Two's complement

Summary

Two's complement is a method for representing signed numbers (negative integers) in binary.
The two's complement of a given binary integer is its negative equivalent.

Detail

Procedural steps

Two's complement divides the available word length (see Binary_encoding) into two subsets: one for negative integrs and one for positive integers.

Take the binary encoding of decimal five (0101). Its complement is 1011.

The procedure for deriving the complement is as follows.

To derive the complement of an unsigned number:

Take the unsigned number and invert its digits: 0 becomes 1, 1 becomes 0
Add one

To derive the unsigned equivalent of a signed number you invert the process but still make the smallest digit 1:

Formal expression

2^n - x

where x is the negative integer in binary that we wish to derive
where n is the word length of the binary system in bits.

Applied to the earlier example we have 2^4 -5 which is:

16 - 5 = 11

When we convert the decimal 11 to binary we get 1011 which is identical to the signed version of the unsigned integer.

We can confirm the correctness of the derviation by summing the signed and unsigned binary values. If this results in zeros (ignoring the overflow bit), the derivation is correct as the two values effectively cancel each other out:

1011 + 0101 = 0000

Applications

Signed_and_unsigned_numbers

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