Two's Complement

Two's Complement is a system used to effectively represent negative integers such that we can treat the negative number as ordinary number and reuse the addition.

The problem arouses from One's Complement. For example, 2 is denoted as "00000010" and "-1" is denoted as "11111110" in One's Complement. Then naive way of doing "2 + (-1)" will get "00000000". That is incorrect. We need to further add a carry bit "1" to the sum. Then we can get "00000001", which is the right answer. However, we need to do an extra addition here.

In Two's Complement, the negative number is denoted by inverting all the bits in a positive number and adding 1. For example, "-1" is denoted as "11111111". Then we can just use the ordinary addition which save one extra addition. Besides, there is only one "0" in Two's Complement, but two in One's Complement (0 and -0).