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Math Help - Determine that a number terminates in base m but will not terminate in base n?

  1. #1
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    Determine that a number terminates in base m but will not terminate in base n?

    Is there a way to determine that for some number a in base m that it will or will not have a terminating equivalent number b in base n?

    For example, when converting number 0.315 in base 10 to base 2, we get the base 2 number 0.01010000101000111101011100001010... I assume this will be nonterminating. How can I actually prove that this number will be nonterminating?

    Thanks
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  2. #2
    MHF Contributor

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    In base b, the numbers which "terminate" are of the form \frac{a}{b^k} for some a,k\in\mathbb{Z}.

    A necessary and sufficient condition for a rational {p\over q}, where p,q are relatively prime, to "terminate" in base b is therefore that the prime divisors of q are prime divisors of b. (which is equivalent to saying that q divides b^k for some k)

    For instance, in base 10, {p\over q} "terminates" iff the only prime divisors of q are 2 and 5 (supposing again that gcd(p,q)=1). In base 2, q has to be even.

    I hope I was clear. If not, just ask.

    Laurent.
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