Pure repeated decimals in base 10 are numbers which can written as [something]/999...999.

For example:

377/999 = 0.377 377 377 377.

1/41 = 2439/99999 = 0.02439 02439 02439 ...

Note that the number of repeated digits is equal to the number of repeated 9's in the denominator.

And the same is true for base 2 numbers. The only difference is that you need to replace 999...999 with a binary 111...111, which is 1 less than a power of two.

So what about 32/100?

Well, 32/100 = 8/25.

The smallest power of two equal to 1 (mod 25) is 2

^{20}. So we have:

2

^{20}-1 = 1048575 = 25x41943.

And:

8/25 = (8x41943)/(25x41943) = 335544/1048575 = 335544/(2

^{20}-1).

The denominator would be 11111111111111111111

_{2} (that's 20 1's). This means that 8/25 will have 20 repeated digits.

Converting 335544 to binary we get 1010001111010111000

_{2}. So:

8/25 = 335544/(2

^{20}-1) = 1010001111010111000

_{2}/11111111111111111111

_{2}.

Now because "1010001111010111000" has only 19 digits, we need to add a leading zero to get a period of 20. So the final answer is:

0.32 = 8/25 = 0.01010001111010111000 01010001111010111000 01010001111010111000 ...

And

17.32 = 10001.01010001111010111000 01010001111010111000 01010001111010111000 ...

P.S. Xenomortis, nice to see a fellow Eulerean here.