onescomplement

One's complement is a binary representation for signed integers in which negative numbers are obtained by flipping all bits of the corresponding positive value. In an n-bit system, a value is represented by either the original bit pattern or its bitwise complement. The sign is indicated by the most significant bit: numbers with a leading 0 are nonnegative, and those with a leading 1 are negative.

A key feature is that zero has two representations: +0, which is all zeros, and -0, which is all ones. The representable range in n bits is from -(2^(n-1)-1) up to +(2^(n-1)-1), with two distinct zeros. For example, in 4 bits, +7 is 0111 and -7 is 1000, while +0 is 0000 and -0 is 1111.

Arithmetic follows unsigned binary addition with an end-around carry. After adding two n-bit values, if a carry exits the most significant bit, that carry is added back into the least significant bit. This yields a result in one's complement form. For instance, 0101 (+5) added to 1100 (-3) gives 0001 with a wrap-around carry that yields 0010 (+2), which matches 5 + (-3) = 2.

Negation is simply inverting all bits. Subtraction A − B can be performed as A + (~B) with end-around carry. Because there are two zero representations, comparisons may require normalization to a canonical form before deciding equality.

Historically, one’s complement was used in some early computer systems but was largely superseded by two’s complement, which has a single zero and simpler hardware for addition and subtraction.