Converting repeating decimal in base b to a fraction the same base The repeating decimal .36666...

Augustus Acevedo 2022-07-08 Answered
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.
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Answers (2)

Brendan Bush
Answered 2022-07-09 Author has 14 answers
0.3 6 ¯ 8 = 3 8 + 6 ( 1 8 2 + 1 8 3 + ) = 3 8 + 6 8 2 ( 1 + 1 8 + 1 8 2 + ) = 3 8 + 6 8 2 1 1 ( 1 / 8 ) geometric series = 3 8 + 3 28 = 27 56 = 33 8 70 8 .
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Mylee Underwood
Answered 2022-07-10 Author has 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 . Then
10 x = 3.666 = 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 × 7 = 25 and 6 / 7 = 0.66
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