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............
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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............]
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Check out this page from MathWorld.
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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).