Ones' complement

Three-bit ones'-complement integers
Bits	Unsigned value	Ones' complement value
000	0	0
001	1	1
010	2	2
011	3	3
100	4	−3
101	5	−2
110	6	−1
111	7	−0

8-bit ones'-complement integers
Bits	Unsigned value	Ones' complement value
0000 0000	0	0
0000 0001	1	1
0000 0010	2	2
0111 1110	126	126
0111 1111	127	127
1000 0000	128	−127
1000 0001	129	−126
1111 1101	253	−2
1111 1110	254	−1
1111 1111	255	−0

The ones' complement of a binary number is the value obtained by inverting (flipping) all the bits in the binary representation of the number. The name "ones' complement"^[1] refers to the fact that such an inverted value, if added to the original, would always produce an "all ones" number (the term "complement" refers to such pairs of mutually additive inverse numbers, here in respect to a non-0 base number). This mathematical operation is primarily of interest in computer science, where it has varying effects depending on how a specific computer represents numbers.

A ones' complement system or ones' complement arithmetic is a system in which negative numbers are represented by the inverse of the binary representations of their corresponding positive numbers. In such a system, a number is negated (converted from positive to negative or vice versa) by computing its ones' complement. An N-bit ones' complement numeral system can only represent integers in the range −(2^N−1−1) to 2^N−1−1 while two's complement can express −2^N−1 to 2^N−1−1. It is one of three common representations for negative integers in binary computers, along with two's complement and sign-magnitude.

The ones' complement binary numeral system is characterized by the bit complement of any integer value being the arithmetic negative of the value. That is, inverting all of the bits of a number (the logical complement) produces the same result as subtracting the value from 0.

Many early computers, including the UNIVAC 1101, CDC 160, CDC 6600, the LINC, the PDP-1, and the UNIVAC 1107, used ones' complement arithmetic. Successors of the CDC 6600 continued to use ones' complement arithmetic until the late 1980s, and the descendants of the UNIVAC 1107 (the UNIVAC 1100/2200 series) still do, but the majority of modern computers use two's complement.

^ Knuth, Donald E. "4.1. Positional Number Systems". The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd ed.). Detail-oriented readers and copy editors should notice the position of the apostrophe in terms like 'two's complement' and 'ones' complement': A two's complement number is complemented with respect to a single power of 2, while a ones' complement number is complemented with respect to a long sequence of 1s.

[1] Knuth, Donald E. "4.1. Positional Number Systems". The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd ed.). Detail-oriented readers and copy editors should notice the position of the apostrophe in terms like 'two's complement' and 'ones' complement': A two's complement number is complemented with respect to a single power of 2, while a ones' complement number is complemented with respect to a long sequence of 1s.

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