Let x1 and x2 be (b + 1)-bit binary numbers with magnitude less than 1. To compute the sum of x1 and x2 using two's-complement representation we treat them as (b + 1)- bit unsigned numbers, we perform addition modulo-2 and ignore any carry after the sign bit.
(a) Show that if the sum of two numbers with the same sign has the opposite sign, this corresponds to overflow.
(b) Show that when we compute the sum of several numbers using two's-complement representation, the result will be correct, even if there are overflows, if the correct sum is less than 1 in magnitude. Illustrate this argument by constructing a simple example with three numbers.