Adders & Subtractors — worked example

Example 1: (+5) + (−3) in 4-bit two's complement

1. Encode the operands. Positive values are their plain binary patterns; negate by inverting and adding 1.
   $$+5_{10} = 0101_2 \quad -3_{10}: \;\sim 0011 + 1 = 1100 + 1 = 1101_2$$
2. Column-add LSB to MSB. Treat the bits as unsigned for the add; the two's-complement interpretation comes back at the end.
   $$\begin{array}{rrrrrl}& 1 & 1 & 1 & 0 & \;\text{(carries)} \\ & 0 & 1 & 0 & 1 \\ + & 1 & 1 & 0 & 1 \\ \hline \\ 1 & 0 & 0 & 1 & 0\end{array}$$
   The bit that falls off the MSB is the carry-out.
3. Read the flags. The 4-bit result is $0010_2 = +2_{10}$. Carry-out = $1$ (a bit fell off the top).
4. Overflow check. Operand signs are $+, -$ — they disagree, so overflow is impossible. Equivalently, the carry into the MSB column is $1$ and the carry out is $1$; $V = C_{in} \oplus C_{out} = 0$.

Example 2: (+7) + (+1) in 4-bit two's complement

1. Encode the operands. Both positive, both already in binary.
   $$+7_{10} = 0111_2 \quad +1_{10} = 0001_2$$
2. Column-add.
   $$\begin{array}{rrrrrl}& 1 & 1 & 1 & 0 & \;\text{(carries)} \\ & 0 & 1 & 1 & 1 \\ + & 0 & 0 & 0 & 1 \\ \hline \\ 0 & 1 & 0 & 0 & 0\end{array}$$
3. Read the flags. 4-bit result is $1000_2$. As an unsigned value that's $+8$, but as 4-bit signed it's $-8$. Carry-out = $0$.
4. Overflow check. Operand signs are $+, +$ but the result sign is $-$. Signed overflow. Carry into MSB = $1$; carry out of MSB = $0$; $V = 1 \oplus 0 = 1$.