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One's Complement

Alternatively, a system known as ones' complement can be used to represent negative numbers. The ones' complement form of a negative binary number is the bitwise NOT applied to it — the complement of its positive counterpart. Like sign-and-magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (−0).

As an example, the ones' complement form of 00101011 (43) becomes 11010100 (−43). The range of signed numbers using ones' complement in a conventional eight-bit byte is −12710 to +12710.

To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to add any resulting carry back into the resulting sum. To see why this is necessary, consider the following example showing the case of the addition of −1 (11111110) to +2 (00000010).

        binary    decimal
       11111110     -1
    +  00000010     +2
   ............    ...
     1 00000000      0   <-- not the correct answer
              1     +1   <-- add carry
   ............    ...
       00000001      1   <-- correct answer

In the previous example, the binary addition alone gives 00000000, which is incorrect. Only when the carry is added back in does the correct result (00000001) appear.


One's Complement