Editing Signed number representations (section)

== Two's complement ==
{{Main|Two's complement}}

{|class="wikitable" style="float:right; width: 20em; margin-left: 1em; text-align:center"
|+ Eight-bit two's complement
|-
! Binary value
! Two's complement interpretation
! Unsigned interpretation
|-
| 00000000 || 0 || 0
|-
| 00000001 || 1 || 1
|-
| ⋮ || ⋮ || ⋮
|-
| 01111110 || 126 || 126
|-
| 01111111 || 127 || 127
|-
| 10000000 || −128 || 128
|-
| 10000001 || −127 || 129
|-
| 10000010 || −126 || 130
|-
| ⋮ || ⋮ || ⋮
|-
| 11111110 || −2 || 254
|-
| 11111111 || −1 || 255
|}

In the ''two's complement'' representation, a negative number is represented by the bit pattern corresponding to the [[bitwise NOT]] (i.e. the "complement") of the positive number plus one, i.e. to the ones' complement plus one. It circumvents the problems of multiple representations of 0 and the need for the [[end-around carry]] of the ones' complement representation. This can also be thought of as the most significant bit representing the inverse of its value in an unsigned integer; in an 8-bit unsigned byte, the most significant bit represents the 128ths place, where in two's complement that bit would represent −128.

In two's-complement, there is only one zero, represented as 00000000. Negating a number (whether negative or positive) is done by inverting all the bits and then adding one to that result.<ref>{{cite web|title=Two's Complement|url=http://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html|publisher=[[Cornell University]]|date=April 2000|author=Thomas Finley|access-date=15 September 2015}}</ref> This actually reflects the [[ring (mathematics)|ring]] structure on all integers [[modular arithmetic|modulo]] [[power of two|2<sup>''N''</sup>]]: <math>\mathbb{Z}/2^N\mathbb{Z}</math>. Addition of a pair of two's-complement integers is the same as addition of a pair of [[signedness|unsigned numbers]] (except for detection of [[integer overflow|overflow]], if that is done); the same is true for subtraction and even for ''N'' lowest significant bits of a product (value of multiplication). For instance, a two's-complement addition of 127 and −128 gives the same binary bit pattern as an unsigned addition of 127 and 128, as can be seen from the 8-bit two's complement table.

An easier method to get the negation of a number in two's complement is as follows:

{|class="wikitable"
!
! Example 1
! Example 2
|-
| 1. Starting from the right, find the first "1"
|align="center"| 0010100'''1'''
|align="center"| 00101'''1'''00
|-
| 2. Invert all of the bits to the left of that "1"
|align="center"| '''1101011'''1
|align="center"| '''11010'''100
|}

Method two:

# Invert all the bits through the number.  This computes the same result as subtracting from negative one.
# Add one

Example: for +2, which is 00000010 in binary (the ~ character is the [[C (programming language)|C]] [[bitwise NOT]] operator, so ~X means "invert all the bits in X"):

# ~00000010 → 11111101
# 11111101 + 1 → 11111110 (−2 in two's complement)

{{clear}}