Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Functional programming, data structures and algorithms, procedural programming, and digital hardware / 10 / a / i / Solution

An $n$-bit two's complement word $b_{n-1}\cdots b_1{b}_0$ represents the integerThe leading bit has negative weight. Words beginning with zero encode the nonnegative values; words beginning with one encode the negative values. Equivalently, interpret the whole word as unsigned and subtract $2^n$ when its leading bit is one. For example, the four-bit words 0111, 1000, and 1111 encode $7$, $-8$, and $-1$ respectively. The range is asymmetric: the most negative value has no positive counterpart in the same number of bits.

For a nonzero representable magnitude, complementing all bits and adding one obtains its negative modulo $2^n$. Fixed-width addition can therefore use the same low-order binary numeral system addition circuitry for signed and unsigned operands; interpreting the result as signed requires detecting integer overflow.