multiplying complex numbers

Multiplying complex numbers is just like multiplying two binomials with the FOIL method — multiply everything by everything — with one extra step at the end: wherever an i^2 appears, replace it with -1. That single substitution is the only thing that distinguishes complex multiplication from ordinary algebra.

Working it out, (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i, after using i^2 = -1 to turn bdi^2 into -bd. The real part of the product is ac - bd and the imaginary part is ad + bc.

There is a beautiful geometric meaning hiding here: multiplying by a complex number both stretches and rotates the plane. The moduli multiply and the arguments add, so |zw| = |z||w| and arg(zw) = arg(z) + arg(w). Multiplying by i, for instance, leaves the size unchanged but rotates everything a quarter turn counterclockwise. Polar form makes this rule effortless.