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0.999...

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Nomination I feel that this article should be nominated because it extremely extensive, provides a wealth of information, and we haven't really had a math-related FA in a while. --Captain538 02:15, 28 September 2006 (UTC)[reply]

  • CommentSupport I don't think my math skills are up to doing a peer review of this article, but it appears well referenced, and it is definitely well-written, charming, even. I think I'll stick around and see what happens, but I'm disposed towards support.--Paul 02:30, 28 September 2006 (UTC)[reply]
  • Comment Wow! Okay, let's see where this goes. I'd especially like to hear ideas about the article's organization, length, and level of detail. I don't think the prose is ready yet, but if anyone has constructive criticism, I'll get right on it. The content of the article is almost complete as far as I'm personally concerned, although there are a few more small history- and education-related details I've yet to add. Melchoir 02:37, 28 September 2006 (UTC)[reply]
  • Support, but sort out 1. the position of the calculator picture (in my - Firefox - display it covers some of the text) and 2. the one "citation needed" tag (it is about the number of topics on a maths newsgroup). Excellent work otherwise, however, and really nicely referenced. Batmanand | Talk 09:57, 28 September 2006 (UTC)[reply]
OK I have now tried three times to upload a screenshot of it but the Upload File thing keeps screwing me over; basically, it is covering a small portion of the text above it in my browser. Anyone having this problem? Batmanand | Talk 13:46, 1 October 2006 (UTC)[reply]
Hmm... you could try applying some of the HTML/CSS tricks and templates at Wikipedia:Picture tutorial and see if one of them works in "Show preview". Melchoir 16:59, 1 October 2006 (UTC)[reply]
Eh, the calculator picture has been removed now, and I'm tired of promoting it. Whatever. Melchoir 16:07, 4 October 2006 (UTC)[reply]

There is at least one other alternative demonstrated at eswiki using the font tag: "0,<font style="text-decoration:overline">9</font>" generates "0,9". Is that good enough? Melchoir 17:35, 28 September 2006 (UTC)[reply]

  • One objection to the title is that who is going to search for "0.999... = 1", or even worse: "0,9 = 1"? Is there a name for this identity? Also, are the notes in the correct format? They refer to authors in the references. Don't we suggest a full cite the first time a reference is used, alowing reversion to author & page number in subsequent cites? --Paul 18:10, 28 September 2006 (UTC)[reply]
  • I think SteveRwanda is only suggesting that we insert another notation in the opening, not that it replace the existing notation or the article title. I'm not aware of a name for the identity, alas. As for the notes, I think there are some advantages to the current format, especially when there are so many notes and so many references. Melchoir 18:25, 28 September 2006 (UTC)[reply]
  • Object (but I'm something of an ignoramus when it comes to maths, so I'm happy to be overridden/ignored). First of all, this seems to me to be a fork of Recurring decimal. Aren't they one and the same? Secondly, I'm not sure this level of depth is even within Wikipedia's scope. Is such deep analysis the domain of an enyclopedia or should it be on Wikibooks? --kingboyk 16:51, 28 September 2006 (UTC)[reply]
    • Well... In mathematics, 0.999… is unique among recurring decimals (that start with 0) in that it represents an integer; this fact has an application in the first paragraph of 0.999...#Applications, and one author also argues that it is relevant psychologically; see the first bullet point in 0.999...#In education. The proofs are unique in that many of them would require substantial modification to work for other recurring decimals, and the ambiguous-subdivision proof wouldn't make sense. Only slightly more generally, recurring decimals with repeating nines are unique because they represent numbers which also have a second, more standard representation. Educationally, 0.999… is different from other recurring decimals because students don't accept it; it may be the single most misunderstood concept in mathematics. And for Wikipedia's purposes, 0.999… is special just because there exist sources that single it out for study. It is neither a trivial matter, nor does it involve original research, to specialize one's focus to the single recurring decimal 0.999…; the sources do it for us. I realize that Recurring decimal needs some help, but ideally the relationship between the two articles could follow Wikipedia:Summary style. Melchoir 18:06, 28 September 2006 (UTC)[reply]
    • I'm not sure what parts of the article you're worried about for depth. Generally, my feeling is that Wikipedia should be as in-depth as possible while still following the content policies. This article isn't the result of crazed editors taking a simple concept and running with it; the ideas it summarizes are a cross-section of what's actually out there in the literature. And the article doesn't bleed dry its subject for details, either; there's still a lot more one can learn by following the internal links and looking up the references. Melchoir 18:17, 28 September 2006 (UTC)[reply]
  • Support well written article Trashking 21:20, 28 September 2006 (UTC)[reply]
  • Comment, I don't like the ibid-like references (I know it is not the same, but for a casual user it is the same). When a user is reading the article and comes to, in example, the Applications section, reference 42, he will click it and be carried to a line that says Leavitt 1984 p.301. He will try to go upwards, finding nothing, and then downwards, until finding The College Mathematics Journal. By the time he finds the reference, he has lost his reference number. I don't like the restricted-access texts, but I don't object them as they are acceptable. -- ReyBrujo 05:01, 29 September 2006 (UTC)[reply]
  • Support This is a well referenced and well written article. Wikipedia needs more articles like this. Mercenary2k 08:01, 29 September 2006 (UTC)[reply]
  • Object - A few things spring to mind:
    • As I said above, the proper mathematical notation for this number is 0.9 in American usage and in British (see here for a reference for this). I know the latter one looks bad and can ruin sentence structure but official titles should appear somewhere in the lead. — OK, I've altered it slightly again, and I'm happy with this. SteveRwanda 13:15, 30 September 2006 (UTC)[reply]
    • Not sure the lead is long enough, and probably needs to summarise the content of the article more thoroughly.
    • The prose needs a copy edit - it reads more like a lecture than an encyclopaedia article at present. Phrases like: "a sometimes mysterious concept" don't really belong here.
      • comment I like the prose, its light and fun. Much of Wikipedia is seriously dull. A bit of irreverent sparkle is a good thing.--Paul 16:26, 29 September 2006 (UTC)[reply]
      • That's never been a terribly popular sentence; I've just removed it. The tone of the article is admittedly inconsistent, so I'm not sure exactly which part you think sounds like a lecture. Could you please point out a specific section or sections that are bad, and possibly a section you think is good? Melchoir 16:54, 29 September 2006 (UTC)[reply]
    • Why are there so many non-inline reference books listed at the bottom? Does each one offer something that the others don't? It would be nice to see page numbers as well, since most of them are general books in which 0.999... is presuambly only a small section.
      • Yes, the books do offer different material. This might be surprising, given that so many of them are intended for the same purpose, but there are many different approaches to the real numbers and even more approaches to decimal expansions. This article attempts to address all of the approaches that are relevant to 0.999…, but a given textbook is usually more interested in getting to the truth as quickly as possible by its own preferred route. As for page numbers, I guess I could do that... is it important to you? Melchoir 17:02, 29 September 2006 (UTC)[reply]
        I may need a second opinion on this... It seems to me that these general references should be a list of books which anyone wanting to know more about this subject can go and borrow from their local library. As it stands they're going to end up with an enormous pile of books, without any clue which part of them to read in order to further their understanding of this mysterious 0.999... business. I think further reading on the in depth stuff, such as Cauchy sequences, can be obtained either from that article or from the inline refs. SteveRwanda 13:31, 30 September 2006 (UTC)[reply]
        Oh, we appear to have a misunderstanding. The sources listed under "References" all have inline citations in the article. Melchoir 06:50, 1 October 2006 (UTC)[reply]
        Ah, I see. I hadn't realised that. It seems a slightly odd way of doing it as it doubles the amount of space the references take up, but no matter. I'll have another look at the lead and the prose when I have some time... Cheers — SteveRwanda 17:28, 2 October 2006 (UTC)[reply]
  • Anyway, I think by and large the article is good and I'm still very new in the world of analysing articles so I will be very happy to defer to others' opinions and withdraw these comments if people feel they're unjust! Cheers — SteveRwanda 10:59, 29 September 2006 (UTC)[reply]
  • StrongObject. Despite the hard work that went into this article, I find it quite unsatisfying: it seems to be written by mathematicians for the entertainment of other mathematicians, and at a level appropriate for advanced college students if not graduate students. Given that, the whole article concept is especially odd: people able to read an article at this level are already quite familiar with the 0.999... = 1 idea. The article goes on and on about how this concept fits into a variety of areas in mathematics, without ever giving an idea why that is important to explore. It feels to me like this article is really about the convergence of sequences and limits, only with the example 0.999... = 1 filled in everywhere. This article is, at best, an entertaining article for math geeks (like me, BTW -- I found the section on p-adics rather interesting, as I've always been a little shaky on p-adics)... and at worst, a rambling essay. At the very least, this article needs a major restructuring, and the real topic needs to be made clear, and all the text needs to be clear in its relevance to that topic. To me, it's very far away from being a featured article. Sorry. :( Mangojuicetalk 15:44, 29 September 2006 (UTC)[reply]
    • Yeah, parts of the article are hopelessly esoteric. But every section of the article has been ordered to make sure that those parts occur last. The "Proofs" arc, consisting of the first three sections on the meaning of 0.999… and why it equals 1, starts with "Digit manipulation" at a middle-school level; "Infinite series and sequences" is at an advanced high-school level; "Nested intervals and least upper bounds" starts into university territory; and finally "Rational constructions" demands a mathematics degree. In "Generalizations", the first paragraph is obvious; the second is clear to anyone who's been introduced to alternate bases; for the next, non-integer bases occur in recreational mathematics; and we only get into topology at the end. "Other number systems" is just intrinsically hopeless; sorry! "Applications" begins by messing around with 1/7; the next section includes the magic word "fractal", so at least it should pique the laity's interest; and again the most abstract bit is at the end. "Skepticism" requires almost no background at all. There's a lot of material that's understandable for readers who aren't familiar with 0.999… equaling 1. Melchoir 17:26, 29 September 2006 (UTC)[reply]
    • About giving an idea why it is important to explore the subject: perhaps I can improve the lead section in that direction... Melchoir 17:33, 29 September 2006 (UTC)[reply]
    • The real topic of the article is simply everything one can say about 0.999…. It's long and varied; what part do you think needs its relevance to 0.999… clarified? Melchoir 17:36, 29 September 2006 (UTC)[reply]
      • Let me put it another way. Everything that's written in the article is relevant to 0.999... = 1. But some of it (a lot of it) seems like a digression: the point of the section about the p-adics, for instance, is to talk about the p-adic integers, and how they relate to 0.999... = 1. To give an analogy, consider an article about George Washington that spent most of its space talking about other people for whom George Washington was important. All the information is relevant, but the article has ceased to really be about George Washington: it's more like an article about George Washington as a role model, or something. That's what I mean when I say I'm not really sure what this article is all about. Maybe a different title? I don't know. Mangojuicetalk 19:14, 29 September 2006 (UTC)[reply]
Okay, I have two answers for this objection: First, the mathematics answer: Nothing in mathematics is interesting or worth investigating by itself. Ideas gain meaning through their relationships with other ideas; objects become useful through their interactions with other objects. (In fact, there's a tradition, at least in the field of algebra, that the best way to investigate an object is through its interactions.) 0.999… is a perfect example. By itself, there's nothing to say except "#redirect 1 (number)". But if you're going to have an article titled "0.999...", this is it.
My second answer is editorial: that even the digressions in this article are interconnected, and therefore not really digressions. The p-adic bit immediately connects back to its parent section by discussing 1-0.999…, and it mentions educational issues twice: 0.000…1 and the seventh-grader. The brief mention of the definition of the p-adics contrasts with the Cauchy sequence section above, and two of the three proofs that …999 = -1 are variations on proofs that 0.999… = 1 given above. Melchoir 20:00, 29 September 2006 (UTC)[reply]
(1): By reductio ad absurdum, that's true. But in reality, it's not that hard to explain why topics within mathematics are interesting. See Group (mathematics) for what I consider a good example of laying out the context of the mathematical topic and giving an idea to a lay reader why it should interest them, and why it has interested others. There may be a limit to how well that can be done in this case, but I do think it can be done better. (2): I don't mean to say the article doesn't have coherence. I think it does. I think, though, that the article isn't exactly about "0.999...", it's more about concepts in mathematics related to "0.999...", especially real numbers and convergence of sequences. An article on the topic of "0.999...", I would expect, would describe the main question, and give the importance of it, and some of the history of attempts to understand the issue, and other issues closely related to it. The digressions have to be controlled: as an example, the "Dedekind cuts" subsection works out the concept of Dedekind cuts to the point where it feels like Dedekind cuts is the topic (though there is an article on that topic already, and it's already been linked), whereas it might suffice to say, simply, that the numbers 1 and 0.999... define the same Dedekind cut (incidentally, I think there's a bit of an error there: the description lapses back to thinking of 0.999 as a Cauchy sequence, doesn't it? For Dedekind cut's, I'd say that any rational number less than 1 is also less than .999..., and (obviously) vice versa.) Mangojuicetalk 20:48, 29 September 2006 (UTC)[reply]
I would love to be able to write about the importance of 0.999… and its history. Unfortunately, it isn't important, and it has no history. I'm not kidding. Mathematicians don't care about 0.999…. It isn't good for anything. Even for the scant applications listed in the article, there are much, much better ways to prove the results. No one knows who first wrote down "0.999…", or who first decided it was 1. There has never been a disagreement between mathematicians on the issue. It was never an unsolved problem, and it hasn't motivated any research. I say all this not because I lack an imagination, but because I have dug through three research libraries, every possible search on every search engine I know, and countless bookstores, and there is simply nothing else to say about 0.999…. There are a couple minor variations on the cited ideas, but what you see is the length and breadth of the topic.
I'm sorry. We can shuffle content around, we can summarize or elaborate ideas, and we can build a much better lead section, but objections about missing content can't be addressed.
About Dedekind cuts, please try reading that section again. The first sentence is the only sentence that builds the idea of Dedekind cuts, and it's already summarized to the point where it's mathematically inadequate. The first five sentences read: "Dedekind cuts. 1. Decimals. 0.999…. 0.999… = 1." It gets to the point really quickly. If we made it any briefer, it wouldn't be a proof. Melchoir 21:26, 29 September 2006 (UTC)[reply]
I've expanded the lead section; I hope it speaks to you somehow. Melchoir 22:20, 29 September 2006 (UTC)[reply]
You're right, the Dedekind cuts are introduced rather minimally, if you are to give a proof (though I still think the proof takes 0.999... to be defined via a Cauchy Sequence. A proof really relying on Dedekind cuts would be to say that if (a/b) < 1, then note that 1-(a/b) >= 1/b > 1/10^b, so (a/b) < 1 - 1/10^b which is clearly less than 0.999... since it's less than a terminating version.) Anyway, I'm striking the "strong" part of my objection: what I see here is more a failure of being clear in the writing and structure than a lack of material. I do think the connection between the "Applications" and 0.999... is clear, but it doesn't seem to relate much to the 0.999... = 1 question, so it seems a bit tangential. Similarly, the section on p-adics (cool as it is) also seems a bit tangential, though this article presents a good opportunity for introducing the p-adic numbers, which don't get enough exposure in general. I like the new lead a bit better, but I think we should lose that "has been taught in textbooks for centuries" bit unless there's a direct source on it (it sounds vaguely peacockish to me). Bring the Skepticism section forward, it helps motivate why all those proofs are given. And I think Zeno's paradox should be mentioned earlier; it gives a bit of historical context (it might be worth giving a brief history of the development of understanding of convergent series). Also, the narrative needs to be connected a bit better. It seems to me that the important thing is for the proofs to address different notions of understanding of what 0.999... is -- is it simply an infinite decimal? The limit of a sequence? The article tries to do this, but it isn't that clear, and the section headers speak more to mathematicians than the less informed: for instance "Rational constructions" might be better as "Concepts of real numbers". Two nitpicky points: the equation in the p-adics section should be redone in full-equation style, it shows up funny on my computer. Also, surely there is an article somewhere on nested intervals and the nested interval theorem, isn't there? Those shouldn't be redlinks. Mangojuicetalk 18:00, 30 September 2006 (UTC)[reply]
Now you're speaking my language! I can act on most of these suggestions, but it'll take some time and probably some discussion on the talk page. Melchoir 06:49, 1 October 2006 (UTC)[reply]
On the one hand, no, the Dedekind cut proof is perfectly okay, and it works as stated without any sequences, limits, or epsilons. On the other hand, I like the simplicity of your idea better, so I'll put it in. Richman never specifies exactly how one should finish off the equality of the Dedekind cuts, considering it obvious, so there's plenty of room to manuever. As an added bonus, the whole argument now fits in a single paragraph. Melchoir 22:58, 1 October 2006 (UTC)[reply]
I've reordered and renamed some sections; hopefully this deals with Applications being tangential, Skepticism being too late, and the section on constructions being opaque. The connecting prose is now weaker, not stronger, but it should be easier to fix. Melchoir 17:04, 2 October 2006 (UTC)[reply]
I've addressed the issues relating to history with these edits. I think the body echoes the intro enough that the latter doesn't need citations of its own. Melchoir 05:05, 4 October 2006 (UTC)[reply]
(resetting indents) I think things look better now, but I'm still concerned about two things: I think the p-adics section is a digression, and I think the applications section is either actually off-topic, or its connection to the 0.999...=1 issue needs to be clarified. Section titles need to do a better job of setting up the narrative: I'm still not sure when I read "digit manipulation" what that section is really about (which is rudimentary proofs that 0.999...=1). Mangojuicetalk 14:53, 5 October 2006 (UTC)[reply]
Yeah, we still have to do a better job of setting up the sections. For the p-adics, I've already tried to explain, here and in the article, how it's a natural outgrowth of 0.999…. Maybe I should put it this way: the Skepticism section states that students expect a last 9, even though there are infinitely many 9s in 0.999…; and students' expectations are sometimes borne out in other number systems. If, after all that, the article didn't discuss …999, something would be missing.
In an encyclopedic article about X, the applications of X can't be off-topic. I've tried to make it clearer: [1]. Melchoir 18:00, 5 October 2006 (UTC)[reply]
By the way, which equation were you talking about that you wanted displayed apart? Melchoir 19:01, 5 October 2006 (UTC)[reply]
That was the one right before citation number 39. Mangojuicetalk 22:03, 5 October 2006 (UTC)[reply]
Uh... that one is already in math-mode and on a line of its own:

and so one can re-use the familiar formula:

[1]

(Compare with the series above.)

I suppose I don't know what "redone in full-equation style" means. Melchoir 23:08, 5 October 2006 (UTC)[reply]
Huh! You're right! I'm surprised; the font looks entirely different to me than some of the other equations, I guess, because the stuff is relatively simple. For me the \cdot and \cdots show up as boxes. I reformatted to match the equation in the Infinite sequences and series section. Font is still different but it's okay as long as the symbols show up. Mangojuicetalk 14:04, 6 October 2006 (UTC)[reply]
  • Question: Why would you want this to be a Featured Artcile anyway? What difference does a symbolic gesture of recognition by people who know nothing about mathematics and a little star on the upper-right hand corner of the article make to anything? Why are Featured Articles about abstruse topics in mathematics important to Wikipedia? --Francesco Franco 18:53, 29 September 2006 (UTC)[reply]
    That is a pretty close-minded view of Wikipedia, if I may say so without it being taken as an insult. Wikipedia aims to offer useful introductory information of high quality to casual users. Featured articles are understood to be that, and if anyone feels an article is of high standard, he is most welcomed to nominate. Why we have featured articles about an anime character when only children know who he is? Why we have featured articles about a painting that probably very few know about its existence, lest see it? Why we have featured articles about events that may be important only for a country? Or Star Wars, Pokemon or some TV show? -- ReyBrujo 19:10, 29 September 2006 (UTC)[reply]
    • Who decides what Wikipedia should ot should not aim to offer. Most of my articles are non-introductory information of high quality intended for people with a realtively advanced education. You're missing my point compltely. Why is it important to the nominators of this article that it be a Featured Article? What does it mean? I already have two fatrured articles. What did I get out of it? Nothing. several people shit on them, vandalized then and insulted them when they were omn the main page anyway. What's the point?
Ah, you are not against this featured article, but against featured articles in general. Well, this is not the place to discuss that. -- ReyBrujo 16:58, 30 September 2006 (UTC)[reply]
I like ReyBrujo's answer. If you want something more concrete, there's this: if the article gets Featured, it'll take a turn on the Main Page. Melchoir 20:42, 29 September 2006 (UTC)[reply]
Who cares? --Francesco Franco 07:52, 30 September 2006 (UTC)[reply]
These ideas are false in the real numbers, as can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly.

I think it's misleading and missing the point. It's not that the misconceptions are false, it's that they're misconceptions. In other words it's not so much that 0.999... can be proven to be equal to 1 by construction. What's important is that when you construct the reals from the rationals in the usual way, then the reals denoted by 1 and by 0.999... are identical. The next sentence probably adds to the confusion even more. We should try to get one of these vocal skeptical students to peer-review this article! Pascal.Tesson 11:00, 9 October 2006 (UTC)[reply]

I don't understand the problem. What's the operative difference between a false idea and a misconception? And what's the significance of emphasizing "in the usual way"? Melchoir 17:19, 9 October 2006 (UTC)[reply]
Ok. I did write that in a haste. But here's a more coherent objection. These ideas are false in the real numbers, as can be proven by explicitly constructing the reals from the rational numbers assumes that the meaning of 0.999... is somehow independent of the construction and that the construction just proves that this number is 1. But it's more like the other way around: the formal notation 0.999... only has a meaning once you construct the reals as limits of rational sequences. And by that construction, the reals represented by the notation 0.999... and 1 are equal. See what I mean? Pascal.Tesson 22:01, 9 October 2006 (UTC)[reply]
Well, first of all, it's not true that you have to construct the reals from the rationals in order to give meaning to 0.999…; see any of the half-dozen equivalent developments back in the "Calculus and analysis" section. Second, the behavior of the real numbers is indeed independent of whichever construction of them one chooses to work with. Melchoir 01:27, 10 October 2006 (UTC)[reply]
  1. ^ Fjelstad pp.14-15
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