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Two's complement is the successor to ones' complement. That is, take the ones' complement of a value and add 1. In signed notation, the two's complement of any value negates that value.

For example:

The ones' complement of 01011010 is 10100101. To get the two's complement, add 1, thus 10100110. The process is reversible: the ones' complement of 10100110 is 01011001. Add 1 to get 01011010, which is the original value.

In unsigned notation, 01011010 is 90 decimal while 10100110 is 166. In signed notation, 10100110 is -90. This is because the most-significant bit (bit-7) indicates the sign, but also has the decimal value 128. Thus if bit-7 is set, the remaining 7 bits (38 decimal in this case) are added to -128, which is -90. Thus 10000000 is -128+0, which is -128, while 11111111 is -128+127, which is -1.

Originally, the ones' complement was used to negate values, but this is rarely used today because 1111111 would be -0, but 0 is neither signed nor unsigned, and you certainly don't want two separate zeroes. An alternative form of negation simply flips bit-7 but this had the same problem with 10000000 being -0. Two's complement doesn't have this problem so counting from -128 to +127 is greatly simplified because we can start at 10000000 (-128) and iteratively increment by 1. When we reach 11111111 (-1), adding 1 wraps around to become 00000000 and we continue until 01111111 (127). Add 1 again and we get 10000000, which takes us back to -128.

As to what is the solution for two's complement program in cpp, I do not know. I wasn't even aware there was a two's complement problem that required a solution. Perhaps if you could specify the problem (using the discussion page) we may be able to update this answer accordingly.

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