The ones' complement of a fixed-width binary numeral is obtained by interchanging its digits 0 and 1. Equivalently,
if it has 
bits and unsigned value 
, its ones' complement has unsigned
value 
. In an 
-bit ones'-complement representation,
a positive integer has leading digit
0, and the representation of the corresponding negative
integer is the ones' complement of its string. Thus the 8-bit
representation of 
is 
, while that of 
is 
.
The representable integers satisfy 
. Both the all-zero and
all-one bit strings represent zero,
so this value has two representations. addition uses
an end-around carry. If an addition
overflows beyond the leading digit, the overflow is returned
to the least significant digit. For example, 
gives
 |
The nines' complement is the analogous operation on fixed-width decimal numerals,
replacing each digit 
by 
(Knuth 1998).
The apostrophe positions in "ones' complement" and two's complement reflect different definitions. An 
-bit ones' complement is taken with respect to the string of
ones, whose value is 
, whereas the latter is taken with respect to the
single power 
. The variant spelling "one's complement"
treats one as a numeral by analogy with the latter term.
The numeral spelling "1's complement" is common in engineering, while the
unpunctuated form "ones complement" occurs when typography omits apostrophes.
All these forms for the ones' complement denote the same operation or representation.
