I need to determine whether fraction is recurring decimal or not (what are conditions for it?), find period and output it as $\frac{1}{3}=0.\bar{3}$.
If it is not recurring, then I already have algorithm.
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The algorithm always outputs, True. Should it also output the decimal? – Peyton Jul 29 '17 at 17:44
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All rational numbers (fractions of integers) have a decimal expansion that is repeating from some point on and vice versa. Example: $12.345\overline {678} = 4111111/333000$. – md2perpe Jul 29 '17 at 17:48
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1Long division is an algorithm – spaceisdarkgreen Jul 29 '17 at 17:57
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Notice that if you have a repeating decimal $$n=0.\overline{D_1D_2...D_k}$$ where each $D_i$ is a digit, you can say that $$10^k n=D_1D_2...D_k.\overline{D_1D_2...D_k}$$ and so $$10^k n-n=D_1D_2...D_k$$ and $$n=\frac{D_1D_2...D_k}{10^k-1}$$ This means that $n$ can only be a repeating decimal if $n$ can be expressed in the form $$n=\frac{a}{10^b-1}$$ and so if you are given a number in the form $$I+\frac{h}{j}$$ Where $I$ is an integer and $j\gt h$, then it can only be a repeating decimal if $j$ evenly divides $10^k-1$ for some $k$. For example, for $j=1,2,5,10$ it will never be a repeating decimal.
Does this help?
Franklin Pezzuti Dyer
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