How can you convert numeric data between different number systems using a floating point?

Answer 1

Here is an example: 3.1416, which is pi rounded to the nearest ten thousandth, converted to binary:

The highest power of 2 in the number is 2^1 (2), so we start with 1-. Subtracting 2 leaves 1.1416. The next power of 2 is 2^0 (1). The left over difference is greater than 1, so we have 11. so far. Subtracting 1 from the previous difference leaves 0.1416. The next power of 2 is 2^(-1), or 1/2. Our difference is less than 1/2, so our next bit (short for binary digit) is 0. So far we have 11.0. The next power of 2 is 2^(-2), or 1/4. Our difference is less than a quarter, so we have 11.00. Next is 2^(-3), or 1/8. Our difference is greater than an eighth, so we have 11.001 so far, and we subtract 0.125 from 0.1416, leaving 0.0166. That's less than 1/16, so we have 11.0010. It's also less than 1/32, so we're up to 11.00100. It's greater than 1/64, so we have 11.001001. 0.0166 - 0.015625 = 0.000975. That's less than 1/128 (so we have 11.0010010), it's less than 1/256 (11.00100100), it's less than 1/512 (11.001001000), it's less than 1/1024 (11.0010010000), but it's greater than 1/2048 (taking us to 11.00100100001). Subtracting 0.00048828125 leaves 0.00048671875. Subtracting 1/4096 and 1/8192 takes us to our original precision (one ten thousandth) and brings our number to 11.0010010000111. Since the next bit is also a 1, I will round the number up to 11.0010010001000. And there you have it: binary pi rounded to the 8192nds place.

Going from binary to decimal is easy: you just add together the powers of 2 for every place in the number that has a 1. There are 1's in the 2s place, in the 1s place, in the 8ths place, in the 64ths place and in the 1024ths place. 2 + 1 + 1/8 + 1/64 + 1/1024 = slightly more than 3.1416 (because we rounded up).