The product between two complex numbers can be performed with only three real multiplications. This is an application of Karatsuba’s algorithm. Indeed,

x = a + i * b; y = c + i * d; real(x * y) = a * c - b * d; imag(x * y) = (a + b) * (c + d) - a * c - b * d;

Of course, the question is: *can we perform the product between two complex numbers with less than three real multiplications*?

The answer is **NO** and is provided by Winograd’s theorem in: Winograd, “On the number of multiplications required to compute certain functions”, *Commun. Pure Appl. Math.* 23 (1970), 165-179.

**NB. The minimum number of multiplications required in the computation of the product between two complex numbers is three.**