two's complement

Two’s complement is a method of representing signed integers in binary.

For a $n$-bit long bit string in two’s complement, a $1$ at the first bit (or the $n$-th from the right) represents the most negative value possible ($-2^{n-1}$) for this string. The rest of the bits represent the original (positive) values assigned to them in binary.

The signed integer represented by the binary string is therefore: $N={\sum }_{i=1}^{n-1}{x}_i{2}^{n-1}-2^{n-1}{x}_n$ where $x_i$ is the $i$-th bit counting from the right and $n$ is the length of the binary string.

Two’s complement for 8-bit signed integers:

Decimal	Two’s Complement Binary
0	0000 0000
1	0000 0001
2	0000 0010
126	0111 1110
127	0111 1111
−128	1000 0000
−127	1000 0001
−126	1000 0010
−2	1111 1110
−1	1111 1111

To negate a signed number in two’s complement, flip all bits and add one (~n + 1).

Therefore, to convert a negative decimal number to two’s complement:

• Find the absolute value of the number in binary
• Negate the absolute value (flip all bits and add one—“taking the two’s complement”).