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I would like to know if there is a way to find out the type of trailing decimals a fraction would create i.e. terminating or non terminating repeating decimals, without actually doing a division.

Like $14641/256 = 57.19140625$ which has a terminating decimal.

Especially Lime
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Shashvat Shah
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  • if the fraction can be rewritten with a denominator of 10^n and have the numerator be integer then it can have a terminating decimal otherwise no. –  Jul 29 '17 at 18:46
  • Note this question assumes number is expressed in base 10 – Χpẘ Jul 29 '17 at 20:22

2 Answers2

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First put it in its lowest terms. Then the fraction terminates if and only if the denominator is in the form $2^a5^b$ (where either $a$ or $b$ may be zero).

Especially Lime
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After reducing the fraction to lowest terms, if the last digit of the denominator is an even number or a 5, then the decimal representation will not repeat. Otherwise it will.

This assumes that the fraction is represented in base 10. Other bases would have similar tests derived from the prime factors of the base. For example, in base 2, if the last bit is 0, then the binary representation will not repeat. Otherwise it will.

David
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