Editing Two's complement (section)

===Subtraction from 2<sup>''N''</sup>===

The sum of a number and its ones' complement is an {{mvar|N}}-bit word with all 1 bits, which is (reading as an unsigned binary number) {{math|2<sup>''N''</sup> − 1}}. Then adding a number to its two's complement results in the {{mvar|N}} lowest bits set to 0 and the carry bit 1, where the latter has the weight (reading it as an unsigned binary number) of {{math|2<sup>''N''</sup>}}. Hence, in the unsigned binary arithmetic the value of two's-complement negative number {{math|''x''*}} of a positive {{mvar|x}} satisfies the equality {{math|1=''x''* = 2<sup>''N''</sup> − ''x''}}.{{efn|For {{math|1=''x'' = 0}} we have {{math|1=2<sup>''N''</sup> − 0 = 2<sup>''N''</sup>}}, which is equivalent to {{math|1=0* = 0}} modulo {{math|2<sup>''N''</sup>}} (i.e. after restricting to {{mvar|N}} least significant bits).}}

For example, to find the four-bit representation of −5 (subscripts denote the [[radix|base of the representation]]):
{{block indent|{{math|1=''x'' = 5<sub>10</sub>}} therefore {{math|size=120%|1=''x'' = 0101<sub>2</sub>}}}}
Hence, with {{math|1=''N'' = 4}}:
{{block indent|{{math|1=''x''* = 2<sup>''N''</sup> − ''x'' = 2<sup>4</sup> − 5<sub>10</sub> = 16<sub>10</sub> − 5<sub>10</sub> = 10000<sub>2</sub> − 0101<sub>2</sub> = 1011<sub>2</sub>}}}}
The calculation can be done entirely in base 10, converting to base 2 at the end:
{{block indent|{{math|1=''x''* = 2<sup>''N''</sup> − ''x'' = 2<sup>4</sup> − 5<sub>10</sub> = 11<sub>10</sub> = 1011<sub>2</sub>}}}}