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I just have a question about rational numbers and their decimal expansion. I have x, which is an rational number and that $x=\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is not equal $0$. How can I show that the decimal expansion of $x$ repeats?

Michael Rozenberg
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Ben
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    Welcome to Math SX! Study carefully the division algorithm. There are only a finite number of possible remainders at each step, hence, if the algorithm never stops, … – Bernard Mar 03 '17 at 01:44
  • The decimal expansion does not necessarily repeats, take for example $a=1$ , $b=2$ the decimal expansion is $.5$ and it does not repeat itself, unless you mean that after the $5$ the $0$ repeats itself. – Jonathaniui Mar 03 '17 at 01:44
  • If $a/b$ is rational then it normalized decimal expansion is periodic. – Masacroso Mar 03 '17 at 01:45
  • @Jonathaniui you are not considering the infinite zeros or the fact that $0.4999999...=0.50000...$ – Masacroso Mar 03 '17 at 01:46
  • Certainly, I forgot that. – Jonathaniui Mar 03 '17 at 01:47

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