Complex Numbers

Multiplying a Complex Number by a Real Number

Find the product $4(2+5 i)$.

Solution

Distribute the $4$.

$\begin{aligned}4(2+5 i) &=(4 \cdot 2)+(4 \cdot 5 i) \\&=8+20 i\end{aligned}$

TRY IT #4

Find the product: $\frac{1}{2}(5-2 i)$.

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 $-1$.

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

HOW TO

Given two complex numbers, multiply to find the product.
1. Use the distributive property or the FOIL method.
2. Remember that $i^{2}=-1$.
3. Group together the real terms and the imaginary terms

EXAMPLE 5

Multiplying a Complex Number by a Complex Number

Multiply: $(4+3 i)(2-5 i)$.

Solution

$\begin{aligned}(4+3 i)(2-5 i) &=4(2)-4(5 i)+3 i(2)-(3 i)(5 i) \\&=8-20 i+6 i-15\left(i^{2}\right) \\&=(8+15)+(-20+6) i \\&=23-14 i\end{aligned}$

TRY IT #5

Multiply: $(3-4 i)(2+3 i)$.