Two's complement: Difference between revisions - Wikipedia

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==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^{''N'' − 1} − 1)}} inclusive and the upper half to be {{math|−2^{''N'' − 1}}} to −1 inclusive. The upper half (again, by the binary value) can be used to represent negative integers from {{math|−2^{''N'' − 1}}} to −1 because, under addition modulo {{math|2^''N''}} they behave the same way as those negative integers. That is to say that because {{math|1= ''i'' + ''j'' mod 2^''N'' = ''i'' + (''j'' + ''k'' 2^''N'') mod 2^''N''}} any value in the set {{math|1= { ''j'' + ''k'' 2^''N'' {{!}} ''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