Repeating fractional binary to decimal.

Could someone explain to me how to convert the binary decimal (0.1001)2 (repeating) to a decimal in base 10? I know how to do this for regualr binary fractions but not for repeating ones. Thanks.

Answer 1

[Roger Stafford]

It isn't clear to me whether you mean a binary number with a pattern that is repeated some finite number of times, or one that is infinitely repeated. I will assume you mean infinitely repeated, as in, say,

 .11010110101101011010.....

which repeats infinitely every five binary digits. Let x be the value of the first pattern (.11010 in the example) and n be the number of binary digits in it. Then the value of the infinitely repeating pattern would be:

 y = x/(1-2^(-n))

In the above example it would be y = (26/32)/(1-2^(-5)) = 26/31 = .8387096774

There is a similar formula for one repeated a finite number of times.

[David]

Thanks for that. What's the formula for the finite version?

[Roger Stafford]

If m is the number of times the pattern is given, then

 y = x*(1-2^(-n*m))/(1-2^(-n))

In the example I gave, if m = 4 for .11010110101101011010, it would be

 y = 26/32*(1-2^(-5*4))/(1-2^(-5)) = 26/31*(1-2^(-20)) = .8387088776

Note: These answers assume the binary point is just to the left of the first pattern. If it is elsewhere make the necessary adjustment with the appropriate multiple of a power of 2.

[David]

Thanks for the response. Great help.