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    Moderators' note: I've merged this thread with the last couple of threads on this topic - just to emphasize how often this comes up, and just how long (and pointless) the resultant discussion usually is.
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    (Original post by shamrock92)
    You write like Wittgenstein without a proof-reader.
    ... your like a cockroach - you can kill them but man they keep coming back
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    Incidentally this is one of those reasons why the sum of the infinite series of fractions is smaller than the sum of the infinite series of decimals.

    \aleph_{0}=\aleph_{1}<\aleph_{2}

    Where 0,1,2 are the cardinal, fractional, and decimal systems respectively.
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    I have had it with this issue.

    FACT  0.9999999...  =1

    If you are going to post something which you think shows that this is not true then don't because you are wrong.

    If you don't understand why you are wrong after being shown multiple proofs of this fact then leave the thread, do another year of a Mathematics course, the review the proofs again. If you still do not understand, do another year of mathematics. When you eventually understand, then you can come here and say... yes... I understand. Until then, do not post, you are wasting your time.

    Btw, if you go more than say, 3 years of mathematics, and still do not understand this fact, then do not bother, you are a lost cause.
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    (Original post by DeanK22)
    ... your like a cockroach - you can kill them but man they keep coming back
    LOL
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    (Original post by JMonkey)
    Incidentally this is one of those reasons why the sum of the infinite series of fractions is smaller than the sum of the infinite series of decimals.

    \aleph_{0}=\aleph_{1}<\aleph_{2}

    Where 0,1,2 are the cardinal, fractional, and decimal systems respectively.
    Sorry, but no.

    Aleph0 is the cardinality of any countably infinite set. So the natural numbers, the integers, the rationals all have cardinality aleph0.

    Aleph1 is the next biggest cardinal (it, under no circumstances, is equal to aleph0). The cardinality of the reals - "decimals" - is usually denoted c for continuum. If you accept the generalized continuum hypothesis then c equals aleph1 and alpeh2 is the cardinality of the set of subsets of the reals.

    And your use of the word "sum" is incorrect. You can't "sum" the rationals (fractions) - that word tends to be reserved for series.
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    This is sort of a duplicate of my post earlier in the thread, so I've deleted that one since it's more relevant here.

    The problem of 0.999... occurs because we find it so hard to detach our minds from our silly base-10 number system. If you were thinking in about it in base 3, it'd be a nice neat 1 (since 0.1 + 0.1 + 0.1 = 1 in base 3).

    It's better to write it as an infinite sum, letting 9N = 0.999...:

    \displaystyle \newline N = 0.111... = \sum_{r = 1}^{\infty} \frac{1}{10^r}

    Then you can be a bit more specific with the whole thing:

    \displaystyle \newline 10N = 10\sum_{r=1}^{\infty} \frac{1}{10^r}\newline

\Rightarrow 9N = 10N - N\newline

 = 10\sum_{r=1}^{\infty} \frac{1}{10^r} - \sum_{r=1}^{\infty} \frac{1}{10^r}\newline

 = 10\left( \frac{1}{10} + \frac{1}{10^2} + \cdots + \lim_{n\to\infty} \frac{1}{10^n}\right) - \left( \frac{1}{10} + \frac{1}{10^2} + \cdots + \lim_{n\to\infty} \frac{1}{10^n}\right)\newline

= 1 + \frac{1}{10} + \frac{1}{10^2} + \cdots + \lim_{n\to\infty}\frac{1}{10^{n-1}} -  \frac{1}{10} - \frac{1}{10^2} - \cdots - \lim_{n\to\infty} \frac{1}{10^n}\newline

= 1 - \lim_{n\to\infty} \frac{1}{10^n}\newline

= 1 - 0\newline

= 1

    This tackles the argument that "you're missing off an infinitely tiny bit" - yes, because the infinitely tiny bit is zero, so it makes no difference.

    I think there should be a sticky that summarises all these common questions, along with "what is 0^0" and all the dividing by zero threads we get.
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    (Original post by nuodai)
    I think there should be a sticky that summarises all these common questions, along with "what is 0^0" and all the dividing by zero threads we get.
    I'm trying to keep the number of stickies down to a reasonable number, but it could potentially get added to the "Guide to Posting". People probably won't read it though, so don't get your hopes up.

    (And besides - the reason these threads go on for ever is not because there's no explanation available. They go on for ever because people don't (can't) understand the explanation. There's a saying about wrestling pigs that comes to mind...)

    Edit: Sorry to say this, but your argument isn't actually terribly good; terms like \displaystyle \left( \frac{1}{10}+\frac{1}{10^2}+\dot  s+\lim_{n\to\infty}\frac{1}{10^n  }\right) look decidedly dubious to me.
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    Add 0.999(...) to the spam filter. It's the only way to stop it!
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    \aleph_0 etc. are well-defined (and hardly obscure) symbols which do not mean what you said.
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    (Original post by .matt)
    but 1 - 0.\dot{9} = 0.\dot{0}1...

    :awesome: :awesome:
    0.\dot{0}1 doesnt make any sense
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    (Original post by JMonkey)
    No I gave them a definition beneath, it's easier if you use some obscure symbol,
    You mean, a symbol so obscure that it is the standard one used when discussing the size of infinite sets. And you just happened to define \aleph_0 to equal the size of the natural numbers as well?

    Yeah, right.

    And it is true that the infinite sum of fractions and the sum of integer numbers are equal.
    To quote Pauli: "That's not right. It's not even wrong".

    Plus it's also true that the decimals are a greater infinity. You probably need to study up on it or something?
    If there's one person on TSR I'm confident doesn't need to study up on it, it's RichE...
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    (Original post by DFranklin)
    You mean, a symbol so obscure that it is the standard one used when discussing the size of infinite sets. And you just happened to define \aleph_0 to equal the size of the natural numbers as well?

    Yeah, right.

    To quote Pauli: "That's not right. It's not even wrong".

    If there's one person on TSR I'm confident doesn't need to study up on it, it's RichE...
    I believe cantor proved on a 1 to 1 basis that all fractions are equal to the integer numbers, and then went on to show that because decimals can have an infinite string of numbers they are greater.

    Are you trying to tell me that I'm not allowed to define terms because you say so as well, interesting.
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    (Original post by JMonkey)
    Incidentally this is one of those reasons why the sum of the infinite series of fractions is smaller than the sum of the infinite series of decimals.

    \aleph_{0}=\aleph_{1}<\aleph_{2}

    Where 0,1,2 are the cardinal, fractional, and decimal systems respectively.
    The sum of all rationals (and therefore of the reals) is not absolutely convergent. Therefore, you can get any answer for the sum that you want, by summing in a different order.
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    (Original post by The Bachelor)
    The sum of all rationals (and therefore of the reals) is not absolutely convergent. Therefore, you can get any answer for the sum that you want, by summing in a different order.
    Interesting, I believe Cantor was just going for different. I include transcendentals in with any other type of decimal though.

    I saw it on a program and it peaked, my interest when they said that you can establish greater infinities. So if it is not true I apologise. Needless to say this sort of set theory or number theory is beyond my level of study, so if anyone can show why it is incorrect to make that statement I can't promise I'll understand it but I would be interested.
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    (Original post by JMonkey)
    Interesting, I believe Cantor was just going for different. I include transcendentals in with any other type of decimal though.

    I saw it on a program and it peaked, my interest when they said that you can establish greater infinities. So if it is not true I apologise. Needless to say this sort of set theory or number theory is beyond my level of study, so if anyone can show why it is incorrect to make that statement I can't promise I'll understand it but I would be interested.
    I'm not sure what you're talking about. Cantor largely has nothing to do with this.

    http://en.wikipedia.org/wiki/Riemann_series_theorem

    Edit: Actually, I don't think the rationals are even conditionally convergent. Can you find an ordering of the sum that makes the sum converge?
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    (Original post by The Bachelor)
    I'm not sure what you're talking about. Cantor largely has nothing to do with this.

    http://en.wikipedia.org/wiki/Riemann_series_theorem
    Could of been Reimann it was a while ago.

    I think it may have been both I don't know but there was a famous statue involved, where you could show one to one an integer and its fractions. They may have moved on to later minds though?
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    (Original post by JMonkey)
    I believe cantor proved on a 1 to 1 basis that all fractions are equal to the integer numbers
    No, Cantor showed on a 1 to 1 basis (technically, a bijection), that the cardinality of the set of all fractions is equal to that of the set of integers. That is, he established a 1-1 correspondence between the two sets.

    and then went on to show that because decimals can have an infinite string of numbers they are greater.
    Not exactly.

    Are you trying to tell me that I'm not allowed to define terms because you say so as well, interesting.
    No. You can define terms as you wish. However, it is somewhat unwise to choose terms with standard meanings, and then claim they mean something else.

    Using those standard terms and then claiming "Oh, I just picked some obscure symbol - it's not my fault other people interpret them differently", particularly when you seem to have at least slight knowledge of Cantor's work, is probably somewhere beyond unwise.

    @The Bachelor: No, I can't see that the rationals are conditionally convergent. The terms aren't even going to tend to 0. So you can't find a convergent rearrangement. (I saw this before, but didn't think it worth mentioning it, as it seemed a bit off-topic).
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    (Original post by DFranklin)
    No, Cantor showed on a 1 to 1 basis (technically, a bijection), that the cardinality of the set of all fractions is equal to that of the set of integers. That is, he established a 1-1 correspondence between the two sets.
    I see then perhaps the guy who made the preposition was being a bit disingenuous. It was the history of maths from ancient to modern times, was on BBC4. Quick google reveals it was called The Story of Maths. Obviously I can't argue the toss, so I'll bow to greater knowledge. Thanks for the information.

    No. You can define terms as you wish. However, it is somewhat unwise to choose terms with standard meanings, and then claim they mean something else.

    Using those standard terms and then claiming "Oh, I just picked some obscure symbol - it's not my fault other people interpret them differently", particularly when you seem to have at least slight knowledge of Cantor's work, is probably somewhere beyond unwise.
    I just picked it as the first latex symbol that came to my head. Noted.
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    (Original post by The Bachelor)

    Edit: Actually, I don't think the rationals are even conditionally convergent. Can you find an ordering of the sum that makes the sum converge?
    Yes, this is the bigger problem. There is no listing of the rationals where the sum makes sense as a finite sum, because the nth term could never tend to zero. You could take some listing though where it makes sense for the sum to tend to infinity (and other listings where it tended to negative infinity).
 
 
 
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