Converting repeating decimal in base b to a fraction the same baseThe 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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Converting repeating decimal in base b to a fraction the same baseThe 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&#039;m not too sure how to convert this decimal in this base to the fraction in the same base.

Brendan Bush · Accepted Answer

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                          .

Mylee Underwood · Accepted Answer

You could just do all your thinking in base 8. To save writing all the subscripts in the following computations I&#039;ll omit the base 8 designation. Legal digits are   0 through   7. It&#039;s a little mindbending, but only because we&#039;re used to base 10.Let   x  =  0.3666  …. Then  10  x  =  3.666  …  =  3  +  6      /    7  =  (  25  +  6  )      /    7  =  33      /    7so   x  =  33      /    70I used the facts that multiplying by 10 just shifts the &quot;decimal&quot; point,   3  ×  7  =  25 and   6      /    7  =  0.66  …

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

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