 Solution

ID: 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

A = a_{n - 1} \dots 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 are

bin (A) = i = 0 \sum n - 1 a_{i} 2^{i}, twoc (A) = - a_{n - 1} 2^{n - 1} + i = 0 \sum n - 2 a_{i} 2^{i} .

Equivalently,

twoc (A) = 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.

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