Two's Complement

Definition
Two's complement is a way of storing signed integers in binary format. It uses the mechanism of integer overflow  to make the sum of all negated values equal a single zero value. (compare to One's complement which has both positive and negative zero).

The primary way to convert a two's complement number to its negative counterpart is to logically invert it and then add 1.

'Example:'

In 16 bit systems the number five is represented as
 * 0x0005 = 0000000000000101

Logical negation (replace ones with zeros and vice versa)
 * ~0x0005 = 1111111111111010

And the addition of one results in negative 5
 * -5 = 1111111111111011

To check our work we can add them together

carry | 1111111111111111 5   |  0000000000000101 -5    |+ 1111111111111011      | 0     |  0000000000000000

How to use it
As stated above, to find the two's complement negation of a number, use the bitwise negation and then add one.
 * -x = ~x+1

This can be used in systems that may not have explicit support for signed integers.