Even the most powerful computer computes using a vanishingly tiny subset of all possible numbers. This follows from the fact that there are infinitely many numbers, so to specify every possible number would require an infinitely long string of bits (ones and zeroes), which no real computer can contain. Therefore some system must be adopted that allows us to compute adequately using a finite number of numbers. Two such systems--fixed-point arithmetic and floating-point arithmetic--are in common use.

The number of numbers that a register (computer circuit for remembering a string of bits) can represent depends on how many bits the register contains; N bits can represent 2^N numbers. Which 2^N numbers they represent is up to the designer. Consider a 16-bit word, which can symbolize 2¹⁶ (or 65,536) numbers. If we wish to be able to count from 1 to 65,536, we might let each possible combination of 16 bits stand...