Exercise 4.2.7

Show how to multiply the complex numbers a+bia + bi and c+dic + di using only three multiplications of real numbers. The algorithm should take aa, bb, cc and dd as input and produce the real component acbdac - bd and the imaginary component ad+bcad + bc separately.

Simple enough. The three multiplications are:

A=(a+b)(c+d)=ac+ad+bc+bdB=acC=bd A = (a + b)(c + d) = ac + ad + bc + bd \\ B = ac \quad\qquad C = bd

The result is:

(BC)+(ABC)i (B - C) + (A - B - C)i