By Jariya Phongsai

A two's-complement system is a system in which negative numbers are represented by the two's complement of the absolute value; this system is the most common method of representing signed integers on computer. In such a system, a number is negated (converted from positive to negative) by computing its two's complement.

The two's-complement system has the advantage of not requiring that the addition and subtraction circuitry examine the signs of the operands to determine whether to add or subtract. This property makes the system both simpler to implement and capable of easily handling higher precision arithmetic. Also, zero has only a single representation, obviating the subtleties associated with negative zero, which exists in one’s complement systems.

Example

Suppose we're working with 8 bit and suppose we want to find how -28 would be expressed in two's complement notation. First we write out 28 in binary form.

0 0 0 1 1 1 0 0

1 1 1 0 0 1 0 0

1 1 1 0 0 0 1 1

Then we invert the digits. 0 becomes 1, 1 becomes 0.

Then we add 1.

That is how one would write -28 in 8 bit binary.

The system is called two's complement because the negative of a number is formed by complementing each bit (subtracting it from 1, the "one's complement") and then adding 1 to the result:

  Because of the way the numbers are

assigned, adding one in binary still adds one to the negative number; in fact, when you add one to -1, you get zero (ignoring the carry). To add any two signed numbers, you just add the binary values, and again ignore the carry, if any