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# Decimal.....

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This decimal is rational: 0.12515622210319585788252097839671.....How do I convert it to proper fraction a/b. In other words, what are "a" and "b"? Thanks for help.

Jul 8, 2018

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One very simple but "foolproof" method of converting rational decimals to proper fractions is the "continued fraction" method, which goes as follows:
Take the reciprocal of the fraction: 0.12515622210319585788252097839671, and you should get:
7.9900142653352353780313837375178. Subtract the integer part, or 7, and take the reciprocal of the fraction and you should get:
1.0100864553314121037463976945245. Subtract the integer part, or 1, and take the reciprocal of the fraction and you should get:
99.142857142857142857142857142857. Subtract the integer part, or 99, and take the reciprocal of the fraction and you should get:
7. Once it converges to a whole number, then do this:
Multiply: 7 x 99.142857142857142857142857142857 =694. Then multiply this: 694 x 1.0100864553314121037463976945245 = 701. Since this is the second last multiplication, then this must the NUMERATOR.
Finally, multiply this NUMERATOR x the first number that we listed and you should get:
701 x  7.9900142653352353780313837375178=5,601 - This is the DENOMINATOR. So, the proper fraction of this rational decimal is:
701 / 5,601.
Note: This decimal fraction does not repeat its digits for 933 decimal places. Or, Its "period" is 933 digits long.

Jul 8, 2018