Editing Signed number representations (section)

== Ones' complement ==
{{Main|Ones' complement}}

{|class="wikitable" style="float:right; width: 20em; margin-left: 1em; text-align:center"
|+ Eight-bit ones' complement
|-
! Binary value
! Ones' complement interpretation
! Unsigned interpretation
|-
| 00000000 || 0 || 0
|-
| 00000001 || 1 || 1
|-
| ⋮ || ⋮ || ⋮
|-
| 01111101 || 125 || 125
|-
| 01111110 || 126 || 126
|-
| 01111111 || 127 || 127
|-
| 10000000 || −127 || 128
|-
| 10000001 || −126 || 129
|-
| 10000010 || −125 || 130
|-
| ⋮ || ⋮ || ⋮
|-
| 11111101 || −2 || 253
|-
| 11111110 || −1 || 254
|-
| 11111111 || −0 || 255
|}

In the ''ones' complement'' representation,<ref>{{Cite patent|title=Array multiplier operating in one's complement format|gdate=1981-03-10|country=US|number=4484301|url=https://patents.google.com/patent/US4484301A/en}}</ref> a negative number is represented by the bit pattern corresponding to the [[bitwise NOT]] (i.e. the "complement") of the positive number. Like sign–magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 ([[−0]]).<ref>{{Cite patent|title=One's complement cryptographic combiner|gdate=1999-12-11|country=US|number=6760440|url=https://patents.google.com/patent/US6760440B1/en}}</ref>

As an example, the ones' complement form of 00101011 (43<sub>10</sub>) becomes 11010100 (−43<sub>10</sub>). The range of [[signedness|signed]] numbers using ones' complement is represented by {{nobr|−(2<sup>''N''−1</sup> − 1)}} to {{nobr|(2<sup>''N''−1</sup> − 1)}} and ±0. A conventional eight-bit byte is −127<sub>10</sub> to +127<sub>10</sub> with zero being either 00000000 (+0) or 11111111 (−0).

To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to do an ''end-around carry'': that is, add any resulting [[carry flag|carry]] back into the resulting sum.<ref>{{cite journal |last1=Shedletsky |first1=John J. |title=Comment on the Sequential and Indeterminate Behavior of an End-Around-Carry Adder |journal=IEEE Transactions on Computers |date=1977 |volume=26 |issue=3 |pages=271–272 |doi=10.1109/TC.1977.1674817|s2cid=14661474 }}</ref> To see why this is necessary, consider the following example showing the case of the addition of −1 (11111110) to +2 (00000010):

<pre>
    binary    decimal
   11111110     −1
+  00000010     +2
───────────     ──
 1 00000000      0   ← Incorrect answer
          1     +1   ← Add carry
───────────     ──
   00000001      1   ← Correct answer
</pre>

In the previous example, the first binary addition gives 00000000, which is incorrect. The correct result (00000001) only appears when the carry is added back in. 

A remark on terminology: The system is referred to as "ones' complement" because the [[Negation#Programming|negation]] of a positive value <var>x</var> (represented as the [[bitwise NOT]] of <var>x</var>) can also be formed by subtracting <var>x</var> from the ones' complement representation of zero that is a long sequence of ones (−0). Two's complement arithmetic, on the other hand, forms the negation of <var>x</var> by subtracting <var>x</var> from a single large power of two that is [[Congruence relation|congruent]] to +0.<ref>Donald Knuth: ''[[The Art of Computer Programming]]'', Volume 2: Seminumerical Algorithms, chapter 4.1</ref> Therefore, ones' complement and two's complement representations of the same negative value will differ by one.

Note that the ones' complement representation of a negative number can be obtained from the sign–magnitude representation merely by [[bitwise complement|bitwise complementing]] the magnitude (inverting all the bits after the first). For example, the decimal number −125 with its sign–magnitude representation 11111101 can be represented in ones' complement form as 10000010.