Augustus Acevedo

2022-07-08

Converting repeating decimal in base b to a fraction the same base
The repeating decimal .36666... in base 8 can be written in a fraction in base 8. I understand simple patterns such as 1/9 in base 10 is .1111.... so 1/7 in base 8 is .1111. But I'm not too sure how to convert this decimal in this base to the fraction in the same base.

Brendan Bush

$\begin{array}{rl}0.3{\overline{6}}_{8}& =\frac{3}{8}+6\left(\frac{1}{{8}^{2}}+\frac{1}{{8}^{3}}+\cdots \right)\\ & =\frac{3}{8}+\frac{6}{{8}^{2}}\left(1+\frac{1}{8}+\frac{1}{{8}^{2}}+\cdots \right)\\ & =\frac{3}{8}+\frac{6}{{8}^{2}}\frac{1}{1-\left(1/8\right)}& \text{geometric series}\\ & =\frac{3}{8}+\frac{3}{28}\\ & =\frac{27}{56}\\ & =\frac{{33}_{8}}{{70}_{8}}.\end{array}$

Mylee Underwood

You could just do all your thinking in base 8. To save writing all the subscripts in the following computations I'll omit the base 8 designation. Legal digits are $0$ through $7$. It's a little mindbending, but only because we're used to base 10.
Let $x=0.3666\dots$. Then
$10x=3.666\dots =3+6/7=\left(25+6\right)/7=33/7$
so
$x=33/70$
I used the facts that multiplying by 10 just shifts the "decimal" point, $3×7=25$ and $6/7=0.66\dots$

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