Adding unsigned integers

• For $x$ and $y$ such that $0\le x,y<2^w$:

$$x{+}_w^{\mathrm{u}}y=\{\begin{array}{llc}x+y,& x+y<2^w& \mathrm{Normal}\\ x+y-2^w,& 2^w\le x+y<2^{w+1}& \mathrm{Overflow}\end{array}$$

• This forms an Abelian group

Visual demonstration

Detecting overflow

• The sum of two unsigned numbers, $s=x{+}_w^{u}y$ can be detected as an overflow if $s<x$ (or equivalently, $s<y$).

Sources

• Computer Systems - A Programmer’s Perspective

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