twoscomplement

Two's complement is a method for encoding signed integers in binary using a fixed number of bits. In an n-bit system, the representable integers range from -2^(n-1) to 2^(n-1)-1. Nonnegative values (0 to 2^(n-1)-1) are represented as their ordinary binary form, while negative values are represented by patterns in the upper half of the range. Arithmetic is performed modulo 2^n, so binary addition and subtraction wrap around within the n-bit space.

To obtain the two's complement (the negative) of a positive number x, you can either invert all bits and add 1, or compute 2^n - x and then take the result modulo 2^n. For example, in 8-bit two's complement, 127 is 01111111, while -1 is 11111111 and -128 is 10000000. This system ensures that adding a number and its two's complement yields zero in modulo 2^n arithmetic, and that subtraction is performed by adding a negated value.

Two's complement has several advantages: it uses a single, uniform representation for all integers; there is no separate sign-magnitude or end-around carry handling; and hardware addition can be designed generically without special cases for negative numbers. Overflow occurs when a calculation produces a result outside the representable range, which can be detected by examining the carry into and out of the sign bit. It is the most widely used signed-integer representation in modern computer architectures.