Bool Algebra in Computer Science

Algebra.md

Except from Computer Systems: A Programmer's Perspective, Second Edition.

For “binary(B) to unsigned(U),” length w:

For “binary(B) to two’s-complement(T)” length w:

A signed number x and its unsigned counterpart:

Accordingly:

An unsigned number u and its signed counterpart:

Accordingly:

When truncating a w-bit number to a k-bit number, we drop the high-order w−k bits. The effect of truncation for unsigned numbers is:

while the effect for two’s-complement numbers is:

Performing addition on two w-bit unsigned values:

Performing addition on two w-bit two’s-complement values:

The inverse of unsigned value x:

Two’s-Complement Negation:

The effect of the w-bit unsigned multiplication operation is:

The effect of the w-bit two’s-complement multiplication operation is:

Note: the low-order bits of the two products (unsigned and two’s complement) are identical.