Editing Two's complement (section)

===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.