Question

36. If m is a positive integer less than $2^{n-1}$, how is the one’s complement representation of −m obtained from the one’s complement of m, when bit strings of length n are used?

Short answer

The one’s complement of$-m$is the one’s complement of$m$with every 1 changed to 0 and every 0 changed to 1.

Step-by-step solution

Step 1:

$m$is positive, thus the one’s complement of $m$starts with a 0(first digit).

$-m$is negative, thus the one’s complement of $-m$starts with a 1(first digit) and thus we need to change the first digit from 0 to 1.

Ignoring the first digit, the one’s complement of $m$is then the binary representation of $m$

The one’s complement of$-m$(without the first digit) is then the complement of the binary representation of m and thus every 1 is changed to a 0 and every 0 is changed to a 1.

We then note that the one’s complement of$-m$is the one’s complement of$m$with every 1 changed to 0 and every 0 changed to 1