Talk:0.999.../Archive 7

message: I've reverted the wholesale conversion of HTML entities, like "δ", to UTF-8 encoded Unicode characters, like "δ". Actually, the one conversion to which I really object is frasl, the fraction slash, "⁄". The problems with the ⁄ conversion are that (depending on font) it can be too easily confused with a plain slash, "/", and that the kerning effect can be a nuisance in the edit window (though it's lovely in the article). --KSmrqT 09:49, 5 April 2006 (UTC)
thread_title:  Entities and Unicode 

message: Noe substituted this:
* And like integers, for most numbers a different series of digits means a different number. The notable exception is numbers that can be represented as a terminating decimal, e.g. 1. Without changing the value, one may either add trailing zeros (as in 1.00 or 1.000…), or decrease the last non-zero digit by one and add recurring trailing nines (0.999…).
for this:
* And like integers, for most numbers a different series of digits means a different number (ignoring trailing zeros as in 0.250 and 0.2500). The one notable class of exceptions is numbers with trailing repeating 9s.
It incidentally raises a curious point: Why is it that nobody complains about 0.25 and 0.250 and 0.2500 representing the same real number, but the unending 9s provoke such a reaction?
Technically, the amendment is correct (including, subtly, the fact that zero has no 9s version). Rhetorically, it is less effective at drawing attention to the psychologically salient point, the theme of the article, trailing 9s. Thus I prefer to keep the original. If others have a strong preference for the extended language, please note that it's better to avoid Latin abbreviations such as "e.g." — exempli gratia, and that an example like 0.25 (with internal decimal separator) is clearer than 1. In the latter case the decimal point and end of sentence coincide, or else the decimal separator is omitted, mixing notations. (The European version would be either "1." or "1,." — which highlights the difference. Notice that the article always says "decimal separator", never "decimal point", to facilitate translation.) And we don't want to introduce 0.24999… until the end of the article, to keep focus on the title proof. --KSmrqT 17:10, 21 April 2006 (UTC)
replies: text: Well, I still like my version better. I think the arguments for using 0.25 rather than 1 here, while absolutely not mentioning 0.24999…, are weak. And (being a little polemical), as the text now stands, first we are told to ignore numbers that can be written as a terminanting decimal, then we are told that the notable exception is the numbers that end in repating 9's, as if that is a completely different thing - while in fact it is exacly the same numbers. - By the way, the reason nobody objects to 0.25 and 0.2500 being equal is that everyone understands (or have been indoctrinated into believing) that zero is nothing.--Niels Ø 21:59, 2 May 2006 (UTC)

text: (I've been enjoying the use of popups for rolling back, but they provide no opportunity to comment the change. Sorry to leave you wondering.) With 0.25 versus 0.24999… the equality requires an endless sequence of 9s. The point about trailing zeros is distinct from the issue of endlessness, and we don't want to confuse the two. So it actually matters that 0.250 and 0.2500 are both finite, both distinct from 0.25 and from each other, and yet represent the same real value. It also matters as psychological preparation. Many discussions about 0.999… act as if having two distinct representations is unusual and suspicious, and the use of endless repetition is even more suspicious. Having two perfectly innocuous examples of added zeros with no repetition reinforces the point that we use multiple representations all the time, and never fret about it. I strongly prefer the existing wording for all these reasons. --KSmrqT 23:03, 1 June 2006 (UTC)
thread_title:  Simple but effective 

message: I've now moved this article to "Multiple decimal representations", as per a suggestion made by someone else in the AfD discussion. -- The Anome 21:43, 5 May 2006 (UTC)
replies: text: Oh, come on -- if you are just going to revert the move, at least please discuss it here. The edit comment for the reverse move was "moved Talk:Multiple decimal representations to Talk:Proof that 0.999... equals 1: Rvt unilateral move disapproved on talk." Which talk page, where, please? Perhaps I haven't looked hard enough, but I can't find any reference to this on this talk page or its archives. -- The Anome 21:44, 5 May 2006 (UTC)

text: *Oh ,feel free to be bold until reverted; but the subject of this article is this proof. A general and rigorous discussion could well be put under Multiple representations; but it wouldn't be this one. Septentrionalis 21:52, 5 May 2006 (UTC)

text: *I saw it just today; ask KSmrq. My apologies for assuming it was this talk page. Put this on WP:RM if you like; I will limit myself to opposing. Septentrionalis 21:52, 5 May 2006 (UTC)
thread_title:  Move to "Multiple decimal representations" 

message: "3 × 0.3333… equals 0.9999…; "???  the only way you could come upon that is with round-off error.  71.213.29.198 04:02, 6 May 2006 (UTC)
replies: text: Seems obvious enough to me; even my mother buys that argument. In any case, the method you refer to is preceded by the caveat, "Elementary proofs assume that manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition." Now, if you'd like to dispute those assumptions, perhaps you'll be more interested in Talk:Proof that 0.999... equals 1/Arguments. Melchoir 04:12, 6 May 2006 (UTC)
replies: text: I suppose we may need a caveat about intuitionism here <sigh>: Not all mathematicians agree that 0.999... is the limit of the sequence {0.9, 0.99, 0.999...}. Will this do? Septentrionalis 16:00, 7 May 2006 (UTC)
replies: text: No, I find that misleading. A mathematician who defines "0.999..." as anything other than the limit (Richman again?) must be aware that he is doing a different mathematics from everyone else, whether it's intuitionism or what. Even Richman admits that the usual definitions lead to the usual result. Melchoir 20:01, 7 May 2006 (UTC)
replies: text: As I understand intuitionism, it argues (more or less) that statements about limits that (in principle) involve infinite calculation are not well-defined, which seems to be what Richman is saying. But I would be perfectly happy to keep it out of the article, unless mentioning it averts edit wars. Septentrionalis 23:08, 8 May 2006 (UTC)

text:  No, we do not need to mention this or all the other variations on the standard reals, because the article explicitly begins with assuming standard reals and because we discuss some alternatives at the end. Imagine the effect on all our articles if we started down that path! Gruesome. So relax. --KSmrqT 20:06, 7 May 2006 (UTC)
thread_title:  factuality disputes 

message: Just out of curiosity, for those disputing this article, assuming we knew nothing of the rigorous proofs given here, does it really matter if 0.999...=1 or not?  It's a widely accepted idea that this is so, and in the real world, being right is only worth one point out of ten. It's like when your dad's fixing the door because it's too tall. You try to tell him that he's sawing off the wrong end, but when he doesn't want to hear it, being right doesn't do you a lick of good, it won't change what he's doing. I remember a girl at a physics symposium who had a chance to discuss limits of functions versus their values at a given point. She very correctly stated that \lim_{x\to\infty}{1\over x}=0 and that one over infinity was not really zero. She pushed herself on anyone who would debate this topic with her and left a lasting impression of stubborness in their minds. People that could have made her life easier in a couple of years and from whom she could have learned many things, now only remember they never want to see her again. What I'm trying to say is, being right won't change anything, it just makes for a lot of unnecessary stress.
Guardian of Light 03:34, 9 May 2006 (UTC)
replies: text: Yes, it does matter. If you're doing mathematics and you assume one inconsistency, you can prove anything, including 1=2. Being vague about limits, in particular, kept mathematicians producing nonsense (of the form -1 > infinity) throughout the eighteenth century. Septentrionalis 05:10, 9 May 2006 (UTC)
replies: text: You raise one of the most important philosophical questions for a human being. Is it necessary to do the right thing or only the easy thing. There is no right or wrong answer. Loom91 07:05, 9 May 2006 (UTC)

text: Well, Guardian of Light, assuming that your audience isn't going to show up and answer your question, maybe I can. Historically, the deniers on this talk page have had axes to grind with mathematicians and educators, and they exult in being smarter than all us sheep. So, there's the personal side. Then, some of them also seem to think that all of analysis is wrong, so they imagine they're fighting a larger battle than just this one point. And finally, at least one user has been confused but actually willing to learn. Melchoir 07:19, 9 May 2006 (UTC)

text: Ah, then you would love the Indiana Pi Bill. Assistance like this we do not need, thanks just the same. While it is certainly true in human affairs that being right is not the same as being persuasive, mathematics demands both. It can be a stressful discipline. But here's a little insight, courtesy of Richard Feynman: “For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.” He was investigating the Space Shuttle Challenger disaster. If you think being right can cause stress, try being the decision-maker who was wrong. --KSmrqT 08:15, 9 May 2006 (UTC)
thread_title:  Thought on the discussion 

message: I have a more rigorous proof grounded in series, but this one will do for now. All you have to do to demonstrate that .9~ =/= 1 is refute the following: All repeating decimals have a corresponding rational representation. What is the rational representation of .9~ if it is not 9/9? - Corbin ∫ 1 ɱ p s ɔ ♫ Rock on, dude! 14:14, 16 May 2006 (UTC)
replies: text: That is like making a law that pi is exactly equal to three.  All real numbers have more than one representation, and that has been defined in mathematics for a really long time. —Mets501talk 11:08, 26 May 2006 (UTC)
replies: text: Now that comparison is not correct. The ratio of the circumference of a circle to its diameter is what it is; "defining" it to have a different value would introduce a contradiction. We (mankind) have no choice here - though of course, one may define the letter \pi to represent any value one likes, and then call the ratio something else. With the decimals, the situation is different: Disallowing 0.999..., we would be contradicting the vast majority of mathematicians, but the system would still work. Here, mankind had a choice, though we wikipedians do not: We have to report what mathematicians do. If we fail to understand such differences, and fail to express ourselves clearly about them, our articles will be open to attack, and, given the open nature of wikipedia, they will be attacked. Let's aim at an article that is irrefutably correct, and that answers the most common objections.--Niels Ø 14:09, 26 May 2006 (UTC)
replies: text: Exactly. Paul August ☎ 17:29, 26 May 2006 (UTC)
thread_title:  Corbin's Proof 

message: Ladies and gentlemen,
I propose an addition to the elementary proofs section.
The proof I would like to add is as follows:
:===Another fraction proof===
:Any repeating single digit is equal to that digit over nine; for example, 0.444... is equal to 4⁄9. (This can be proved by dividing 4 by 9; the result is 0.444...) Therefore, 9⁄9 must be equal to 0.999..., but any number over itself equals one, so 9⁄9 = 0.999… = 1.
If anyone sees a reason why not to add it or if we should change it, please leave comments. Otherwise, it makes sense and is a fairly simple explanation. The only argument against it I can see is that the article has already made the point; but I think that even if we can never have enough proof to satisfy the nonbelievers, we might as well put more.
(Actually, I already added that section to the article, but then removed it once I saw the extensive talk section -- I figured I'd better put it on here first, or else I'd end up getting a bunch of angry mails ;-) .) Bobburito 04:40, 4 June 2006 (UTC)
replies: text: I think that it's too redundant with the 1/3, 2/3, 3/3 thing to be included.  Maybe both in the same paragraph would work though. —Mets501talk 05:40, 4 June 2006 (UTC)

text: Please do not add this. Experience shows that it raises more questions than it answers. A typical objection is that a proper computation of 9 divided by 9 will never give the repeating decimal. An advantage of the 1⁄3 argument already in the article is that it shows how repeating 9s can naturally occur in a routine computation, and that the result needs to be 1 for consistency. (Thanks very much for asking first!) --KSmrqT 08:45, 4 June 2006 (UTC)
thread_title:  An addition to the elementary proofs section: