| If you start with the idea that you want to store every number in the same finite number of n bits, you’ve got a problem: you can store (at most) 2^n different numbers in n bits, but there are infinitely many numbers. So, if you want to hold on to that idea, the question becomes what numbers best to pick. If you go for rational numbers, you soon discover that addition of rationals is slow because you have to [1] multiply the denominators and find a greatest common divisor (example: 1/3 + 1/9 = (19 + 13)/(3*9) = 12/27 = 4/9) You also will find that your denominators can rapidly get large, and that, in real life, you need to represent a fairly wide range of numbers (say at least between 10^-3 and 10^6). That means you either accept large, random rounding in calculations, or have to pick a large number of bits Also, the math you have to do to find the closest rational for square roots and results of goniometric functions makes those functions slow. So, you either have to give up the idea to store rationals, or the idea to store every number in the same finite number of n bits. IEEE chooses the former for performance reasons. [1] assuming you want to have a single representation for each rational. You likely want, as you don’t want to waste space and because you want equality testing to be simple. |