Multiplying two complex numbers using only three multiplications of real numbers

Question

I have problem given below.

Show how two complex numbers $(a+ib)$ and $(c+id)$ may be multiplied using only three multiplications of real numbers, where $i=\sqrt{-1}$. You may use any number of additions and subtractions.

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Please help me. How to solve this?

Answer 1

You are interested in two numbers : $\alpha_1 = ac-bd \text{ and } \alpha_2 = ad+bc$. You can compute 3 products, viz. $P_1 = ac, P_2 = bd, \text{ and } P_3 = (a+b)(c+d)$. Then $\alpha_1 = P_1 - P_2, \text{ and } \alpha_2 = P_3 - P_2 - P_1$.

This process is often called Karatsuba multiplication, and is used in algorithm design quite frequently.