| Converting number bases: fractions | ||||||||||
| Given: decimal fraction representation | ||||||||||
| Find: binary representation | ||||||||||
| value: | 0.375ten | |||||||||
| b-1 | b-2 | b-3 | b-4 | b-5 | ||||||
| weight | 2-1 | 2-2 | 2-3 | 2-4 | 2-5 | |||||
| 0.5 | 0.25 | 0.125 | 0.0625 | 0.03125 | ||||||
| digit | 0. | 0 | 1 | 1 | 0 | 0 | ||||
| To convert a whole number, we successively divided by 2 | ||||||||||
| What should we do with a fraction? | ||||||||||
| 0.375ten | = b-1 * 2-1 + b-2 * 2-2 + b-3 * 2-3 + b-4 * 2-4 + etc. | |||||||||
| If we multiply by 2, | ||||||||||
| 0.750 | = b-1* 20 + b-2 * 2-1 + b-3 * 2-2 + b-4 * 2-3 + etc. | |||||||||
| Result is still less than 1, so b-1 must be 0, since we assume all digits are positive | ||||||||||
| Multiply by 2 again, | ||||||||||
| 1.50 | = b-2 * 20 + b-3 * 2-1 + b-4 * 2-2 + etc. | |||||||||
| Now we know that b-2 must be 1, since the rest of the expression is a fraction: | ||||||||||
| 0.50 | = b-3 * 2-1 + b-4 * 2-2 + etc. | |||||||||
| Repeat the process: | ||||||||||
| 1.00 | = b-3 * 20 + b-4 * 2-1 + etc. | |||||||||
| 0.00 | = b-4 * 2-1 + b-5 * 2-2 | |||||||||
| What are all the bits to the right? | ||||||||||
| 0 | ||||||||||