TABLE OF CONTENTS What is the best way to represent numbers on the computer? Let us start by considering integers. Typically, integers are stored using a 32-bit word, so we confine our attention to this case. If we were concerned only with nonnegative integers, the representation would be easy: a bitstring specifying the binary representation of the integer. For example, the integer 71 (see (2.1)) would be stored as
The nonnegative integers that we can represent in this way range from 0 (a bitstring of 32 zeros) to 2 32 ? 1 (a bitstring of 32 ones). The number 2 32 is too big, since its binary representation consists of a one followed by 32 zeros.
In fact, we need to be able to represent negative integers in addition to positive integers and 0. The most obvious idea is sign-and-modulus: use one of the 32 bits to represent the sign, and use the remaining 31 bits to store the magnitude of the integer, which may then range from 0 to 2 31 ? 1. However, nearly all machines use a more clever representation called 2's complement. [5] A nonnegative integer x, where 0 ? x ? 2 31 ? 1, is stored as the binary representation of x, but a negative integer ? y, where 1 ? y ? 2 31, is stored as the binary representation of the positive integer
For example, the integer
