Two's Complement

Two's complement is a mathematical operation on binary numbers, as well as a representation of signed binary numbers based on this operation. The two's complement of an N-bit number is defined as the complement with respect to 2N, in other words the result of subtracting the number from 2N. This is also equivalent to taking the ones' complement and then adding one, since the sum of a number and its ones' complement is all 1 bits. The two's complement of a number behaves like the negative of the original number in most arithmetic, and positive and negative numbers can coexist in a natural way.

In two's-complement representation, negative numbers are represented by the two's complement of their absolute value; in general, negation (reversing the sign) is performed by taking the two's complement. This system is the most common method of representing signed integers on computers. An N-bit two's-complement numeral system can represent every integer in the range −(2N−1) to +(2N−1 − 1) while ones' complement can only represent integers in the range −(2N−1 − 1) to +(2N−1 − 1).

The two's-complement system has the advantage that the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers (as long as the inputs are represented in the same number of bits and any overflow beyond those bits is discarded from the result). 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 ones'-complement systems.

The method of complements can also be applied in base-10 arithmetic, using ten's complements by analogy with two's complements.

Read more about Two's Complement:  Potential Ambiguities of Terminology, Two's Complement and Universal Algebra

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