Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 1 / a / Solution

Write an $n$-bit binary string as $A=a_{n-1}\cdots a_1{a}_0$, where each $a_i\in \{0,1\}$ and $a_{n-1}$ is most significant. Its unsigned interpretation and two's complement interpretation areEquivalently, $\mathrm{twoc}(A)=\mathrm{bin}(A)-a_{n-1}{2}^n$. The ranges are $0$ through $2^n-1$ and $-2^{n-1}$ through $2^{n-1}-1$, respectively. The same stored bits can therefore denote different integers depending on the chosen interpretation.