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Given any fraction where both the numerator (N) and denominator (D) are both positive and are both whole numbers.

Without manually dividing N by D, is it possible to pre-determine if the resulting value represented in decimal would be a repeating value? (e.g. 44÷33 is 1.3333333333....)

I believe the value of N ÷ D will NOT be a repeating decimal if and only if D is any of the following

  1. D is equal to 1 OR
  2. D's prime factors only consist of 2's and/or 5's. (includes all multiples of 10)

Otherwise, if none of the two rules above hold true, then the positive whole numbers N and D will divide into repeating decimal.

Correct, or am I missing a case?

  • 0
    @BenMillwood, ya,sorry for the mistyping. The rectified version : in base $B$, if $d_i\mid B$, then $\frac{N}{\prod d_i^{r_i}}$ is terminating in base $B$.2012-09-16

3 Answers 3

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Correct.

From wikipedia:

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000…" one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational numbers of the form $k/(2^n5^m)$.

http://en.wikipedia.org/wiki/Repeating_decimals

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    Ah, I see I neglected that. Thanks! ;)2012-09-16
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Decimals of fractions are always eventually periodic (eventually because e.g. $\frac1{300}=0.00\overline 3$, i.e. the period need not start immediately). Your exceptions only summarize the cases when the period consists of zeroes (and hence can be left out for convenience), e.g. $\frac18=0.125\overline0$. This will happen iff multiplication by a suitable power of ten makes the fraction an integer, i.e. if the denominator contains only 2's and 5's, as you correctly state.

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    Yeah, I suppose that's a fair point, on both counts. I don't, however, think that the statement "a terminating decimal consists of only finitely many digits" is an unreasonable one - the interpretation in which it's true may be slightly more complicated than the one in which it isn't, but both are valid and arguably one is less inclined to throw about infinities and limits where they aren't required.2012-09-17
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There is a way to do this. You carry out the long division so as to produce remainder ratios and it either terminates or you get a ratio you've had before, in which case the expansion is repeating. Proof that every repeating decimal is rational