TITLE: Converting repeating decimal in base b to a fraction the same base
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: 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 $x=0.3666\ldots$. Then 
$$
10x = 3.666\ldots = 3 + 6/7 = (25 + 6)/7 = 33/7
$$
so $x=33/70$. 
I used the facts that multiplying by 10 just shifts the "decimal" point, $3 \times 7 = 25$ and $6/7 = 0.66\ldots$.