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Math Help - ir/rational expansion

  1. #1
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    Tricky one

    Which rational numbers between 0 and 1 have finite decimal expansions? Give a brief description, with some explanation in your answer.

    Example 3/40 = 0.075, however 2/3 = 0.66666666666............


    Help I am
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  2. #2
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    Which rational numbers? There are infinitely many. Is that everything the question says?
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  3. #3
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    The question I have been given is essencially that:

    Give a brief description, with some explanation, of which rational numbers between 0 and 1 have finite decimal expansions? [An example is 3/40 = 0.075, however 2/3 = 0.66666666666............]

    I am truly
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  4. #4
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    Question - what kind of course is this for? Do you think your teacher/professor wants a rigorous proof or a general explanation?
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  5. #5
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    I think a general proof would do, obviously the more I put down the better
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  6. #6
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    Check out this page from MathWorld.

    http://mathworld.wolfram.com/DecimalExpansion.html

    Tell me if it helps. If not, I'll post more.
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  7. #7
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    I had already looked at this site up, thanks anyway.

    I always use google before mathhelpforum actually. It does help a little but I think I need more... sorry for being awkward.
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  8. #8
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    They are the numbers of the form m/n, where m<n, n=2^j*5^k for some nonnegative integers j and k, and m is relatively prime to n. The reason for this is that the algorithm for division will terminate after j of k steps, whichever is greater, because 2 and 5 are factors of 10 and therefore 10^max(j,k) is divisible by n, but if n contains any other prime factor, it will not divide 10^a for any integer a, and the algorithm for division will repeat itself (it has to repeat because of the finite sets involved; the numbers that you get at each stage of the division come from a finite set, and each number determines the next one).
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