Editing Two's complement (section)

==Why it works==
Given a set of all possible {{mvar|N}}-bit values, we can assign the lower (by the binary value) half to be the integers from 0 to {{math|(2<sup>''N'' − 1</sup> − 1)}} inclusive and the upper half to be {{math|−2<sup>''N'' − 1</sup>}} to −1 inclusive. The upper half (again, by the binary value) can be used to represent negative integers from {{math|−2<sup>''N'' − 1</sup>}} to −1 because, under addition modulo {{math|2<sup>''N''</sup>}} they behave the same way as those negative integers. That is to say that, because {{math|1= ''i'' + ''j'' mod 2<sup>''N''</sup> = ''i'' + (''j'' + 2<sup>''N''</sup>) mod 2<sup>''N''</sup>}}, any value in the set {{math|1= { ''j'' + ''k'' 2<sup>''N''</sup> {{!}} ''k'' is an integer } }} can be used in place of&nbsp;{{mvar|j}}.<ref>{{cite web
 | url = http://www.cs.uwm.edu/~cs151/Bacon/Lecture/HTML/ch03s09.html
 | title = 3.9. Two's Complement | work = Chapter 3. Data Representation
 | date = 2012-12-03 | access-date = 2014-06-22
 | publisher = cs.uwm.edu
 | archive-url=https://web.archive.org/web/20131031093811/http://www.cs.uwm.edu/~cs151/Bacon/Lecture/HTML/ch03s09.html
 | archive-date=31 October 2013
 | url-status=dead
}}</ref>

For example, with eight bits, the unsigned bytes are 0 to 255. Subtracting 256 from the top half (128 to 255) yields the signed bytes −128 to −1.

The relationship to two's complement is realised by noting that {{math|1=256 = 255 + 1}}, and {{math|(255 − ''x'')}} is the [[ones' complement]] of&nbsp;{{mvar|x}}.

{|class="wikitable floatright" style="width:14em;"
|+ Some special numbers to note
!Decimal
!Binary (8-bit)
|-
|style="text-align:right"|  127 ||0111 1111
|-
|style="text-align:right"|   64 ||0100 0000
|-
|style="text-align:right"|    1  ||0000 0001
|-
|style="text-align:right"|    0  ||0000 0000
|-
|style="text-align:right"|   −1 ||1111 1111
|-
|style="text-align:right"|  −64 ||1100 0000
|-
|style="text-align:right"| −127 ||1000 0001
|-
|style="text-align:right"| −128 ||1000 0000
|}

===Example===
{{hatnote|In this subsection, decimal numbers are suffixed with a decimal point "."}}
For example, an 8&nbsp;bit number can only represent every integer from −128. to 127., inclusive, since {{math|(2<sup>8 − 1</sup> {{=}} 128.)}}. {{nowrap|−95. modulo 256.}} is equivalent to 161. since

{{block indent|−95. + 256.}}
{{block indent|{{=}} −95. + 255. + 1}}
{{block indent|{{=}} 255. − 95. + 1}}
{{block indent|{{=}} 160. + 1.}}
{{block indent|{{=}} 161.}}
<pre style="width:25em">
   1111 1111                       255.
 − 0101 1111                     −  95.
 ===========                     =====
   1010 0000  (ones' complement)   160.
 +         1                     +   1
 ===========                     =====
   1010 0001  (two's complement)   161.
</pre>
{|class="wikitable floatright" style="width:14em;text-align:center"
|+ Two's complement 4&nbsp;bit integer values
!Two's complement
!Decimal
|-
| 0111 ||style="text-align:right"|  7. 
|-
| 0110 ||style="text-align:right"|  6. 
|-
| 0101 ||style="text-align:right"|  5. 
|-
| 0100 ||style="text-align:right"|  4. 
|-
| 0011 ||style="text-align:right"|  3. 
|-
| 0010 ||style="text-align:right"|  2. 
|-
| 0001 ||style="text-align:right"|  1. 
|-
| 0000 ||style="text-align:right"|  0. 
|-
| 1111 ||style="text-align:right"| −1. 
|-
| 1110 ||style="text-align:right"| −2. 
|-
| 1101 ||style="text-align:right"| −3. 
|-
| 1100 ||style="text-align:right"| −4. 
|-
| 1011 ||style="text-align:right"| −5. 
|-
| 1010 ||style="text-align:right"| −6. 
|-
| 1001 ||style="text-align:right"| −7. 
|-
| 1000 ||style="text-align:right"| −8. 
|}

Fundamentally, the system represents negative integers by counting backward and [[modular arithmetic|wrapping around]]. The boundary between positive and negative numbers is arbitrary, but by [[Convention (norm)|convention]] all negative numbers have a left-most bit ([[most significant bit]]) of one. Therefore, the most positive four-bit number is 0111&nbsp;(7.) and the most negative is 1000 (−8.). Because of the use of the left-most bit as the sign bit, the absolute value of the most negative number (|−8.| = 8.) is too large to represent. Negating a two's complement number is simple: Invert all the bits and add one to the result. For example, negating 1111, we get {{nowrap|0000 + 1 {{=}} 1}}. Therefore, 1111 in binary must represent −1 in decimal.<ref>{{cite web
 |first=Thomas |last=Finley
 |date=April 2000
 |title=Two's Complement
 |series=Class notes for CS&nbsp;104
 |publisher=Cornell University |department=Computer Science |place=Ithaca, New York
 |url=http://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html |access-date=2014-06-22
}}</ref>

The system is useful in simplifying the implementation of arithmetic on computer hardware. Adding 0011&nbsp;(3.) to 1111&nbsp;(−1.) at first seems to give the incorrect answer of 10010. However, the hardware can simply ignore the left-most bit to give the correct answer of 0010&nbsp;(2.). Overflow checks still must exist to catch operations such as summing 0100 and 0100.

The system therefore allows addition of negative operands without a subtraction circuit or a circuit that detects the sign of a number. Moreover, that addition circuit can also perform subtraction by taking the two's complement of a number (see below), which only requires an additional cycle or its own adder circuit. To perform this, the circuit merely operates as if there were an extra left-most bit of 1.