Talk:0.999.../Archive 2

message: :::No, a Dedekind cut cannot represent more than one real number, because between any two real numbers there are some rational numbers.  And people would be more confident in what you write if you could understand the phrase "for all intents and purposes". Michael Hardy 22:47, 29 November 2005 (UTC)
replies: text: This is true, but in the absence of a definition of "real number", your argument is circular. I haven't tried to force a definition upon 70.110.81.253, since it will not accept the reasonableness of any definition until its existing misconceptions are cleared up. I have infinite respect for you, but I don't think you're helping. Melchoir 23:10, 29 November 2005 (UTC)

text: I seem to have failed. Everyone else, please do not repond to the above comment. Melchoir 23:23, 29 November 2005 (UTC)
thread_title: Melchoir cordons the following off so that 70.110.81.253 is not tempted to respond:

message: If there is no rational number between 0.999... and 1, and rationals are dense in the reals, seems to me like 0.999... is equal to 1. — JIP | Talk 12:21, 28 November 2005 (UTC)
replies: text: Of course you're right, but my whole point is that we don't need to call on any theorems from analysis or topology. The simplest link between the reals and the rationals is a definition. Melchoir 17:15, 28 November 2005 (UTC)

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

message: You may not need to call on any theorems because they won't help you.
I know where this is leading: If  68.238.98.71 says no, then you will try
to say that
{(-inf,1);[1,inf)} = 1 and {(-inf,1);[1,inf)} = 0.999... therefore they
must be the same.
However, by saying this you would concede that 0.999... is less than 1
since (-inf,1)
does not contain 1 and I don't think you are quite ready to say that
0.999... is a member of [1,inf), are you? 70.247.52.236 18:13, 28 November 2005 (UTC)
replies: text: I am unfamiliar with the { ; } notation, but I'll address your other points. No, of course (-inf,1) doesn't contain 1. But I find your second claim very strange. Where do you base your claim that we are "not quite ready to say that" 0.999... is a member of [1,inf)? We've been saying 0.999...=1 all along, therefore it is of course a member of [1,inf). Do you want us to say "no, 0.999... is not a member of [1, inf)"? If we say that, then it's less than 1, and that would prove your argument. So essentially you're asking us to say that we are "not quite ready to say that" 0.999...=1. — JIP | Talk 18:59, 28 November 2005 (UTC)

text: 70.247.52.236, please create an account. I cannot respond to you because, for all I know, you are 68.238.98.71. Melchoir 22:31, 28 November 2005 (UTC)
replies: text: The two IPs do come from the same area... - Fredrik | tc 23:12, 28 November 2005 (UTC)
thread_title: To which yet another IP says:

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

message: I'm not really comfortable with this section, how do you guys here feel about changing it?, maybe something like:
Another kind of proof adapts to any repeating decimal. Let the number 0.999… be called c, then 10×c = 10×0.999… = 9.999…; If we substract c, we have 10c - c = 9c and 9.999… - 0.999… = 9, thus 9c = 9. Dividing both sides by 9 completes the proof: c = 0.999… = 1
Please edit this entry, so we can have a 'definite' version of this section. Or am i the only one who has trouble with the current wording?  Thank you. Jesushaces 05:05, 30 November 2005 (UTC)
replies: text: I strongly oppose such a change. Your rewording is less meticulous than that currently in the article, and therefore liable to attacks that have plagued this topic both here and elsewhere. --KSmrqT 06:16, 30 November 2005 (UTC)

text: I support the vague idea of a compromise between the current wording and Jesushaces'. The current wording suffers from a pretention to precision that isn't really justified, and the fact that it's more detailed than the previous proof (using 1/3) seems to suggest that the latter proof stands on firmer ground, which it doesn't. Now, I don't think there is a danger of attack if we switch to a less meticulous wording, since the article already has disclaimers about elementary proofs. However, even if there were an increased danger of attack, one must always attempt a balance between readability and precision, even in a textbook or a journal article, to say nothing of an encyclopedia. Right now it's unreadable. Melchoir 07:04, 30 November 2005 (UTC)
replies: text:  I will not agree to any wording that does not include the present details. Previously this article was constantly being revised. It degenerated into a horrible hash. Everything that the proposed revision omits has been shown to lead to attack and confusion. The current version of the article is by far the most stable ever. The algebra proof pretends nothing; it honestly points out subtle issues. I guarantee you that if you strip out the safeguards the clamor will begin anew. Such a "compromise" will bring not a risk of attack, but a certainty; and the consequences will be ugly. Maybe the current wording is a little awkward, but that's nothing compared to the mess it replaced, and that you want to "risk" again. I think that would be a damned foolish choice. --KSmrqT 09:03, 30 November 2005 (UTC)
thread_title:  Algebra Proof 

message: You are either sincerely mistaken or you are lying. You challenged logamath1 to suggest a construct for the reals. He suggested the Decimal construct. You then tried to show this is true and were not able to do so. 70.110.92.137 19:57, 2 December 2005 (UTC)
replies: text: 70.110.92.137, I am beginning to wonder again whether or not you are logamath1, despite logamath1's statement otherwise. I will assume you are different people for now. You have insulted me for the last time. I will not respond to anything further you say, and all of your future edits will be cordoned off from discussions with other users. Melchoir 20:14, 2 December 2005 (UTC)

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

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

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

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

message: Melchoir writes: "...any careful definition of the real numbers based on decimal 
expansions SETS 0.999... equal to 1."  - Melchoir
Not only is this an outright lie but it is clearly demonstrated to be false:
1) The Indians did not use a decimal point. This was an invention of Scotland.
 The notation we use today first appeared in a book called "Descriptio" by the Edinburgh 
 mathematician, John Napier, Laird of Merchiston, in the 1616. He used a decimal point 
 to separate the whole number part from the decimal number part. Known as 
 'Marvellous Merchiston", he published many other treatises including 
 "Mirifici logarithmorum" (1614) and Rabdologia (1615) on systems of arithmetic using 
 calculation aids known as Napiers Bones. 
2) Fractions were invented long before the decimal point.
In the 10th cent. A.D. Arab mathematicians extended the decimal numeral system to 
include fractions.
Melchoir will try to justify himself by using terminology such as
any careful definition of the real numbers - this is entirely subjective and is
open to debate.
I asked who came up with the idea that 0.999... = 1. No one on Wiki's board of idiots
was able to answer this. Why? The answer is simple: they are all products of today's
institutions that are run by a bunch of baboons masquerading as Phds. My apologies to
the few Phds who really earned their diplomas but the majority are to be regarded
as ignoramuses who are spineless and incapable of thinking on their own.
: It's not a concept that is unique to the Wiki in any sort, and it's a simple application of Calculus. If 1 != .999..., then several concepts in Calculus are completely false — and I've yet to see evidence that anything in Calculus is false. And your talking about number systems that don't use decimals says nothing about number systems that do. Melchior talked about "any careful definition of the real numbers based on decimal expansions." And insulting my intelligence, or implying that I'm just "going along with the crowd" (a laugh and a half from anyone that knows me in the real world) is not appreciated. Your complaints against the institution that defined the decimal number system is ludicrous at best. Sure, there are other logically consistent number systems, but they're not the one used by most of the world today. And, in the decimal system, an infinitely recurring decimal is defined as the limit of the sum of its digits. And, as it just so happens, the limit of the sum of the digits in .999... is 1.0. --HeroicJay 16:41, 5 December 2005 (UTC)
replies: text: We have. Multiple times. Most of them clearly seen above. We've shown that the limit of the series .9, .99, .999, etc. (which is how a repeating decimal such as .999... is defined) is 1. We've shown that there must be a real number between any two unequal real numbers, but there's nothing between .999... and 1. We've shown that simple algrebra or arithmetic lead to .999... = 1, although these aren't as convincing as the Calculus stuff. You just simply refuse to acknowledge any of it. --HeroicJay 22:44, 5 December 2005 (UTC)

text: I'm not re-adding it, but I'm going to reply to the "points" made by Mr. IP. (While, yes, he is being abusive, I don't think deleting any his posts other than the insults is really accomplishing anything.) The limit of a a sum of an infinite number of terms is an infinite summation, so .999... is the limit is of the sequence .9, .99, .999... and the sum of the series .9 + .09 + .009 + ... (the result is the same either way.) To say that something can be the definition of a term and not be a property of that term is just mind-bogglingly silly. And if there is a number between .999... and 1, could you tell us what it is or describe it? The answer is no, you can't, because it would require that, somewhere along the line, one of the digits would have to be larger than 9. For instance, for something between the numbers .333... and .4, a number in between would have a digit larger than 3 in some place other than the first. (such as .34, .33501, .333333338, etc.) However, there is no digit larger than 9, and the nines repeat forever, so there isn't any other digit to increase (without resulting in 1 or something larger.) --HeroicJay 02:19, 6 December 2005 (UTC)
replies: text: Okay, I've restored the non-abusive bits of anon's latest post. I hope that my deletions do accomplish something; I hope that they convince would-be insult throwers to reconsider their methods. Melchoir 02:35, 6 December 2005 (UTC)
replies: text:  No complaints with deleting the insults. None whatsoever. But deleting his arguments, shaky as they are, kinda gives the vibe that we don't want to (or can't) respond to him, or are simply trying to silence him for whatever goals, which is kind of exactly what he implies with his conspiracy theories and derision at the word "PhD" and insults. Last thing I want to do is make him think his silly conspiracy theories are true. :p --HeroicJay 03:36, 6 December 2005 (UTC)
replies: text: Indeed. Melchoir 05:30, 6 December 2005 (UTC)

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

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

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

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

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

text: Whoa! I was wrong! --HeroicJay 20:19, 6 December 2005 (UTC)
thread_title: Here we go again!