet some editors here want to say, "We know what you're probably thinking, and here's what's wrong with it." That's one problem; here's another. Some readers may think that just because they can understand the statement of a theorem, that makes them qualified to understand the proof. Sorry, the proof of Fermat's last theorem should be a sufficient counterexample. (The statement of the theorem involves only elementary concepts: multiplication and addition of integers!) It's too bad our advanced proofs require advanced mathematics, but that's something we all must learn to accept. I feel great sympathy for those who wish it were otherwise; I have a few topics of my own I wish had more accessible proofs! But the fact is, we don't know what's in people's heads, and there is no royal road to mathematics. It does not serve the article, nor Wikipedia's readers, to pretend otherwise. --KSmrq^T 14:21, 16 May 2006 (UTC)

Aaargh. The whole point of adding the "bogus disproofs" section is to make clear why a significant minority of people find this difficult to accept, and why they're wrong. The fact that so many people find this hard to grasp, and that understanding it properly requires some mathematical sophistication, is also a good reason why this topic is important to have in an encyclopedia. The elementary proofs give the correct mathematical intuition, but rigorous proofs are required for proper mathematics: the existence of elementary, but wrong, arguments is one of the major motivations for rigor.

The wrong arguments criticised here are the classic mistaken arguments made by 99% of people who get this wrong. Ignoring wrong arguments will not make them go away; acknowledging them, and dealing with them briefly, will. Where I do agree with you is that lengthy argument on this should be kept to a separate page: there is no need to give crank arguments undue prominence.

Just to clarify further: on re-reading, I think what may have annoyed you may have been my accidentally ambiguously-worded sentence:

"Opponents of the concept that 0.999... = 1 often state that they can disprove it, with arguments that are just as valid as the elementary proofs presented above."

I think you read this as

"Opponents of the concept that 0.999... = 1 often state that (they can disprove it), with arguments that are just as valid as the elementary proofs presented above."

and thus read this as my stating that the wrong arguments are just as valid, which was not my intention: on the contrary, the meaning I intended was:

"Opponents of the concept that 0.999... = 1 often state that (they can disprove it, with arguments that are just as valid as the elementary proofs presented above.)"

I've reworded this completely, to remove any ambiguity. -- The Anome 13:29, 16 May 2006 (UTC)

Please do not keep adding objectionable material. The "revisions" mollify none of my objections. Now you're trying to say you think you know "what may have annoyed" me, assuming both that you know I'm annoyed and that you know why, which is presumptuous, preposterous, in opposition to what I've explicitly said, and flat wrong. Nor do you know what's in reader's heads. --KSmrq^T 14:26, 16 May 2006 (UTC)

In the Comment discusion above, the near consensus was that such a section was warranted in the article, with Dv82matt, Melchoir, Septentrionalis, and myself supporting the idea. I'm adding The Anome's edits back in.--Trystan 17:14, 16 May 2006 (UTC)

I agree with keeping the content there, because otherwise you may ask "If 0.9... = 1 so obviously, why go to so much effort to prove it?" or "If you can prove the equality even with elementary mathematics, even if it isn't as rigorous as it could be, why do you need to find even more complicated proofs?" The answer to those questions is "because at first glance the result is counterintuitive, and even after seeing the elementary proofs not everyone is convinced it's true". The best way to explain that is to show why some people may feel it's wrong, and then show specifically why their understanding is mistaken. Confusing Manifestation 04:34, 17 May 2006 (UTC)

I find it offensive that material is being added to the article despite an ongoing discussion about what (if anything) should be added and where. Please stop.

We do not insert content like this in any of our other proofs, and I don't think we should do so here. I have indicated that I might be amenable to cultural comments at the end of the article. I will not accept this non-mathematical speculative crap inserted at the beginning or in the middle of the proofs. This is a mathematical proof, not a philosophical debate. And this is an encyclopedia, concerned with the clear and accurate presentation of facts. That does not include guessing what readers are thinking.

For the record, I think the cultural phenomenon is mildly interesting. I also think it has multiple facets. One facet is the people who are intelligent, curious, and confused. They don't understand the fact, but are open to persuasion. Another facet is the people who "know" the theorem is false, and will not be persuaded. A third facet is the semi-knowledgeable explainers who feel compelled to sort everybody out, despite not really fully understanding the issues themselves. A fourth facet is the very rare individuals who know enough to see that 99% of the "proofs", as offered by the previous parties, are unreliable. A fifth facet is the even more rare individuals who know enough, and can explain well enough, and are willing to get involved.

I have watched endless idiotic "debates" on this page between people who were not interested in being persuaded and a recurring series of "helpers" who didn't really understand what they were explaining nor the futility of trying to do so. Now I see people trying to "help" the article. Shall we take a poll? How many were familiar with a Dedekind cut before seeing the term in this article? How many understand it, and the proof in which it appears, now? How many have experience with non-standard reals?

The recent vote demonstrated clearly a broad consensus that the article is not broken. Stop trying to "fix" it. These misguided attempts make the article worse, not better. A cultural section at the end might be of interest. Mindreading and debating with the crazies is not. --KSmrq^T 11:59, 17 May 2006 (UTC)

There is indeed an ongoing discussion. You can read it above. At the moment, most participants in it disagree with you. Would you also recommend that we took all discussion other than a strict proof out of the squaring the circle article, an article about a topic even more loved by crazies than this one? -- The Anome 12:18, 17 May 2006 (UTC)

And, by the way, although I keep forgetting to point it out, the intro of the article already includes the following paragraph:

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

Sound familiar? --KSmrq^T 12:13, 17 May 2006 (UTC)

Then let's work together to improve the intro, rather than deleting all mention of the topics discussed above. Simply stating that "I am right, and all the rest of you are wrong" does not constitute consensus support. -- The Anome 12:28, 17 May 2006 (UTC)

Stop adding material to the article, The Anome (and cohorts). I have written a great deal here, but you have not responding to a single one of my points in a substantive way. And the "squaring the circle" article supports my view. It was famous within mathematics for the question of whether it was possible, which is mentioned. And at the end there is a brief cultural discussion. But nowhere does it try to raise and counter all the false "proofs". --KSmrq^T 14:21, 17 May 2006 (UTC)

Anome, I think I understand what you're trying to do, but I'm afraid you are not doing it properly. If you want to explain why people find it counter-intuitive, that's fine, but going down to mathematical reasoning for that is absurd. I'm not sure such section is needed, and if so, it should be properly explained. BTW, anyone can understand the explanation that three times one-third is one, no need to be a rocket scientist. Mariano(t/c) 13:07, 17 May 2006 (UTC)

KSmrq, you've made your objections very clear, and they are certainly compelling. However, in the above discussions, nearly everyone who has commented on having a section outlining the reason this result is counterintuitive has been in favour. The fact that you will never be convinced does not constitute ongiong discussion. The sections The Anome added are new, and naturally can be improved in various ways. I certainly would support moving them to the end, since I don't really care for interrupting the proofs. It is very difficult for users to improve a section that is constantly being reverted.--Trystan 14:30, 17 May 2006 (UTC)

To more substantively address your concerns, it doesn't require mind reading to outline the most commonly repeated objections. Nor is anyone but you suggesting that such a section would cover all counter arguments. The two main flawed objections I have seen are that two different decimal expansions can not be equal, which is simply begging the question, and the line of reasoning that states that since 0.9 < 1, 0.99 < 1, and so on, then 0.999... must be less than 1. As you stated, we already had a single sentence on this second point. Essentially all the new section is trying to achieve is to explain more fully why that line of reasoning is invalid.--Trystan 17:11, 17 May 2006 (UTC)

I will never be convinced by an argument that has never been stated. And if my objections are so compelling, why do you ignore them? The sections The Anome added should not be there. Don't stick a needle in my eye and then ask "How can we make you more comfortable?" Since no one else has been willing to discuss or write a culture section, I have taken it upon myself. Perhaps this makes clear what I've been saying. As for the "two main flawed objections", the introduction already addresses both. The second you now acknowledge; the other is the sentence immediately before:

• "It should be no surprise that a notation allows a single number to be written in different ways. For example, ¹⁄₂ = ³⁄₆."

Pardon me if this sounds insulting, but I really get the impression that people are commenting without having studied the article nor the history of objections and ill-conceived revisions. There is a great deal of very carefully worded content packed in here, yet new editors plow in like a bull in a china shop. I suppose that's typical of Wikipedia, but it's especially unhelpful with an article like this one. Folks repeatedly demonstrate that they don't know what the article says, they don't understand the subtleties of the mathematics, and they don't understand the psychology of their audience; but they're determined to "improve" the article anyway. Oh, joy.

But let me take a moment to truly enjoy the use of the talk page for purposes of improving the article. It took a long time to get to this point, and I'd like to hope it persists. --KSmrq^T 20:07, 17 May 2006 (UTC)

I suggested earlier, in the Comment section, that the sentence, "The 9s case does surprise...," should be the basis of the "Objections" section. It's existence in the article isn't new to me, I just think it warrants expanding upon to illustrate the point better. Also I think it would help to explicitly state that assuming real numbers have unique decimal expansions is begging the question.

As for why people don't respond to your arguments, in the Comment section, which led to Anome's edits, your entire contribution consisted of referring to your earlier post in the Where we go wrong section as an example of consensus against the idea. It also probably doesn't help to repeatedly question the education or reading comprehension of anyone who disagrees with you and refer to the edits of other users as "crap" and needles in your eye.--Trystan 20:44, 17 May 2006 (UTC)

I share most of the concerns expressed by KSmrq. Paul August ☎ 20:53, 17 May 2006 (UTC)

KSmrq, your arguments are very clear but with one notable exception they are not compelling in the least. You say, "The recent vote demonstrated clearly a broad consensus that the article is not broken. Stop trying to "fix" it." This is not an arguement it’s rhetorical garbage. I'm surprised that someone of your obvious intelligence would advance it.

Elswhere you say, "Yet some editors here want to say, "We know what you're probably thinking, and here's what's wrong with it." What! Surely you realize that the objections are often communicated vocally or in written form and it does not require mind reading to discern them.

Then you say, "That's one problem; here's another. Some readers may think that just because they can understand the statement of a theorem, that makes them qualified to understand the proof." In what way does this relate to including a section on misconceptions?

Elswere you say, “Other proof pages don't stoop to this level of nonsense, and neither should this one.” First of all it isn’t nonsense. Many people really do have trouble with the concept. But despite your clear bias there is a grain of truth to this. Other pages dealing with mathematical proofs (at least the ones I’ve looked at) don’t delve into common misconceptions in any detail. So the question is whether this article warrants being an exception or not. I think it probably does as the misconceptions are the main reason this concept generates so much interest.

On a final note, I find it offensive that well intentioned edits are being reverted wholesale in the article despite an ongoing discussion. Please stop. --Dv82matt 23:14, 17 May 2006 (UTC)

While I agree that it would be a bad idea to present counter-arguments, certainly this page could stand to present a section covering why an article of this nature needed to exist at all when there is no article on why 1/2 = 0.5. While the proof is clear, the public phenomenon of this debate is noteworthy in its own right, and might deserve a little attention here (as opposed to yet another article just for that). A brief note on infinitesimals vs. Archimedes, limits vs. equality, and process over time vs. a static value, might explain to a casual visitor exactly why this article is even necessary, without the need for any troublesome statements like, "Opponents of 0.999... = 1 claim that..." --The Yar 21:59, 19 June 2006 (UTC)

Concerning misconceptions

Thank you everyone who finally decided to use the talk page rather than the article to examine our perceptions, intentions, and actions. My comments here are a continuation of the dialog in the earlier section, but I'm starting a new section just for convenience because the other one was getting long.

• Item one: In the history of Wikipedia, certain articles have drawn extraordinary attention among editors, and one of the ways Wikipedians have evolved for coping is to use the talk page, rather than shoot first and ask questions later. In part it is courtesy, and in part it is merely a practical grasp at stability. This is one of the milder versions, but still warrants that protocol.
• Item two: In some discussions we find participants who are only interested in disruption or pushing a fixed view without listening. That has happened here in the past, but I am honestly warmed to say that I have seen only good-faith efforts in the current discussion.
• Item three: I am not emotionless, and I don't expect others to be, but I hope we can set that aside as we try to agree on what the best article should be. I would like to base both the discussion and the article on facts and reason.
• Item four: We all bring different knowledge, skills, and experience to the party. There is no shame in knowing less, and no pride in knowing more; those feelings only get in the way of honestly working with our strengths and weaknesses to get the job done.

So. What does the article say or not say (if anything) that cries for improvement, and why? --KSmrq^T 11:06, 18 May 2006 (UTC)

I would like to support everything that KSmrq has written above. As for the present state of the article, it seems fine to me. Perhaps the article, Mathematical Beliefs and Conceptual Understanding of the Limit of a Function by J.E. Szydlik, mentioned by Trystan above might be added as a supporting reference for the new "The proof in popular culture" section? — Paul August ☎ 16:24, 18 May 2006 (UTC)

That article, unfortunately, will not be available to most readers. However, I was able to track down a more recent related document without that limitation at the MAA web site, and have added it to the External links. It is not as closely targeted in its content, but perhaps will serve as a surrogate for now.

For those who do not know, teaching undergraduate calculus is a major service industry in mathematics departments, and for those who view it as an opportunity rather than a just a burden, there is some interest in how to be more effective. Limits are a known trouble spot.

Mathematicians may not realize the extent to which they speak a different language, where words and expressions are used in ways not understood by others. Overcoming this barrier is a constant challenge for Wikipedia mathematics writers.

I found a nice web site by Charles Wells with a page on real numbers, but also pages on proofs. This site or some of its pages seem like helpful links, yes? --KSmrq^T 19:02, 18 May 2006 (UTC)

As the discussion in the Comment section demonstrates, the faulty intuitive "gut reaction" is what makes this proof interesting to most readers, and if it warrants formal study, I think it warrants further discussion in the article. At a minimum, the specific justifications described in both Tall and Szydlik's works as common explanations for the initial rejection of the equality, which I listed above in the Comment section, should be either added to the pop culture section or to a section of their own. They would serve to support the assertion in the introduction that the result is "surprising." --Trystan 19:41, 18 May 2006 (UTC)

I think the new section is a reasonable compromise and I'm pleased with it. In regards to improving it, I agree with Trystan's comment above. --Dv82matt 23:36, 18 May 2006 (UTC)

As there seems to be no further objections, I've added the quote from the Tall paper, described in the Comment section above, following the sentence mentioning his studies.--Trystan 03:40, 27 June 2006 (UTC)

Significance

This article needs a section Entitled

== Significance ==

Or

== Real life Applications ==

Has there been any applications of this proof? Has anything been built of of this proof or is it a dead end?

Also for those of us that use "0.999 repeating" as a physical representation of an abstract number "approaching but not equal to" one we need a new variable. Has a new variable been given (No pun in tended).

For instant although .999¯ equals one ... when we graph the number line (-∞,1),(1,∞) or (x≠1) How do we illustrate the "conceptual number" before and after 1 or any other whole number. Before we proved this we just used .999¯ and .000¯1 to say this concept. Now that we Proved this number actaully does not exist we need a new way of saying what we have to say. This Proof is hard to Grasp becose we don't have an alternitive to issustrate the idea.

Remember that "1" is a symbol for a number. And if ".999¯" equals one then it is just a #REDIRECT [[1]]. We now need a new Title for XYZ.--E-Bod 22:38, 11 June 2006 (UTC)

Your "variable" would be worthless. It varies. You need a constant. That would be 1 - dx, where dx is an infinitessimal. Either way, can you possibly name any use for 1- dx? He Who Is 21:24, 16 June 2006 (UTC) P.S. .000¯1 = 0. Adding womething after repitition has no effect on the value.

Table

n/9

/	.
0/9	0.000
1/9	0.111
2/9	0.222
3/9	0.333
4/9	0.444
5/9	0.555
6/9	0.666
7/9	0.777
8/9	0.888
9/9	0.999
10/9	1.111

n/9

/	.
0/9	0.000
1/9	0.111
2/9	0.222
3/9	0.333
4/9	0.444
5/9	0.555
6/9	0.666
7/9	0.777
8/9	0.888
9/9	1.000
10/9	1.111

If you make a Table it makes sense. If you try with a proof people will try to think it is a 1=2 proof. Somebody still needs to address what to call the number that approaches 1 but isn't. Just like X^0 is one not zero if you make a table and look at it.

That's the point of the proof: The number that "approaches 1 but isn't" does not exist (at least, not in the standard reals we usually employ). As to the significance, I can't think of any "real world applications" right now. This proof should just show the non-existence of such an "approaching number" and thus clarify our mental image of the reals. If you want to discuss the mathematical details, we should probably do so on the arguments page. Yours, Huon 09:04, 12 June 2006 (UTC)

Sorry Ill move it to arguments]] now. I was Actually Suggesting We move the Charts into The Article. Or a Variant of the chart. I will Further the Discussion under arguments. However I'm not arguing the proof is wrong I was suggesting Article Content.--E-Bod 23:27, 12 June 2006 (UTC)

Note the word "if" in Huon's comment. As long as the discussion is about improvements to the article, it can be held here. -- Meni Rosenfeld (talk) 15:37, 13 June 2006 (UTC)

As noted above, what you want is 1 - dx. (See Infinitessimals He Who Is 21:27, 16 June 2006 (UTC)

Wikify tag

User:Hamish2k added a {{wikify}} tag to the top of the article without rationale. I disagree and have of now removed the tag. Perhaps someone could say what exactly needs to be wikified in this article? —Mets501 (talk) 12:35, 19 June 2006 (UTC)

Geometric series proof

Could someone please check if this proof is correct? I'm a little rusty. Thanks. Supadawg - Talk 23:20, 20 June 2006 (UTC)

Looks good to me! —Mets501 (talk) 23:22, 20 June 2006 (UTC)

By the way, I fixed it up so it uses all n, and not sometimes k and sometimes n. —Mets501 (talk) 23:31, 20 June 2006 (UTC)

Thanks. My old math teacher did the same thing on the projector all the time - imagine seeing i, k, and n, used for the same variable, all in one proof! Supadawg - Talk 23:40, 20 June 2006 (UTC)

We've discussed this line of attack before, and emphatically discarded it. It raises nasty problems and is not helpful. Please do not try to add this to the article. The two advanced proofs already included are careful, correct, and much more delicate than most people seem to realize. Thanks for your understanding and cooperation. --KSmrq^T 23:53, 20 June 2006 (UTC)

What exactly is wrong with the proof? It's short, simple, uses well-grounded formulas, and is (as far as I can tell) correct. The material is taught in any Algebra II or Pre-Calculus class. If you have a problem with it, please disprove it. Could you provide a link to the discussion you've cited, as the archives are rather large? Supadawg - Talk 00:00, 21 June 2006 (UTC)

Well, one problem is that Mets501 seems to have wiped out much of the previous work, presumably by accident. If so, it's ironically representative of some of the worse sections of this tak page's history (of which I share much of the blame), in that the different developments are claimed to be in conflict. Let me take a crack at the article to explain otherwise..... Melchoir 00:08, 21 June 2006 (UTC)

I'm so sorry about that! It was a complete accident (for some reason, Firefox seems to empty large portions of text boxes on me at random intervals). —Mets501 (talk) 00:20, 21 June 2006 (UTC)

Please don't. We've been down that road before. The "geometric proof" is a sloppy version of the existing careful limit proof. --KSmrq^T 00:20, 21 June 2006 (UTC)

You keep saying that, yet you don't provide specifics. Please specify exactly how the proof is incorrect. I like it a lot better than the other advanced proofs because it's expressed in mathematical notation instead of paragraphs, and is a lot cleaner. Honestly, I don't care if you have the page exactly the way you like it. People can choose to study and accept whichever proof they like; this merely widens the scope of their choices. Supadawg - Talk 00:25, 21 June 2006 (UTC)

Okay, I'm mostly done. Here's my commentary, if it's not clear from my editing:

In a sense, the geometric series proof is a descendant of the limit proof. If you take the former and substitute in the proof of the convergence theorem instead of its statement, then you will of course get a proof based on limits. However, applying the theorem straight up, you get a perfectly correct proof that never even mentions the word "limit". It's a bit of a rip-off educationally, because it seems to suggest a shortcut bypassing limits; those of us who have gone through real analysis know otherwise. Rather than declare geometric series invalid, though, it is better to mention them while making clear that they build on previous work. Melchoir 00:43, 21 June 2006 (UTC)

I like it! Let's keep it. —Mets501 (talk) 01:24, 21 June 2006 (UTC)

Great job, Melchoir. Supadawg - Talk 01:34, 21 June 2006 (UTC)

Bear with me, I'm at the end of a long day. For some of you (presumably Supadawg) this is a new conversation; for me it's anything but, and I'm not thrilled about going over the same ground again. Ah well.

A complete proof here requires three basis components. We must define what we mean by real numbers. We must define what we mean by decimal expansions. And we must define what we mean by equality. Then we put those pieces together in a proof. The limit proof does just that, explicitly. So does the order proof. The geometric series proof is sloppy about all three basis components; it is not cleaner, nor really a proof. The essential pieces required to substantiate the geometric series proof are contained in the limit proof.

On this talk page, and in the sci.math newsgroup, we have seen over and over again that a geometric series proof draws attacks. It practically invites them, because the real heart of the question requires explicit consideration of the omitted three basis components. It is too glib, too much of a handwave. It is comfortably familiar for those who do not need proof, but leaves others unsatisfied.

Nor does the sloppy way it is written inspire confidence. I realize it was dashed off in haste, but really: The entire rest of the article uses "0.999…", not ".999…". Never start a sentence with a number like this. There is no good reason to write a series using ".9" instead of "9". The first sentence of the paragraph has a typo. You get the idea.

Don't get me wrong, the geometric proof could be fleshed out to include the missing pieces. It's not an erroneous proof per se. But by the time it's filled in, it just adds unnecessary baggage to the limit proof. Since I'm practically writing this in my sleep, I hope I've made my objections clear. --KSmrq^T 02:49, 21 June 2006 (UTC)

Without entering into a discussion of what makes a proof complete or not, this is not how proofs are communicated between mathematicians, let alone to the general public. When you explicitly call on a theorem, you are implicitly asserting that your definitions are the same as those used by the theorem's author, and that all hypotheses of the theorem are met by the situation at hand. Sometimes to help the reader, you explicitly verify some of these points, such as 1/10 < 1, and especially when one of the steps isn't obvious. But it is not necessary or practical to restate all the definitions.

So, taking up the first of your three components, why should every proof in the article have to define what real numbers are? Isn't it enough to ask the reader to trust that the real numbers are a system in which the cited theorem holds? Melchoir 07:57, 21 June 2006 (UTC)

(Whew. I've had a bit of sleep. Highly recommended!) You implicitly raise a question of some interest to me, which is a closer examination of the nature and role of proofs in mathematics, both in practice and in teaching. One role of a proof is as a kind of bridge. At one end we have the definitions and facts we have already established, and at the other we have the statement we wish to assert. So we could say the purpose of a proof is to get to the other side. If that were the end of it, we'd be hard pressed to explain why we have 196 published proofs for the law of quadratic reciprocity, eight of them by Gauss!

In building the bridge we may state our premises in different ways, or uses different lines of reasoning in reaching our conclusion. Why? Because another important role that proofs play is providing insight. That insight might support better understanding in a student, or a new generalization in a researcher. The proofs in this article are as much about insight as they are about getting to the other side.

The geometric series proof essentially says, you know what we mean by real numbers, you know what we mean by the sum of an infinite series, you know what we mean by convergence; we define the meaning of a decimal expansion as the real number which is the sum of the series, and look, the conclusion is trivial. That completely misses the point! The people who most need this proof don't know what we mean by real numbers, certainly not in any formal sense good for proofs. They demonstrably balk at the sum of an infinite series, which immediately throws both infinity and limits in their face, which they neither understand nor accept. Convergence historically baffled some fine mathematicians, so that's also not making things easier.

In essence, if you can understand this proof, you don't need it! My impression is that for most of the people who champion this approach, it is less a proof than a mnemonic device. If they were challenged to fill in the fundamentals, they would flounder. And they honestly don't see how it fails to communicate with the audience they need to reach.

Anyone who has been around long has seen it play out. The question, the easy answer, the blank stare or the balk followed by the hostile rejection, and each side regarding the other as idiots who just don't get it.

This article, up til now, has carefully avoided that debacle. I respectfully suggest that we continue to do so. --KSmrq^T 11:54, 21 June 2006 (UTC)

Now that you pointed it out, this "proof" is actually more like a definition. If you understand infinite series fully, that is usually how you would define it in your head. —Mets501 (talk) 13:05, 21 June 2006 (UTC)

(rewrap) I agree that the proof provides little insight. There's a similar problem at Euler's identity, which spends seven displays on a derivation that consists of just one logical step and calls on a theorem so powerful that the article is left with nothing to explain. Nowhere is the meaning of exponentiation hinted at, and without that, the article is just performing a formal manipulation without explaining what's going on. Yes, it lets down -- perhaps even betrays -- the reader.

But would I remove that "derivation" from the article? No, not even if the problem were mitigated by the presence of better proofs. What ought to be done is what I've tried to do here: frame the proof intelligently. Explain that while it is correct, it draws upon a background that the reader might not be familiar with, and it does not attack what some consider to be the crux of the matter. We can avoid the blank stare and the hostile rejection if we work at it. And why should we go to all that trouble when omission is easier and safer? Well, here is Melchoir's Positive Defense for Using a Slick Proof:

• It increases interest and awareness in the tools called upon. With any luck, the reader becomes curious about the mathematical development of the topic (and not just its bedrock).
• It instills awe in how easy mathematics is when you're properly armed. (The illusion of ease does not need to deceive, and awe does not have to conflict with understanding, given a decent explanation.)
• It builds the Wikipedia web and improves the comprehensiveness of an article.
• Slick proofs are popular. Readers of Wikipedia may already have seen one, in which case they'll benefit from a discussion. They might expect to find the slick proof and become confused if they don't find it. Worse, they might think their secret knowledge obsoletes the article.
• Slick proofs are popular. Editors of Wikipedia will periodically attempt to insert one if it isn't already there. It can be easier in the long run to head them off.
• Slick proofs are popular. To avoid mentioning popular viewpoints is not NPOV.
• Textbooks and other published sources rarely explain the details behind each and every proof. Although we aren't sheep, a slick proof can be the most verifiable and least original-researchy part of an article. We mathematics editors already ask a lot of leeway on these policies from our Wikipedian peers; the least we can do is throw them a bone.
• Theoretically, it is possible that the reader has developed the background necessary for a given slick proof but hasn't seen the proof yet. That reader might benefit from seeing two and two put together.

With that last one, I'm scraping the bottom of my imagination, so I'll stop there. Melchoir 18:52, 21 June 2006 (UTC)

I'm reminded of Bogie in the Maltese Falcon, listing all the reason he has for turning Miss O'Shaughnessy over to the police: "Maybe some of them are unimportant — I won't argue about that — but look at the number of them." ;-) Paul August ☎ 02:19, 22 June 2006 (UTC)

I'm inclined to stop with your first statement, "the proof provides little insight." Permit me an analogy:

• Q. How do I kill this pesky fly?
• A. Detonate a thermonuclear device next to it.

The answer is correct; the fly will die. (Caveat: The device must be close, and this may not work for wasps]. [1]) It's also useless.

As I've tried to point out, this proof is not so slick if done right. Nor is popularity persuasive. Consider three popular perennials:

1. The pattern. Since d⁄9 equals 0.ddd… for d a digit from 0 to 8, then 0.999… equals 9⁄9, which is 1.
2. The squeeze. If 0.999… is less than 1, then there must be a number somewhere between them; but we can't find one.
3. The series. The notation 0.999… means ∑_k 9/10^k, for k = 1…∞, namely the sum of an infinite geometric series with first term 9⁄10 and term ratio 1⁄10, thus equal to 9⁄10/(1−1⁄10), which is 1.

Although each argument contains a germ of truth (after all, the conclusion is true), not one constitutes a satisfactory proof. Furthermore, each one is dominated by a proof already contained in the article (the fraction proof, the order proof, and the limit proof, respectively). On top of that, the series proof is apparently one of the least persuasive. [2]

My guess is that people remember and offer arguments that first convinced them, or that they can easily remember. But that does not mean we should include all arguments in the article. We want to be accurate, readable, and compelling. An unfortunate side effect of including any of these perennials would be support for the delusion that they are effective arguments. What we do instead is show people what most (including teachers) have never seen before: the rigorous genuine proofs in the advanced section.

I searched the web some time back and could nowhere find a mathematically complete discussion, not even on mathematics forums. Is it any wonder that the general public is confused? I have no interest in perpetuating that tradition here.

The geometric series proof is not only unhelpful, it is actually harmful. And it is poorly written. Therefore it must go. --KSmrq^T 16:29, 22 June 2006 (UTC)

It is helpful for a variety of reasons, and it is not intrisically harmful. I'm really done making those points if it's just one user who disagrees. Melchoir 21:48, 22 June 2006 (UTC)

You have not made a compelling argument that it is helpful. And seeing that no one else has offered any other arguments, and since no attempt has been made to correct the problems of both form and content that I have pointed out, and since the pre-existing article was widely reviewed and approved, I'm going to kill this section. And this time I think it should stay dead permanently. --KSmrq^T 22:43, 22 June 2006 (UTC)

I've reviewed this talk page. Arguments about including the geometric series in the article seem to be confined to the current page, not the archives. My reading is:

• Supported: Meni Rosenfeld, Trystan, Melchoir, Mets501, Supadawg
• Ambiguous: Mdwh, Paul August
• Opposed: KSmrq, CorbinSimpson

This is hardly wide approval for deletion. If you do wish to build a consensus around your opinion, you are welcome to ask the Mathematics Wikiproject or of the usual dispute resolution mechanisms. Pending such a community rejection of the material, I think you should restore it. Melchoir 23:15, 22 June 2006 (UTC)

If you see problems of form and content, why not fix them, instead of deleting the section? Furthermore, it is not our duty to prove to you that the section should stay. It also isn't your duty to keep watch over this article and revert any attempts to edit it that you don't like. You don't have a concensus, so I've restored the section. I defer to Melchoir on the technical aspects of the proof, as he has generalized (so far as I can tell) the proof so much that he's really the major contributor, and he seems to have a much more thorough understanding of the topic than I do. I'm still very much in favor of keeping it, as it seems to be a good middle ground between the "elementary" and the highly formalized "advanced" proofs. Supadawg - Talk 00:30, 23 June 2006 (UTC)

Say what?! Neither Meni Rosenfeld nor Trystan have participated in this discussion, and the last comment from Mets501 was his acknowledgment to me: "Now that you pointed it out, this "proof" is actually more like a definition." I could count as supporters voters in the recent AfD.

I've given you an opportunity to counter my objections to the content, and to clean up the problems I pointed out (such as starting a paragraph with ".999"!), and to garner additional support. You have done none of these. Supadawg is your only clear supporter, and has contributed nothing of substance to the discussion. If you wish to involve the wider community again, may I remind you of their last experience with you concerning this article? You goaded Loom91 into wasting everyone's time with a deletion vote. Surely you must realize that interested parties have this article and talk page on their watchlist?

I deleted the new section for the reasons I have clearly stated. If neither you nor anyone else can be bothered to clean it up, nor can counter my objections, then it has no business being in the article. Lacking any change in these facts I intend to delete it again. --KSmrq^T 01:40, 23 June 2006 (UTC)

You are not the owner of this article, so please stop acting like it. As I said before, we don't have to justify inclusions to you. Just because you still disagree does not mean your objections have not been countered. If these "interested parties" have this on their watchlist, why don't they come and help you out? And please be more careful, as you deleted my previous comment. Supadawg - Talk 01:58, 23 June 2006 (UTC)

I welcome your comment here. I never saw it nor a warning of an edit conflict. Sometimes the servers hiccup and cause problems like this.

As for being the "owner" of this article, on Wikipedia all interested parties are owners. You are acting like one by inserting new material, and I am acting like one by removing it. It's a little weird; but for reasons I don't really understand, it seems to be more productive than the PlanetMath model. Fascinating. --KSmrq^T 13:34, 23 June 2006 (UTC)

I fully understand the Wikipedia concept of ownership: it doesn't exist. You and I are acting as editors, not owners, but this is beside the point. You deleted a section without concensus. Find more people to support your side, and we can have a decent poll. I find sarcasm irritating. Supadawg (talk • contribs) 13:58, 23 June 2006 (UTC)

You have only to search for their names to find statements in the "Intermediate proofs" section above. The AfD does not include the words "geometric" or "series" even once, so it is not relevant here. As for your objections, if you come up with constructive advice I'll consider it, but just calling the proof "unsatisfactory" and mocking my use of decimal points is not going to get us anywhere. If I'm missing a logical step, point it out. If the discussion is incomplete, say how. Melchoir 03:55, 23 June 2006 (UTC)

I was precise in my wording; I counted comments in this discussion. The AfD opinions are just as relevant as the old comments you want to count, if not more so. As for constructive advice, I have offered quite a lot. But here you call my comments about form "mocking", and do nothing to correct them. Want a list? Then let me repeat myself with regard to form, but this time with bells and whistles:

1. The entire rest of the article uses "0.999…", not ".999…". (See Number Style.)
2. Never start a sentence with a number like this. (See Math Style.)
3. There is no good reason to write a series using ".9" instead of "9". (See common sense.)
4. The first sentence of the paragraph has a typo. (Fixed.)

Finally, aside from my previous detailed remarks about content, I think I've made clear the way I want to "improve" this section, namely delete it. I see no point in trying to "put lipstick on a pig", to use a popular expression. --KSmrq^T 13:34, 23 June 2006 (UTC)

One problem with this proof is that it does depend on an (unspecified) proof of convergence. I would naturally prove convergence here as in the limit proof above; and to those who do so, this proof is redundant. Septentrionalis 02:10, 23 June 2006 (UTC)

We could put the convergence proof under a new heading, or simply link to the convergence article. Assuming users aren't lazy :) Supadawg - Talk 02:36, 23 June 2006 (UTC)

I'd have to say I dislike having it in there, at least as an "advanced"-style proof, simply because it assumes things that people who disagree with the article don't like assuming - stuff like what it actually means to sum an infinite number of terms. However, if it's stated as a high-school level proof, or, as someone said, a mnemonic, then I wouldn't mind it staying, as long as it had some kind of statement about "in a more formal setting, mathematicians impose a definition on what the infinite sum means (treating it as the limit of finite partial sums)" or somesuch. Confusing Manifestation 03:15, 23 June 2006 (UTC)

Go ahead and try it! Our pontifications here don't help the reader unless they penetrate into the article. Melchoir 04:09, 23 June 2006 (UTC)

The cited theorem guarantees convergence; logically, that suffices. The whole point of using theorems in mathematics is so that you don't have to reinvent the wheel every time you come across a new situation. (And don't kid yourself: the other proofs call on plenty of locally unproven results.) Of course, for pedagogical effect, it would be better to link to the convergence proof at least. Perhaps you'd prefer it if Infinite geometric series contained a sketch or something? Melchoir 04:07, 23 June 2006 (UTC)

I find the geometric series proof helpful, but would agree with ConMan's suggestions for altering it. It seems to fit notionally between the "Digit Manipulation" and "Algebra Proof" sections, so would make more sense to me to come between them in the article.--Trystan 04:48, 23 June 2006 (UTC)

Someone has altered the section titles (badly), so I'm not sure exactly what you mean. But it would be completely inappropriate to insert a geometric series proof in the section written for young students. --KSmrq^T 13:43, 23 June 2006 (UTC)

"Someone", riiiight. Are you being incivil in an effort to drive me away, or do you just not know better? Melchoir 19:36, 23 June 2006 (UTC)

I think he just didn't check the edit history, Melchoir... Supadawg (talk • contribs) 20:10, 23 June 2006 (UTC)

You know, it doesn't even matter who it was. There's no excuse for that attitude here. Melchoir 20:32, 23 June 2006 (UTC)

...(crickets chirping) Fine, I'll change the headers again. I think it is self-evident that the new titles are more descriptive and less POV than the old ones. Melchoir 16:55, 26 June 2006 (UTC)

I agree; to me it would best fit between the Elementary and Advanced proofs sections (as they have been returned to their original names), in an Intermediate section.--Trystan 16:38, 23 June 2006 (UTC)

Also, I would like to point out that the French, Japanese, Thai, and Chinese Wikipedias all include this geometric series, so I don't see why we cannot (by the way, those are all of the other languages which include this article outside of English). —Mets501 (talk) 13:28, 26 June 2006 (UTC)

Have you noticed that all four of those articles consistently place a 0 before the decimal separator? And that not one of them begins a sentence with a decimal separator? If you want to imitate something, imitate that.

That said, even though I cannot read the text it is apparent that these four articles are essentially regurgitations of the same dubious stuff you find all over the web, and contain none of the real substance and careful advanced proofs of our English-language article. Nor do they include our section on "other number systems", such as non-standard reals, p-adic numbers, and Hackenstrings. I see no reason for us to sink to their level; they should aspire to rise to ours. --KSmrq^T 15:07, 26 June 2006 (UTC)

OK, I added zeros before all of the decimal separators. That should settle at least one problem.

As far as the proof: why should adding more information to our article be considered sinking to a lower level? We do not want to remove any of the other well thought-out proofs, just add another proof, which would appeal to intermediate mathematicians. It is not an utterly pointless section, even if it might be quite obvious to you and other advanced mathematicians. This section does not go against any other proof, it is only additional information, and cannot hurt. It does not raise any more questions then does the actual fact the 0.999...=1. It is, in fact, the same thing, just represented as an infinite series instead of in decimal form. If you fully understand why 0.999...=1 then you understand the convergence theorem, and there are no questions, and if you don't understand why 0.999...=1, then perhaps this proof will help you understand it (sort of as a last resort) or else it won't help, but it can't confuse you more. —Mets501 (talk) 15:25, 26 June 2006 (UTC)

Amazing, someone finally fixed one of the three style problems I listed. I listed two just above. I listed all of them with numbered bullets several days ago, complete with links to relevant style article sections. I do appreciate this one effort, but can't help noticing the originators did nothing.

You go on to make statements that are surely well-meant, but false. You claim that this section raises no questions and does no harm. Did you not read what I wrote about this? Did you not go look though the talk page archives, and the sci.math archives, and the endless web forums where this has been debated? It is simply not credible to state that it raises no questions, for we can see them being asked! It is demonstrably false that it does no harm, because it does raise needless questions, it does cause confusion, and it does provoke rejection of the fact. That is harm, not help.

Is this a last resort? Consider where it might logically be placed. We know it is not suitable for elementary school students, so it can't go with those proofs. We know it lacks the foundations of the advanced proof, so it can't go with them either. If we put it between those two sections then we destroy all the careful work aimed at not raising the "infinite", "limit" handwave objections. If we put it after those two sections then we're proving poorly something we've already proved carefully.

It adds nothing but problems, it has no logical place in the article.

And since not one of the originators has tried to correct any of the flaws in style or substance, despite ample opportunity to do so, they apparently care little for the quality of the article. I do. Therefore, I am deleting this section permanently. --KSmrq^T 18:34, 26 June 2006 (UTC)

The geometric series proof does not lack foundations. It is, in fact, the one argument in the article that is well-founded in the mathematical literature. As long as you continue to insist that it isn't a proof, you cast doubt on everything else you say. And may I remind you that you lack the power to do anything permanently here. Melchoir 19:15, 26 June 2006 (UTC)

It isn't just our responsibility to fix any style errors, it's yours as well. I cannot belive you would go so far as to purposely not fix any errors you find, just because you didn't create the section. I will see what I can do, but since you apparently went through the article so methodically, why didn't you just fix things as you went along? This raises questions as to your motives for being a Wikipedian. You have no concensus; it's just you who wants it deleted. The section stays. Supadawg (talk • contribs) 19:24, 26 June 2006 (UTC)

I appreciate your belated concern for attending to errors. I previously addressed the question of how I would "fix things", and, in fact, I followed through and implemented my fix; as I said, I see no reason to put lipstick on a pig. May I respectfully suggest that trying to cast aspersions on my motives violates WP:NPA? I have done my damnedest to assume good faith of all parties, and will thank you to do the same.

Now that two (still not all) of the style questions have been addressed, how about addressing issues of content?

• The first sentence is unsatisfactory. While there is no doubt that a satisfactory proof requires better tools than are available in elementary school, the question itself is important even then, and does impact us.
• Next we come to the phrase "convergence of limits", which is semantically ill-formed; limits do not converge, sequences do.
• And speaking of sequences, the geometric progression article reminds us that we are summing a geometric sequence, not a series.
• Furthermore, the series itself is semantically ill-formed, because it uses a decimal expansion ("0.9") to define a decimal expansion; and again, we see notation inconsistent with that used previously in the article. Standard form would be

  $0.999\ldots ={\frac {9}{10}}+{\frac {9}{10^{2}}}+{\frac {9}{10^{3}}}+\cdots .\,\!$

• Likewise, the limit should also be written using rationals:

  ${\frac {\frac {9}{10}}{1-{\frac {1}{10}}}}.\,\!$

• The concluding sentence is both stylistically flawed (why is "algebra proof" in quotation marks and limit proof not?), and a peculiar assertion. How can the form of the proof of convergence make this look like the algebra proof? And the limit proof (deliberately!) does not talk about the limit of the series of truncations; instead it considers the limit of the series of differences between the truncations and 1. That is not identical to the geometric series argument.

I have now criticized the proof essentially line-by-line. As written, this section is not up to (English) Wikipedia writing standards, and certainly not up to the level of the rest of the article.

And so now I come again to my larger point, which is that even if this list of problems is corrected, this section harms rather than helps the article. I have argued repeatedly that the limit proof is pedagogically superior, and I stand by that claim. People who do not understand and accept the assertion being proved also do not understand and accept this proof. We know that because they say so, all over the web. The people who are so in love with this proof don't need it; it is for them merely a reminder of facts they already know and accept. Unlike, say, the order proof, it does not teach them anything they do not already know. And, frankly, the limit proof probably handles limits more carefully than they do.

• Shall we compare? The limit proof begins by explicitly and transparently constructing the reals from the rationals. To do so it first defines absolute value using the order property of the rationals. You don't have to be a graduate student to follow that. It defines the reals as Cauchy sequences, using three different methods to explain what it means for a sequence to be Cauchy: (1) a link, (2) a formal definition, and (3) an informal gloss. The formal definition uses absolute value, so this is all self-contained. Again, you don't have to be a graduate student, and you don't even have to follow links and do background reading. The next step is vital, and worded with great care: it is the introduction of the concept of limit. Again, this is self-contained, using only the tools we have with us. There is no "series approaching a limit" language, because we know that builds the wrong mental image. Instead the wording is "a sequence has a limit". It's a static property, not some kind of dynamic process. And what a simple definition it is, one almost anyone might be able to understand. Then equality of reals is defined in terms of limit, again avoiding any misleading "process" language. Finally, it looks at the sequence in question. That sequence, 1/10^k, happens to be a geometric progression, but that is not mentioned because it is not necessary to do so. Instead, the proof merely relies on the fact — obvious to almost anyone — that the sequence gets arbitrarily close to zero. Done!
• Although this is called an advanced proof, it is actually far more accessible to a broader audience than the geometric series proof. Despite its rigor it is self-contained, providing all of its definitions in elementary terms using properties of rational numbers. It is short. It scrupulously avoids well-known "hot buttons". Nowhere do the words "infinite" or "infinity" appear. A discussion of "convergence" is unnecessary, as is a reference to any "convergence theorem". It has no need to beg the question by using controversial phrases like "sum of an infinite series", language which annoyingly assumes that such a sum is well-defined and exists — the very issue we're trying to untangle!

And, by the way, I see no evidence that any of the opposition has put anywhere near this amount of thought into the article. (But I'd love it if they would.) Instead I see poor thinking, poor writing, and poor debate.

So again I say, now with bells and whistles and a big brass band, the geometric series proof is not slick, it is not helpful, it is not harmless, and it is not going to stay. --KSmrq^T 23:28, 26 June 2006 (UTC)

I would remind you that I too have assumed good faith throughout the discussion - you proved me wrong. I did not resort to name-calling, as you charged in your edit summary. You list "inadequacies" from the mundane to the so-long-it-resembles-a-rant, yet you refuse to correct any of them. If I had to pick just one of your arguments, it would be on the "standard form" of recurring decimals. The number 0.999... is being represented as a geometric progression, not a decimal: this is fundamental to the proof. I don't feel like further following your instructions on how to improve the section, since you don't care to follow them yourself. Again, you are the only one in favor of deletion, and that is not a concensus. Please bone up on Wikipedia policy in that area. This discussion has gone stale, and I'm tired of it. Supadawg (talk • contribs) 23:55, 26 June 2006 (UTC)

Perhaps you've noticed that when you make ten complaints at once, I choose to respond only to the weakest of them. It's really the only way we can carry on a linear conversation here. This time I'll be generous and answer four: (1) The quotation marks in the last sentence were fixed hours ago. (2)"Sum of a geometric series" is perfectly good language for which I can supply references if necessary; Geometric progression is unreferenced and probably wrong. Generally, Wikipedia is terrible on language use. (3)"Convergence of limits" is so trivial to fix (and I will) that I wonder why you didn't. (4)The series is not ill-formed. The distinction between finite and infinite decimal expansions is well-known. And, again, if you prefer a different style, you are free to implement it.

Now, about your tone: unless you are a time-traveler from the future, please do not make such claims as "it is not going to stay". Ending all your speeches with a promise to destroy Carthage does not necessarily make it so, and in the meanwhile you unnecessarily raise the stress levels of everyone around you. Melchoir 00:03, 27 June 2006 (UTC)

KSmrq, from this talk page there is clearly no consensus to delete this section. Because of this, I think that you should now actually go to the article and edit it. You say the first sentence is unsatisfactory. Go and make it better, instead of complaining here. You also say that the concluding sentence is "a peculiar assertion". Rewrite the last sentence, then. Make it better. If it's unclear, make it clear. Or remove it and make a new concluding sentence. The rest of your points that you listed in the list above have been fixed. I have changed "sum of an infinite series" to just "an infinite series", because a series is already a sum. The phrase "convergence of limits" has already been changed. The decimals have been changed to fractions in the expansion. If you have other minor points, go ahead and fix them. —Mets501 (talk) 00:44, 27 June 2006 (UTC)

I have stated clearly how I intend to "fix" this.

I gave a detailed list of problems for two reasons. The first is obvious: they all existed, for anyone to see and fix. The second is because there were so many. The claim has been made that this is a "slick" proof that improves the article. Perhaps I should reproduce the first version to show just what Supadawg finds acceptable. Here it is from the history:

---

Geometric series proof

.999… can be defined using the infinite geometric sequence

$a_{n}=.9\times .1^{n-1}\,$

By definition,

$\sum _{k=0}^{\infty }.9\times .1^{k}=.9\times .1^{0}+.9\times .1^{1}+.9\times .1^{2}+.9\times .1^{3}+\cdots \,$

$.999...=.9\times .1^{0}+.9\times .1^{1}+.9\times .1^{2}+.9\times .1^{3}+\cdots \,$

Therefore,

$.999...=\sum _{k=0}^{\infty }.9\times .1^{k}\,$

By the formula for the sum of a geometric sequence,

$\sum _{k=0}^{\infty }.9\times .1^{k}={\frac {.9}{1-.1}}\,$

$\sum _{k=0}^{\infty }.9\times .1^{k}=1\,$

Therefore,

$.999...=1\,$

---

To make matters worse, this was added to "Elementary proofs". Now, I'm sorry if this is going to step on Supadawg's toes, but I guess it's time to speak plainly: To anyone with any sensitivity at all, this is a fat turd laid in the middle of a fine article. I've tried to discuss issues and carefully point out problems, but the big picture has not changed. Supadawg doesn't see it, doesn't want to hear about it, and (by self admission) doesn't care.

Melchoir objects to me listing so many problems; I object to seeing so many! And in any event, the short (three items!) list was ignored also. I don't enjoy spending my time seeing a blatant problem, looking up chapter and verse in a style manual to cite as to why it's a problem, then seeing the proponents of this section ignore it for days. And, frankly, at this point I'm not writing for them. I'm writing for the other people who will come along when I delete the section day after day so they can see that I bent over backwards to give the proponents a chance to improve and debate, to no avail.

I'm sorry, Mets501, if any of my irritation at the poor behavior of Supadawg and Melchoir seems to spill over on you, or others who have behaved in a more civil and helpful manner. I do appreciate that you have tried to respond positively to deal with some of the problems I have enumerated. That you have failed does not reflect badly on you; at least you tried.

I admit to having my patience tested by the behavior of Melchoir, whose past actions with regard to this article have been a recurring problem. Recent examples other than this include goading another user into nominating the article for deletion, and proposing to move the article to "0.999…". Without assuming bad faith, I can still see bad judgment.

The Wikipedia way traditionally involves words on talk pages. I have read those of my opponents and responded; they have stated a disinterest in what I have to say. That makes further use of words pointless. It also makes a mockery of any pretense that consensus applies.

So now I am again going to exercise my judgment, and again delete the section. For all the reasons I have detailed. At length. Repeatedly. --KSmrq^T 02:24, 27 June 2006 (UTC)

This is a diversionary strawman attack. I was the first user to massively clean up the text you just quoted, and I have continued to improve it in response to criticism. The current text is the result of a collaboration between four writers. No one suggests that we go back to the rough early version.

I already apologized over the AfD, which I vehemently opposed at length, and I don't see how proposing to move an article is problem behavior. On the latter point, I am not the only editor who spoke in favor of a move.

Finally, I am not disinterested in what you say, but your tactic of lengthy speeches prevents me from focusing on any one of your concerns. You appear more interested in grandstanding for the record than in coming to an understanding with your critics. If you want to learn the reasons why others do not agree with your reasoning, pick an argument and we'll talk about it. Just one, please, and I don't want to hear about any more damn typos.

Oh, and stop deleting the section. It's getting old. If you're so sure that you're right, that you're being reasonable, and that this page is being watched by the right people, then surely you don't need to do it yourself? Melchoir 02:47, 27 June 2006 (UTC)

OK, I don't think that we're getting anywhere with just us four arguing about whether to include this section or not. I have an idea. Let's all actually work on that section in the article (as if it were going to stay permanently) until we can get that section perfect. Then let's ask for the opinions of other mathematicians at the math wikiproject, and for the opinions of non-mathematician respected members of the Wikipedia community. This discussion is getting out of hand, it's causing people to act incivil who would not normally be, and I don't think it is helpful to anyone if this continues to escalate. —Mets501 (talk) 03:02, 27 June 2006 (UTC)

KSmrq, I am at a loss as to why you pulled up my first draft of the section, something that hasn't seen daylight for quite a while. Although technically correct, you eloquently called it a "turd". It has been improved greatly by Melchoir, Mets501, and even you, indirectly. I would direct you to my request for help at the top of this discussion, where I specifically requested that someone improve it. Accusing Wikipedia editors, namely Melchoir, of past poor behavior does nothing to help your case. I cite Wikipedia:Concensus, something that the deletion of a section requires. Although Wikipedia is not a democracy, I propose a poll on the major issue you seem to object to.

Keep the section, working with KSmrq and others to fix any and all errors. Supadawg (talk • contribs) 03:07, 27 June 2006 (UTC)

We know you want to keep this section, but please, can we not vote for keep or delete until after we have improved the section to all that it can be (that might not even be enough for KSmrq)? Let's just wait on the vote and try to improve the section. —Mets501 (talk) 03:10, 27 June 2006 (UTC)

I like Mets501's idea. Melchoir 03:16, 27 June 2006 (UTC)

I absolutely agree with Mets501's idea - I just wanted to get KSmrq to stop deleting the section so that we could start fixing it. Supadawg (talk • contribs) 03:21, 27 June 2006 (UTC)

OK, that's fine with me. —Mets501 (talk) 03:38, 27 June 2006 (UTC)

Me too. Paul August ☎ 03:30, 27 June 2006 (UTC)

Thanks, Mets501, for continuing to press for improvement.

And when is the "fixing" going to start, and what will it address? I see no reason to waste more of my time detailing problems that are ignored. The only time anyone began to show the slightest interest in the issues I raised was when I deleted the section. Melchoir accuses me of a "diversionary strawman attack", and wants to take credit for a "massive cleanup". Both are fiction. The problems I first listed were after that "cleanup". Will this "fixing" will do better? And then, "I don't want to hear about any more damn typos." How odd; I haven't mentioned even one, unless you count the quotation marks issue as a typo. (But I would if I saw one; a typo is still a mistake that needs correcting, and sometimes hard to spot.)

Is deleting the section getting old? Fascinating. It's quite new to me; I've never had to push this hard to be taken seriously before. But what I do find getting old is repeating myself and being ignored. I'm accused of "grandstanding", and of giving "lengthy speeches". The first is ad hominem bullshit, another attempt to ignore the substance of what I say. Nor am I giving speeches, though I have written at length. Shall I quote Supadawg, from early on in this discussion?

"You keep saying that, yet you don't provide specifics."

I guess I missed the part where he silently added, "Just kidding." And by the way, notice my posts, including my fixit lists, started shorter?

Neither Melchoir nor Supadawg can understand why I mention their past behavior to illustrate their bad judgment. Between the two of them they still could not or would not clean up the mess. Supadawg thinks it has been "improved greatly"; I think it has been improved only slightly. But wait, what about that request that someone help improve it? Sounds good; but I went looking for it. All I could find was "Could someone please check if this proof is correct?" Well, I do like that, as far as it goes. It's humble and collegial, both qualities I admire. And I note with approval that the proof itself referred to the "sum of a geometric sequence", not series. But there is no hint of awareness of serious problems other than correctness.

And, by the way, I checked a copy of Ahlfors, Complex Analysis, 3/e (ISBN 978-0-07-000657-7), and found that his definitions of "sequence" and "series" (pages 33–35) are consistent with what I claimed: a series is a sum, a sequence is an enumeration of values.

If folks want to keep putting lipstick on the pig, I suggest you copy the text to the talk page. Because unless I see my substantive concerns addressed, as well as my stylistic ones, I'm going to kill the pig. And to know my concerns, like it or not, you're going to have to read what I wrote. Maybe even all of it. Who knows? You might learn something about just how much thought goes into careful writing, even a single sentence. I'll be kind: You could start with my in-depth examination of how the limit proof is written, compared to the geometric series proof, and the line-by-line analysis of the latter.

Based on past results, and no insult intended, I don't believe Supadawg, Melchoir, and Mets501 together are up to this task, even with my detailed lists. (Please, just copy my TeX verbatim.) If Paul August wants to jump in and help, great; I've said before more privately that he strikes me as "intelligent, educated, level-headed, and inclined to try to spread oil on troubled waters." I hope I'm right; we could use some of that.

Foolishly optimistic (and because I have other things I want to do), I will not delete the new section again for at least three days. If I see an honest effort to address my concerns, then we can talk. If not, get ready for a luau.

And in case anyone wonders, I still assume good faith of all parties. Please, prove me right. --KSmrq^T 07:22, 27 June 2006 (UTC)

Enough dancing then! One issue. Pick one. Keep it short. Melchoir 07:36, 27 June 2006 (UTC)

You're still stalling. It won't work. I'm waiting to see action. You've got plenty to go on. (Main course at a luau: whole pig.) Tick-tock. --KSmrq^T 15:48, 27 June 2006 (UTC)

This truly is pointless. You act as if you have some power or final say over the article. You will be reverted. The reason other people don't listen to you (as you claim) is because you are one person. We've been over this. You've lost. Please accept it. I suggest reading the policy on Wikipedia ownership, along with the other policies I've mentioned that you seem to be in violation of. As to your quote of me very early on in the discussion, I said that before you started giving "fixit" lists. I am tired, tired, tired of this. Supadawg (talk • contribs) 16:11, 27 June 2006 (UTC)

Do you all really have to continue on like this? Please, we know everyone's opinion. For a little bit just use to talk page to talk about a specific problem if you can't think of how to fix in the article. If you can, fix it in the article. All we're doing is wasting time writing huge responses, and that time could be better spent working on the encyclopedia. —Mets501 (talk) 16:21, 27 June 2006 (UTC)

One issue. No pigs, no clocks, no "see above", no accusations. Focus. Just one! Melchoir 16:32, 27 June 2006 (UTC)

Actual work

I've removed the statement that the terms of the sum go to 0, since it's not strictly relevant to the proof, and it raises issues of convergence that the theorem (now stated) actually allows us to avoid. Melchoir 17:11, 27 June 2006 (UTC)

That's much better. I just did a little bit of formatting, as well. —Mets501 (talk) 17:16, 27 June 2006 (UTC)

I've math-ified the whole theorem, and moved the convergence citation. Supadawg (talk • contribs) 18:04, 27 June 2006 (UTC)

I'm not sure if all the TeX is necessary. Do you two agree with the principles of Wikipedia:Manual of Style (mathematics)#Typesetting of mathematical formulas? —Preceding unsigned comment added by Melchoir (talk • contribs)

I mostly agree with those principles. Now there are two cases of TeX on it's own line (which is acceptable) and one case where it is inline ($r=\textstyle {\frac {1}{10}}$). That seems OK to me. The only thing which would go against the recommendations of that Manual of Style would be the inline text, but here it is used just once to display a fraction, which I think is OK. —Mets501 (talk) 18:39, 27 June 2006 (UTC)

We have a couple of options:

All

${\mbox{If}}~|r|<1~{\mbox{then}}~ar+ar^{2}+ar^{3}+\cdots ={\frac {ar}{1-r}}\,$

None

If |r| < 1 then a + ar + ar² + ... = a / (1 − r)

Some

If $|r|<1$ then $a+ar+ar^{2}+\cdots =\textstyle {\frac {a}{1-r}}$

Or anything in between. I like the first one (but then I again, I made it) because, although large, it's easier to read and doesn't switch formatting mid-sentence, but I can see the advantages of each, and I can definitely live with any of them. Supadawg (talk • contribs) 18:40, 27 June 2006 (UTC)

Okay, then I'll paste in the last option, since it keeps the English the same size. Melchoir 18:55, 27 June 2006 (UTC)

We should either change the theorem or the expansion of 0.999… so that they match. Now we have

Theorem. If $|r|<1$ then $a+ar+ar^{2}+\cdots =\textstyle {\frac {a}{1-r}}.$

$0.999\ldots =9({\textstyle {\frac {1}{10}}})+9({\textstyle {\frac {1}{10}}})^{2}+9({\textstyle {\frac {1}{10}}})^{3}+\cdots ={\frac {9({\textstyle {\frac {1}{10}}})}{1-{\textstyle {\frac {1}{10}}}}}=1.\,$

We either need to change the theorem to

Theorem. If $|r|<1$ then $ar+ar^{2}+ar^{3}+\cdots =\textstyle {\frac {ar}{1-r}}.$

or adjust the expansion of .999…. What do you think? —Mets501 (talk) 19:07, 27 June 2006 (UTC)

Starting a sequence with ".9" was one of KSmrq's objections, and since we don't want another 50-kilobyte speech, I'll change the theorem. Supadawg (talk • contribs) 19:09, 27 June 2006 (UTC)

First paragraph

User:KSmrq prefers this for an introduction:

This article presents background and proofs of the fact that the recurring decimal 0.999… equals 1, not approximately but exactly. More precisely, the standard real number represented by 0.999… (where the 9s recur) is exactly equal to the standard real number 1.

I prefer the following:

The recurring decimal 0.999… equals 1, not approximately but exactly. More precisely, the standard real number represented by 0.999… (where there are an infinite number of 9s) is exactly equal to the standard real number 1.

There are two problems I have with KSmrq's intro (1) "This article presents background and proofs" seems to not fit with the way most WP articles are written (2) "where the 9s recur" does not tell users who do not know much math that "there are an infinite number of 9s". Thoughts? —Mets501 (talk) 23:21, 20 June 2006 (UTC)

I agree with Mets501. Wikipedia articles almost always start with a definition of the subject or at least a statement of fact. If the article is titled "Proof that 0.999... equals 1", chances are that the content of it will be the background and proofs of it. However, I would change "where the 9s recur" to "where the 9s repeat forever" - it's shorter than "there are an infinite number of 9s". Supadawg - Talk 23:37, 20 June 2006 (UTC)

The intro does not need "fixing". Please leave it alone. This article is not about a definition, it is about a proof — and that's what it says! Nor is the point of the article merely to state what it is proving. The intro doesn't fit the form of those other articles because this is not that kind of article.

Someone also tried to change "0.999…" to $0.{\bar {9}}$, which is an ugly abomination that is inconsistent with the title and not at all helpful. It also has been proposed and discarded before.

There are very good reasons for the careful language currently used. We have an article on recurring decimals and want to stick to that language. More importantly, we do not want to say sloppy things like "the 9s repeat forever" or "infinite number of 9s", because it causes trouble. It raises false issues and confuses people.

Read the top of this talk page. This is not an article for bold edits. Those who need more convincing are welcome to read all the archives. If that's not enough, try the thousands (no exaggeration!) of posts to the sci.math newsgroup. --KSmrq^T 00:11, 21 June 2006 (UTC)

How exactly is that language sloppy? It is correct, no? Wikipedia is not a scientific journal; it should be as accessible to the lay reader as possible. In response to the comments you made about the intro, there is only one kind of article: an encyclopedic one, and this article needs to conform to it. I happen to think that this article should have a slightly wider scope than just the proofs, and look! it already does. See the section on "The proof in popular culture". I'm in favor of changing the title to "0.999..." and discussing all aspects of the number in the article, not just the proofs. All articles are open to major revisions at all times. Supadawg - Talk 00:30, 21 June 2006 (UTC)

For the record, I was the one who changed it to $0.{\bar {9}}$ merely because regardless of accompanying explanation, many would frown upon the concept of extension by ellipsis because it is extremely ambiguous. Not to mention, most would agree that the bar notation is the standard notation for representing recurring decimals. With this notation, no explanation is needed so the wording thereof needn't be discussed. Also, if you consider that inconsistent with the title, that's almost equivalent to considering the use of $\pi$ inconsistent with the title, pi. -- He Who Is^{[ Talk ]} 01:49, 21 June 2006 (UTC)

I'm too tired and cranky to discuss this now. But I'll try anyway.

As we have beaten to death previously, ellipsis is not at all ambiguous. It means something has been omitted. In fact, we have an entire Wikipedia article discussing ellipsis, the first sentence of which states that it is Greek for "omission". (Or consult a dictionary.) What has been omitted? The article carefully states what each time it is used. The ellipsis notation is familiar because it is used for the same purpose, omission, outside of mathematics; and it is used frequently, with the same meaning of omission, throughout mathematics.

The bar notation is hardly universal, and will be unfamiliar to many of our readers, especially the younger ones. For example, another way of indicating repetition is with dots. Introducing additional notation like this throws up an unnecessary obstacle.

Since the point of the article is to prove an equality involving 0.999…, we must formally define its meaning eventually to be able to proceed.

And finally, the inline TeX PNG is hideous. --KSmrq^T 03:21, 21 June 2006 (UTC)

OK, if you want to keep the intro as it is now, that's fine. One way cannot satisfy everyone... —Mets501 (talk) 13:02, 21 June 2006 (UTC)

"Not to mention, most would agree that the bar notation is the standard notation for representing recurring decimals."

I had never encountered that notation until yesterday. The notation I am used to uses dots above the numbers. I think that the use of the ellipsis should be kept for the moment. Raoul Harris 13:55, 21 June 2006 (UTC)

Trivial trivia about the linked joke

The Mathematical Gazette joke was published in 1954. The link, however, states that it is "[to be read in the voice of Fractured Fairy Tales]". Fractured Fairy Tales did not air until 1959.

Since there is no way we can edit the quote on the linked site, there's nothing we can do about this mistake ... but I wanted to mention it somewhere, just so I can feel smug. :-) Rpresser 16:05, 26 June 2006 (UTC)

Is this an essay?

This article sounds like an essay, as said in the summary: This article presents background and proofs of the fact that the recurring decimal 0.999… equals 1, not approximately but exactly. More precisely, the standard real number represented by 0.999… (where the 9s recur) is exactly equal to the standard real number 1.

It needs a rewrite. Computerjoe's talk 20:19, 28 June 2006 (UTC)

I completely agree, but this issue got lost in the debate over the series proof. If you want to rewrite it, go ahead (I might actually do it myself). Supadawg (talk • contribs) 21:05, 28 June 2006 (UTC)

No, it is not a lost issue, it is a decided one. The article was reviewed by numerous mathematicians and other interested parties just recently, as documented at the top of this page. It was highly regarded, overwhelmingly. It does not need a rewrite. Furthermore, the history of edits to the article is such that any such attempt will be treated as hostile. The problem is not with the article. --KSmrq^T 23:08, 28 June 2006 (UTC)

I checked the top of the page for discussion concerning the opening sentence. I didn't find anything. There is a section just above this one in which you, KSmrq, were the only party to object, against three who wanted to change it. That hardly qualifies as "numerous mathematicians", and I couldn't find an "overwhelming" concensus on anything. Once again, I ask you to cite your sources and provide specifics. Both Computerjoe and I say the opening sentence is unencyclopedic. Two beats one; I'm changing it, incorporating what was proposed earlier. Supadawg (talk • contribs) 23:41, 28 June 2006 (UTC)

An AfD keep is not a mark of quality. If you're looking for one, why don't you list the article on WP:GA? (But don't be surprised if it's failed for lack of references.) And even if the article were Featured, even that wouldn't be an excuse to treat all edits as hostile.

Now, when someone says "essay" on Wikipedia, they usually mean Original Research. I've scoured bookstores and libraries for material backing up this article, and I have yet to find it. The closest I've gotten is Pugh's Real Mathematical Analysis, which builds decimal expansions through Dedekind cuts in the first chapter, but Pugh explicitly forbids infinite strings of 9s. Other than that, I have never seen any published material that attempts to go straight from the rational arithmetic to decimal expansions; everyone builds up real analysis first, including series.

Of course, I would welcome sources that follow your program. Do you have one? Melchoir 00:11, 29 June 2006 (UTC)

I gave up on this page some months ago, so I'm not sure what you're asking for. Constructing the reals directly from decimal expansions is hard, which is why this is not often done. A paper which does it in great detail, with subtraction as a basic operation, is: De Bruijn, Defining reals without the use of rationals, Indag. Math. 38(2):100-108, 1976. He also explicitly forbits infinite strings of 9s, but it follows immediately from the paper that 1.000… − 0.999… = 0. This argument is given explicitly, but not in full detail, in Griffiths and Hilton, A comprehensive textbook …, page 395. The "limit proof" from the article is given in Baylis, What is Mathematical Analysis?, p.11-14, though for 0.4999… and this is not a fully rigorous book. I can provide more details if any of this might answer your question. -- Jitse Niesen (talk) 09:35, 29 June 2006 (UTC)

Since you have those sources already, do you think you can add them to the references section of the article? (Either citing them in the article or just adding them in the references section) Thanks —Mets501 (talk) 12:53, 29 June 2006 (UTC)

PS: I don't see why it needs a rewrite. To me, it doesn't read like an essay. It does need better sourcing, but that's something different from a complete rewrite. -- Jitse Niesen (talk) 09:48, 29 June 2006 (UTC)

I am also hopeful that the article won't need to be rewritten. One thing I'm not asking for is a source that constructs the reals directly from decimal expansions. The part of the article I'm concerned about constructs the reals as the usual two completions of the rationals, but then attempts to address decimal expansions directly in terms of those constructions. So, for example, in order to make the Order Proof non-OR, we'd need to find a source that actually says "In this way, every number in decimal notation determines a Dedekind cut, which is taken to define its meaning as a real number." I guess if all you know is the Dedekind cut formalism, this is the natural way to get a cut from a decimal expansion. But does anyone actually do just that? Likewise, does your Baylis reference actually say "Truncations of the decimal number b₀.b₁b₂b₃… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number"? Melchoir 19:00, 29 June 2006 (UTC)

Anyway, Jitse Niesen, I appreciate that you're helping out on the OR front, but until this is resolved I'm going to force the issue by applying {{OR}} to the section. Melchoir 20:05, 29 June 2006 (UTC)

IMO, this would be better on Wikibooks. Computerjoe's talk 16:27, 29 June 2006 (UTC)

I disagree, and I have a feeling that this will be the majority opinion here. Anyway, that wasn't the reason why I reverted to the Pmanderson version. The page had been vandalized by User:WAS 4.250 immediately prior to your tagging it. Feel free to add the {{wikibooks}} template again. Silly rabbit 18:35, 29 June 2006 (UTC)

An interesting idea, but I think we should wait until the dust settles on the Wikipedia article to decide. If too many proof methods are lost from any of the camps, then archived versions can certainly be added to Wikibooks. Melchoir 19:41, 29 June 2006 (UTC)

original research

It's not clear to me from the above discussion why we need an {{OR}} tag here. Is someone disputing the fact that the reals can be (and sometimes are) defined in terms of infinite decimal expansions? What exactly is it that we're looking for sources to back up? -lethe ^{talk +} 20:19, 29 June 2006 (UTC)

Sorry, I should have made it clearer that I've gone into detail on this issue in the "Is this an essay?" section above. We can certainly continue down here if you have further questions. Melchoir 20:22, 29 June 2006 (UTC)

I'm sorry, but it's still not clear to me, after having re-read the above conversation. Are you looking for a source which defines real numbers in terms of decimal expansions? It actually looks like you want a source which defines real numbers as Dedekind cuts defined in terms of decimal expansions, which seems quite bizarre to me. One needs either decimal expansions or Dedekind cuts, but not both. Is that really what you're asking for? It's simply not clear to me. Can you summarize clearly and succinctly which bit you think is OR, and what you're looking for a source to verify? -lethe ^{talk +} 20:35, 29 June 2006 (UTC)

No, that's not what I meant at all. Let's just focus on a single example:

• "In this way, every number in decimal notation determines a Dedekind cut, which is taken to define its meaning as a real number."

This definition has to come from somewhere! Making up new definitions is forbidden by WP:NOR. To verify the material, we need a reference that says:

• "Given a decimal expansion ... define the real number represented by that decimal expansion to be the following Dedekind cut: ..."

Okay? Melchoir 20:47, 29 June 2006 (UTC)

I have plenty of references which show the theorem that the field of decimal expansions is isomorphic to the field of Dedekind cuts. Saying that a decimal expansion defines a Dedekind cut is nothing more than an invocation of this theorem. That is not original research. I guess I still do not understand what your complaint is? -lethe ^{talk +} 20:58, 29 June 2006 (UTC)

It sounds like your references define decimal expansions to be mathematical objects in their own right forming their own model of the real numbers, which of course turns out to be isomorphic to the Dedekind cuts. The article defines their only meaning in terms of real numbers as Dedekind cuts right away. Does anyone else take that shortcut?

Anyway, if you believe that one of your references verifies part of the argument, by all means add it as an inline reference. Melchoir 21:20, 29 June 2006 (UTC)

...and if you don't have a citation on hand, please be patient and refrain from removing the {{OR}} tag. I'll restore it. Melchoir 21:37, 29 June 2006 (UTC)

The theorem is a standard result. This isn't original research. I will now remove the tag. -lethe ^{talk +} 21:41, 29 June 2006 (UTC)

There's a reason why I want a source, besides policy. What background goes into the theorem? How can you even develop decimal expansions as independent entities without addressing the equality of 0.999... and 1? Are you sure that one of these sources does it successfully? And if it's such a standard result, why can't we leave the tag up until someone lifts a finger to provide a citation?

Now, I'm not a lawyer, but WP:NOR is very clear on the only way to demonstrate that one is not doing original research: cite sources. So, someone, please do. Melchoir 21:55, 29 June 2006 (UTC)

A reference for decimal expansions as Dedekind cuts would be Hans von Mangoldt, Einführung in die höhere Mathematik, vol. 1, chapter 72. This book's obvious disadvantage as a reference is that it's in German. I don't know whether there exists an English translation. The advantage is that it must be among the very first textbooks (written 1911) to properly cover the subject (including the example 0.090909...=1/11, analogous to the 0.9999...=1 order proof). If there are no objections to having a foreign language reference, I will gladly add it to the article. --Huon 22:03, 29 June 2006 (UTC)

By all means, add it. Another useful reference may be Cohen, L., and Ehrich, G., The Structure of the Real-Number System. Van Nostrand, Princeton, 1963. I believe they do quite a bit with decimal expansions, but it's been a while. Silly rabbit 22:27, 29 June 2006 (UTC)

Damn, I don't read German, and Cohen isn't in my libraries. Well, yes, cite whatever you can, and we'll go from there. Melchoir 22:43, 29 June 2006 (UTC)

(In what follows, a number of times I have self-censored to produce a family-oriented version. My uncensored sentiments include a variety of colorful expletives. Readers are encouraged to supply these for themselves if they like.)

Since Melchoir brings up WP:NOR, let me quote from it:

• "No original research" does not prohibit experts on a specific topic from adding their knowledge to Wikipedia. On the contrary, Wikipedia welcomes the contributions of experts, as long as their knowledge is verifiable.

The interpretation of policy Melchoir seems to be advocating would bring Wikipedia to a standstill. We can't say anything unless we can find essentially those exact words in a reputable source, but we can't use those words because it would be a copyright violation.

If we take the language of a graduate text or beyond and make it accessible to a wider audience by using more familiar words and examples, or by a convenient or helpful or more modern alteration of symbols or notation, by his restriction we're committing the sin of original research. ("Original sin"?) If a source we find takes twenty pages to go from defining A to defining B, including details and subtleties that do not concern us, Melchoir would prohibit us from taking the "shortcut" of omitting the irrelevant material, and perhaps doing a direct substitution. I have a serious problem with that view!

Are we allowed to snip away the helpful bits? How about this: "We can define the real numbers using Dedekind cuts. We can define the meaning of a decimal expansion using real numbers. We can prove 0.999… = 1 using properties of the real numbers. QED." But if we include helpful bits it's OR?

Pardon me for being blunt, but I'm fed up with this nonsense. Nothing in the advanced proofs section is in the least bit controversial among professional mathematicians, including defining both reals and decimal expansions in terms of Dedekind cuts. What's particularly galling is that Melchoir surely knows that. In other words, this OR tag serves only one purpose: disruption. --KSmrq^T 00:52, 30 June 2006 (UTC)

Way to assume good faith there. The OR tag here serves the same purpose it always serves: encouraging users to cite their sources, which we should have been doing all along. And, where sources do not support the article, revising it.

I am not at all convinced that published mathematicians have given the same definitions found in the article. You're going to have to call me a liar if you want to argue with that. And if a book actually defines A as C, which after much work turns out to be equivalent to B, then it is dishonest, logically backwards, and harmful to the reader's understanding to claim that A is defined as B. In this article, of all articles, that mistake must not be allowed. Melchoir 01:35, 30 June 2006 (UTC)

If all you want is references, why don't you use Template:unreferencedsect instead of the rather inflammatory Template:OR? Though I disagree with such a template, it would still be more in keeping with what you claim your goals are, and therefore might be seen as less disruptive. -lethe ^{talk +} 03:35, 30 June 2006 (UTC)

Just because I hope that references can be found to back up the article doesn't mean that I'm betting it's possible. If I were confident that the section could be proven not to be OR through a few citations, then I'd add {{unreferenced}}. Since I suspect that the necessary citations do not exist, and that the section will have to be approached from a different angle, I added {{OR}} instead. This suspicion of mine arises from a significant amount of research and thought, and I'm really sick of having my goals called into question. Can we please just focus on the content for a while? Melchoir 04:55, 30 June 2006 (UTC)

Well, we've discussed the content. I don't know what else there is to discuss, we seem to disagree with no road forward to resolve. I guess the only thing now is to try to get more opinions, let consensus decide. I don't think it qualifies as OR, nor does it even need a source, to rely on sequences of digits defining Dedekind cuts. -lethe ^{talk +} 05:04, 30 June 2006 (UTC)

Interesting. I say everything needs a source. If a reader finds something on Wikipedia and can't determine where it came from, we've failed that reader. Melchoir 05:33, 30 June 2006 (UTC)

[3] Ah, thanks, Jitse Niesen! QA37.2... I don't know if I've looked around there, and I'll have to track that book down. But on the OR issue, I'm afraid the princess is still a couple of castles away. Verifying the definitions is a valuable step, but what about the proofs themselves? Specifically:

• Where do we find the proof that the Dedekind cut generated by {⁰⁄₁, ⁹⁄₁₀, ⁹⁹⁄₁₀₀, ⁹⁹⁹⁄₁₀₀₀, …} is the same cut generated by {1}? Where is the justification for the "or equivalently" at the end of the second paragraph? Whose idea was it to include an induction argument, and where should the curious reader look to find the rest of the argument?
• Where do we find the proof that (⁰⁄₁, ⁹⁄₁₀, ⁹⁹⁄₁₀₀, ⁹⁹⁹⁄₁₀₀₀, …) is co-Cauchy with (1,1,1,1,...)? Whose assertion is it that the last step is "clear by inspection", and where should the reader look for details?

Perhaps I need a disclaimer here before I get yelled at: Yes, I'm serious. Of course the proofs can be made to work, and conceivably every sentence could be individually verified. But that's not what I want, and given WP:NOR#Synthesis of published material serving to advance a position, it's not what policy demands. The question is simple: have these particular proofs been published elsewhere or not? (And if not, what has?)

So, good work everyone, and I'm putting up the tag again. Melchoir 10:01, 30 June 2006 (UTC)

Opus citus, at least for the Cauchy sequences. Let me quote the book so that you don't have to go to the library.

"Finally, we consider the case of an infinite decimal terminating in 9's. To be precise, let $\{\delta _{n}\}$, $\{\delta '_{n}\}$ (where $\delta _{n}=a_{0}.a_{1}a_{2}\cdots a_{n}$, $\delta '_{n}=a'_{0}.a'_{1}a'_{2}\cdots a'_{n}$) be two infinite decimals such that

$a_{n}=a'_{n},\qquad n<m$

$a_{m}=a'_{m}+1,$

$a_{n}=0,\quad n>m$

$a'_{n}=9,\quad n>m.$

Then it is plain that $\delta _{n}=\delta '_{n}+z_{n}$ where

$z_{n}=0,\quad n<m,$

$z_{n}={\frac {1}{10^{n}}}\quad n\geq m.$

Plainly, $\{z_{n}\}\in \Sigma _{N}$, so that $\{\delta _{n}\}$ and $\{\delta '_{n}\}$ determine the same real number (which is, of course, rational, since $\delta _{n}$ is a terminating decimal)."

Here, $\Sigma _{N}$ denotes the set of null sequences (sequences that converge to zero).

For me, this is precisely the proof in the article. The only difference is that the article restricts to a particular case.

Furthermore, you seem to be the only one who thinks that NOR applies in this way. Furthermore, Mathematics seems to say that there is no consensus. So, I replaced the OR tag by Template:Fact after the one paragraph that still seems to be left. -- Jitse Niesen (talk) 11:36, 30 June 2006 (UTC)

Looks perfect! But that quote does no good for our readers if it just sits here on the talk page and we ignore it in the article. Ergo, [4]. Little help? Melchoir 15:55, 30 June 2006 (UTC)

Oops... I missed that. Ok, I won't yell at you now. ;-) Silly rabbit 16:03, 30 June 2006 (UTC)

Uh, right! Well, we still ought to cite the proof quoted by Jitse Niesen above, and I've trimmed the "must be" language. Ultimately, for 1/n->0, it would be nice to point to a proof simpler than Rudin's general theorem; it's really just Archimedes again. Melchoir 17:51, 30 June 2006 (UTC)

I've added a more basic reference, as per your request. It uses Archimedes without explicitly mentioning it, but otherwise it is fairly solid. Silly rabbit 19:22, 30 June 2006 (UTC)

Okay, I'll combine the ref tags. We wouldn't want to give the impression that the step is harder than it is! Melchoir 19:38, 30 June 2006 (UTC)

Melchoir, I'm not sure what you mean with this [edit you mention above. The quote "Finally, we consider the case of an infinite decimal …" is from Griffiths & Hilton. I hope my last edit addressed your concern. -- Jitse Niesen (talk) 05:01, 1 July 2006 (UTC)

Yes, I see now, thanks. I wasn't going to make a big deal either way, because the last couple of references are available at my library. I haven't yet gotten around to checking them out to sort out the details and page numbers, but I'll probably find the time tomorrow. Melchoir 06:18, 1 July 2006 (UTC)

Okay, Griffiths & Hilton is the perfect resource for Cauchy sequences, providing a citation for every step in the development of that section.

But, sadly, Apostol has nothing to do with Dedekind cuts. He never even mentions them. Rather than construct the real numbers or any other set of numbers, he relies on an axiomatic development, including the ordered field axioms and the completeness axiom for the existence of suprema. The cited theorem claims only that any real number can be given finite decimal approximations to any digit. The theorem's proof does not build either a Dedekind cut or a supremum out of a set of arbitrarily fine approximations, as our proof does. He uses the Archimedean property and induction only implicitly, and not in order to prove that 0.999... = 1. In fact, he never proves that or any similar statement. The closest he ever gets to our development is the single sentence "It is easy to verify that x is actually the supremum of the set of rational numbers r1, r2, . . . .", and even after making that observation, when he finally defines what an infinite decimal representation means in the next section, there are no such supremum used (let alone a cut). When Apostol eventually mentions that some numbers can have two representations, he is actually talking about two different definitions of what a representation means, not an equality in one unified development.

Since Apostol is irrelevant to the development here, I am removing that ref and putting up everyone's favorite tag. At some point we should consider just using either suprema or nested intervals. Melchoir 21:22, 3 July 2006 (UTC)

I beg to differ with your judgment to remove the Apostol tag on the relevance grounds. The inductive argument used in the proof is identical to Apostol's inductive construction of the decimal expansion. (Which was, I take it, the only "missing unreferenced link" in our proof.) Anyway, as it is clear that there is no original research in this article, I am removing the tag. Silly rabbit 21:47, 3 July 2006 (UTC)

Apostol is proving that for every real x and every natural n there exists a finite n-digit decimal expansion d_n such that

$d_{n}\leq x<d_{n}+10^{-n}.$

Apostol's proof of that theorem says nothing about 0.999... or upper bounds of such finite expansions. It's so irrelevant to upper bounds of finite expansions that he explicitly mentions them in a separate remark without proof! Do you deny any of this? Melchoir 22:23, 3 July 2006 (UTC)

You know what? I give up. Go find your own references. Perhaps Wikipedia would be better off without me. So long. Silly rabbit 01:36, 4 July 2006 (UTC)

No need for that, you've been very helpful so far. And let me assure everyone that I am trying to find my own references. I exhausted entire shelves of books in the QA37.2 and QA300 ranges today, and I still have yet to find a reference that backs up the "order proof" section. Now, I probably never would have found Griffiths & Hilton, so it's possible that another miracle book is out there, and I don't know where to find it. In that case, I need help, and that's what cleanup tags and discussion are for. Melchoir 02:20, 4 July 2006 (UTC)

Hey, that's great. If you find an awesome reference, we'd love to have it in the article. In the mean time, maybe you could lighten up on this OR business. This crusade is clearly not so good for the community's patience, as you may perhaps notice. -lethe ^{talk +} 02:37, 4 July 2006 (UTC)

How would you like me to deal with original research? I'm just about giving up on finding a reference, and I don't know who else is looking. Is someone going to pledge to do their own search? Can we set a one-month deadline for verification? Should I really become a crusader and just start deleting things now? (Policy says that I can, and we all know I wouldn't be the first. Maybe it's better to get it over with and move on.) How does this get solved? Melchoir 02:55, 4 July 2006 (UTC)

How I would like you to deal with this is to chill out. No month deadlines, no crusades, no deleting campaings, no ultimata, no pledges. Just relax. You think it needs a source. You've said so, and you've even gone to great lengths to find one. Your efforts are important and appreciated, but there's no need to impose your opinions above the disagreements of others. -lethe ^{talk +} 03:13, 4 July 2006 (UTC)

I don't see how policy could be any clearer on its need to be applied over the disagreements of others. But okay, let's say I give up. And then what happens? Melchoir 03:24, 4 July 2006 (UTC)

Then people spend more time writing and less time arguing. -lethe ^{talk +} 03:54, 4 July 2006 (UTC)

What happens to the original research? Melchoir 04:07, 4 July 2006 (UTC)

Hello? Melchoir 07:10, 5 July 2006 (UTC)

What happens to it? Well, if you chill out, then nothing happens to it. It stays on in the article, a product of the community consensus that it is not, in fact, original research, despite your claims. This may be the end result even if you don't chill out, through reverting and arguing, but there more acrimony might be generated in the process. Hence my suggestion. -lethe ^{talk +} 07:18, 5 July 2006 (UTC)

If stasis is the result then I will not give up. You can't cure a section of original research by willing it so and crying consensus; we need a citation that uses this method of proof. No more, but no less. I can't believe I even have to explain such a basic concept.

Months ago, I used to be confused by the pathetic story told at Wikipedia:WikiProject Mathematics#Featured and former featured articles. I didn't understand: how can such an energetic and talented community, commanding such a broad territory, produce so few quality articles? How is it that 2006 has been completely dry? After reading the removal votes and the articles that remain featured, I think you've given up. Wikipedia started taking the content policies seriously and demanding references, and the Math Wikiproject couldn't move with the times. It's so much easier to pretend this is Wikibooks and rail against any of the laity that cry foul. Well, guess what: Wikipedians stand behind our ideals now. That means you don't get to editorialize, and you don't get to promote your favorite approach to an issue. It means you don't propose new solutions to old problems.

The mathematicians around here have forgotten what a Featured Article looks like. I intend to show you. I can feature this article. I can give you your first little golden star in seven months, and the first featured math article to have more than a dozen citations, ever. It will be the only featured math article that would pass by today's standards. I hope some of you are excited about this idea. But it'll be a lot harder if every time I try to take the article a step away from Wikibooks and a step toward Wikipedia I have to go ten rounds with the old guard. I can't draw this much fire just for removing an inappropriate ref tag. So please, whoever isn't serious about crafting a Wikipedia article, you chill out. Melchoir 08:44, 5 July 2006 (UTC)

Please calm down, both of you. (I do not regard a lack of Featured Articles as a discredit, btw: far too many of them have been advocacy-driven NPOV violations with lots of pretty pictures). Septentrionalis 17:51, 5 July 2006 (UTC)

Okay, without making a value judgement, then, do you agree that the FA situation is a symptom of a divergence between the standards of the math community and the standards of the rest of Wikipedia? Melchoir 18:47, 5 July 2006 (UTC)

I think most of the FA situation is due to a divergence between WP:FAC and policy; this may well be a symptom of desperation: FA has to come up with a new article a day, whether there are any or not.

As for the lack of mathematical FA's: how many mathematical articles are nominated? how many are rejected as hopelessly abstruse, even when this is unavoidable? and how many are rejected because the article doesn't have the fine polish sometimes given by a fanatic to his Proclamation of The Truth?

All those being discounted, there is a cultural difference between mathematics and, say, history. In mathematics, if you can prove something, and you didn't take it from some particular paper, you don't need to cite it; your reader can reconstruct the proof. Why shouldn't Wikipedia vary with its material? Septentrionalis 19:11, 5 July 2006 (UTC)

The FARCS I read had nothing to do with being abstruse or unpolished; they were simply unreferenced.

At a certain level, mathematics deals with undeniable truths. These truths do not need citations or any other kind of evidence to be believed by the reader. If Wikipedia were just a big, unusually well-cited textbook meant to impart The Truth onto young impressionable minds, then yes, we might tolerate a bit of unverified original research in the mathematics articles for that reason.

But convincing the reader of a given mathematical statement is not the only reason to verify one's proofs. It isn't even the best reason! By offering a mathematical proof we are asserting that this is how mathematics is done. That this very argument is notable and important enough to be repeated, and a careful review of the literature will find it in the wild. By citing that literature we reassure the reader of those points, and we recommend further reading where surrounding results can be found, often including hundreds of pages of background. We provide historical and personal context. And the reader shouldn't have to reconstruct the proof. We are not in the business of assigning exercises to bright students. That's a job for Wikibooks.

Neither was Laplace, yet his use of "obvious" has become notorious proverbial. Septentrionalis 18:28, 7 July 2006 (UTC)

As for cultural differences, mathematics is not so entirely different from other fields of human study. There are conflicts, both in the way mathematics is done and in the way it is taught. There are partisans of proving everything from axioms, and there are partisans of proving everything by construction. There are methods that exploit what a mathematical object is, and there are methods that exploit how an object interacts with others. There are continuing disagreements over the meaning of rigor and the amount of detail required in a proof. There are countless ways of ordering one's presentation: does one deal with examples of a subfield as soon as possible, or does one defer them until one builds up the standard results of the subfield? What is the role of a simple theorem that gets eclipsed by a stronger but less elegant result? How much of a theorem's background do you have to explain before you're "allowed" to use it?

If we offer novel methods of proof, correct or not, we become participants in these debates. We become the historian who has a brilliant new way of looking at the facts. That's why mathematics isn't exempt from the three big content policies, and why it never will be. It's why I'm asking everyone to please show a little respect for those policies. They're there for a reason. Melchoir 21:07, 5 July 2006 (UTC)

We're not offering "novel methods of proof". In some cases, we're offering "proofs" considered obvious by all reputable sources. Those would be hard to cite. — Arthur Rubin | (talk) 23:30, 5 July 2006 (UTC)

Has anyone who isn't a Wikipedia editor ever thought it appropriate to express 0.999... as a Dedekind cut, compare it to the cut generated by 1, and conclude that they are the same? Has any reputable source really remarked that such a strategy is obvious? In fact, is there any reference to the strategy in the literature at all? Melchoir 23:44, 5 July 2006 (UTC)

"Has any reputable source really remarked that such a strategy is obvious?" Is Arthur Rubin not a reputable source?  :) (That's an article - not a user page...) Lunch 18:00, 7 July 2006 (UTC)

And admirably free from vanity edits... Septentrionalis 18:30, 7 July 2006 (UTC)

Of course I don't mean to attack anyone's reputation. In our capacity as Wikipedia editors, none of us are sources of any kind. So {{cite user}} and {{cite talk}} remain redlinks. Melchoir 19:31, 10 July 2006 (UTC)

division by zero?

If .999 = 1, it follows then that 1/(1-.999...) = 1/0, so it is possible to divide by zero? //// Pacific PanDeist * 05:40, 3 July 2006 (UTC)

You might be more interested in the Division by zero article. As for .999..., we can also make the observation that since sqrt 4 = 2, it follows that 1/(2-sqrt 4) = 1/0; it doesn't seem too helpful to me. Am I misunderstanding you? Melchoir 06:22, 3 July 2006 (UTC)

[Edit conflict] See division by zero. Your argument is absurd in any case; it is the same as saying, "If 1 = 1, it follows then that 1/(1-1) = 1/0, so it is possible to divide by zero?". Specifically, why did you decide it was possible to calculate 1 / (1-0.999...)? This is just as possible as calculating 1 / 0 - which depends on our definition of division. -- Meni Rosenfeld (talk) 06:25, 3 July 2006 (UTC)

To be precise, it is possible to divide by zero in some cases. For instance, the Projected reals allow this because they add an unsigned infinity element, so it can represent both positive and negative infinity, both limits to zero of 1/n. -- He Who Is^{[ Talk ]} 03:37, 4 July 2006 (UTC)

If my argument is "absurd" as you (middle response) say then I don't know it as I am not a mathematician but an artist... I had always thought that 1/0 = ∞ anyway, but I figured that .999... was only one because its variance from 1 was an infinitely small number so that any difference between the two can never be calculated, and therefore that 1/0 and 1/(1-.999...) were both infinite, but perhaps different types of infinite (like one was just there, while the other one you could start chasing down a long path towards infinity before realizing the chase was without end)!! Well, thank you to everyone for responding, I'll check out that dividsion by zero article!! //// Pacific PanDeist * 05:48, 4 July 2006 (UTC)

The article on division by zero gives a perfect reason why its not infinity. Negative numbers divided by n aproach naegative infinity and n aproaches zero. -- He Who Is^{[ Talk ]} 06:05, 4 July 2006 (UTC)

Well, since 0.999... = 1, and thereby 1 - 0.999... = 0, it is of course impossible for 1 / (1 - 0.999...) to be different from 1 / 0. Both (if, as in the real projective line, they are considered to equal ∞) are the type of infinity that "is just there" - the idea of chasing something down an endless path is a common misconception regarding these issues, and is generally something we wish to avoid in order to keep things mathematically accurate. -- Meni Rosenfeld (talk) 13:28, 4 July 2006 (UTC)

Since this discussion does not belong on the article talk page, I replied on the user talk page. In the interests of keeping traffic on this page focused on improving the article, please redirect any further responses. Thanks. --KSmrq^T 19:54, 4 July 2006 (UTC)

Thank you, I'll keep my responses there. //// Pacific PanDeist * 07:04, 5 July 2006 (UTC)