Negation of signed integers

• For $x$ in the range $TMi{n}_w\le x\le TMa{x}_w$, it’s two’s complement negation is given by:

$${-}_w^{t}x=\{\begin{array}{ll}TMi{n}_w,& x=TMi{n}_w\\ -x,& x>TMi{n}_w\end{array}$$

Performing the negation in binary form

There are two methods to do this.

1. Complement the bits and increment the result by 1.
   • -x = ~x + 1
2. Complement each bit that is to the left of rightmost 1

Example

$\vec{x}$	$\vec{x}$	$\text{incr}(\vec{x})=-\vec{x}$
5 [0101]	-6 [1010]	-5 [1011]
7 [0111]	-8 [1000]	-7 [1001]
-4 [1100]	3 [0011]	4 [0100]
0 [0000]	-1 [1111]	0 [0000]
-8 [1000]	7 [0111]	-8 [1000]

Sources

• Computer Systems - A Programmer’s Perspective

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