Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

Multiplying a Complex Number by a Real Number

Let's begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example, $3(6+2 i)$:

Multiplying Complex Numbers Together

Now, let's multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term, $i^{2}$, it equals $-1$.

$\begin{array}{rlr}(a+b i)(c+d i) & =a c+a d i+b c i+b d i^{2} & \\ & =a c+a d i+b c i-b d \qquad \qquad \qquad i^{2}=-1 \\ & =(a c-b d)+(a d+b c) i \qquad \qquad \text { Group real terms and imaginary terms. }\end{array}$