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.

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

Beginner2022-07-09Added 14 answers

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

Mylee Underwood

Beginner2022-07-10Added 3 answers

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

$10x=3.666\dots =3+6/7=(25+6)/7=33/7$

so

$x=33/70$

I used the facts that multiplying by 10 just shifts the "decimal" point, $3\times 7=25$ and $6/7=0.66\dots $

Let $x=0.3666\dots $. Then

$10x=3.666\dots =3+6/7=(25+6)/7=33/7$

so

$x=33/70$

I used the facts that multiplying by 10 just shifts the "decimal" point, $3\times 7=25$ and $6/7=0.66\dots $

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