Binary arithmetic, two's complement and overflow

Adding in binary

Four rules only: $0+0=0$; $0+1=1$; $1+1=10$ (write $0$, carry $1$); $1+1+1=11$ (write $1$, carry $1$). Work right to left exactly as in denary. The carry is the only place mistakes happen, so write it above the column.

Representing negative numbers: two's complement

To make $-n$ in $8$ bits: write $n$ in binary, flip every bit, then add one. So $-45$: $45$ is $00101101$, flipped is $11010010$, plus one is $11010011$. The leftmost bit now acts as a sign: $0$ means positive, $1$ means negative.

Why two's complement and not sign-and-magnitude

Because subtraction becomes addition: $a - b$ is just $a + (-b)$, so the processor needs only an adder. There is also only one representation of zero, whereas sign-and-magnitude has both $+0$ and $-0$.

Overflow

An $8$-bit signed register holds $-128$ to $+127$. If a calculation lands outside that range the result wraps round and is simply wrong — $100 + 50$ gives $-106$. Overflow is detected when adding two numbers of the same sign produces a result of the opposite sign.