Question

Given n is the number of positions in radix-r, how many possibilities are wasted due to the sign position in the signed-magnitude numbering system.

Answer 1

I'm attaching the pic for better understanding:-

As total number possible with n positions = rⁿ (i.e, 0 to rⁿ -1)

But for signed magnitude one bit will be wasted for sign representation. So, it will not be fully utilized.

For signed-magnitude number possible will be = 2 * r^n-1 (i.e, -r^n-1 to r^n-1 -1)

So, the numbers wasted will be-

rⁿ - 2 * r^n-1 = r^n-1 * (r - 2)

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