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Math Help - any number can divides infinity???

  1. #1
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    any number can divides infinity???

    Definition: An integer b is said to be divisible by an integer a \neq 0, in symbols a|b, if there exists some integer c such that b=ac.

    If b is infinity \infty, then what are the possible a?

    I think a can be any number. Is it true?
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  2. #2
    Senior Member nikhil's Avatar
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    Lightbulb according to me

    yes. (b/a) should not be undefined.
    even infinity/infinity is valid though its a inderminate form as infinity can not be compared with other infinity unless we know its generating function.
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    Your definition says that  a, b \mbox{ and } c are integers.
    Can you reasonably say that \infty is an integer, (which would be necessary to include it within your definition) ?
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  4. #4
    Senior Member nikhil's Avatar
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    Lightbulb

    Quote Originally Posted by BobP View Post
    Your definition says that  a, b \mbox{ and } c are integers.
    Can you reasonably say that \infty is an integer, (which would be necessary to include it within your definition) ?
    well if one is talking about integers then infinity will be an integer
    if one is talking about natural numbers then infintity will be an natural number for sure.
    it depends upon the function used.
    note:
    if there are two person A and B. they are talking about natural numbers, both of them then think about infinity then you will never be able to find if
    (infinity suggested by A)/(infinity suggested by B)>1 or <1 this will be totally indeterminate.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by nikhil View Post
    yes. (b/a) should not be undefined.
    even infinity/infinity is valid though its a inderminate form as infinity can not be compared with other infinity unless we know its generating function.
    What do generating functions have to do with this? Perhaps you mean rate of convergence and such.

    well if one is talking about integers then infinity will be an integer
    if one is talking about natural numbers then infintity will be an natural number for sure.
    it depends upon the function used.
    note:
    if there are two person A and B. they are talking about natural numbers, both of them then think about infinity then you will never be able to find if
    (infinity suggested by A)/(infinity suggested by B)>1 or <1 this will be totally indeterminate.
    You have it all wrong. "Infinity" is not a natural number, not a real number, not a complex number. For instance, take the Peano axioms as a foundation for arithmetic; then if \infty \in \mathbb{N} we must have \infty+1 \in \mathbb{N} and (\infty+1)+1 \in \mathbb{N}.. But you will agree that both of these are equal to \infty hence they cannot be natural numbers because the successor function n \mapsto n+1 is injective (it's one of the axioms).

    In other words, if you define \infty as a natural number you end up with things such as 0=2. The cancellation laws for addition are easily derived from the Peano axioms and they do not hold for "infinity".
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    Assuming that b can be infinity, and a is any integer not equal to zero, consider what integer c multiplied to a would equal infinity? If you multiply any two integers, no matter how "large", you'll still get an integer which might be extremely "large" but still not infinity.
    So, to answer your question, a can't be any number ...

    Regarding infinity:

    Wikipedia: "Infinity (symbolically represented by ) refers to several distinct concepts usually linked to the idea of "without end"".
    i.e. Infinity is not a number, it's an idea of being "endless".
    Last edited by Bingk; August 19th 2009 at 04:15 PM. Reason: reorganized
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  7. #7
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    Quote Originally Posted by Bingk View Post
    Assuming that b can be infinity, and a is any integer not equal to zero, consider what integer c multiplied to a would equal infinity? If you multiply any two integers, no matter how "large", you'll still get an integer which might be extremely "large" but still not infinity.
    So, to answer your question, a can't be any number ...

    Regarding infinity:

    Wikipedia: "Infinity (symbolically represented by ) refers to several distinct concepts usually linked to the idea of "without end"".
    i.e. Infinity is not a number, it's an idea of being "endless".
    On that note, thread closed.
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