Each of a and c is a positive integer and each of b and d is an integer with b ≥ d.

Determine all possible pairs (a + bi, c + di) of complex numbers whose sum equals their product.

First the easy one:  (2+0i) and (2+0i)

Looking at the imaginary parts of the sum equaling the product we have:
ad+bc=b+d

Clearly if b and d are both positive or both negative the only possibility is a=c=1
Looking at the real parts
ac - bd = a + c
becomes 1 - bd = 2
bd=-1 which yields the solution
(1+1i) and (1-1i)

If instead b and d are opposite signs
ac - bd = a+c
is only possible if ac < a+c (since -bd becomes a sum)
which is only possible if a=1 or c=1
so letting c=1
ad + b = b + d
ad = d
a=1
which yields the already found solution.

Posted by Jer on 2012-08-29 13:57:00