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Multiplying Complex Numbers using FOIL

Author: Sophia

what's covered
This tutorial covers how to multiply complex numbers, through the definition and discussion of:

Table of Contents

1. Imaginary and Complex Numbers

To review, the square root of a negative number is non-real, or an imaginary number. The imaginary unit i is defined as the square root of -1.

formula to know
Imaginary Number
i equals square root of short dash 1 end root

If you were to square both sides of this equation, you would have i^2 on the left side, and -1 on the right side, so you also know that i squared is equal to -1.

formula to know
Imaginary Number
i squared equals short dash 1

A complex number is a value in the form below, in which a and b are real numbers, and i is the imaginary unit.

did you know
Complex numbers are used in fields such as engineering and physics.


2. Multiplying Complex Numbers using FOIL

When multiplying complex numbers, use the FOIL method, because of the addition or subtraction that occurs between the real and imaginary parts of complex numbers. Therefore, multiplying complex numbers together is similar to multiplying binomials together.

hint
You may recall that FOIL is an acronym to remember the steps for distributing factors in binomial multiplication:
First
Outside
Inside
Last

EXAMPLE

Suppose you want to multiply the following complex numbers.

open parentheses 8 minus 4 i close parentheses open parentheses 2 plus 6 i close parentheses

Using FOIL, you start by multiplying according to the steps for distributing the factors.

open parentheses 8 close parentheses open parentheses 2 close parentheses plus open parentheses 8 close parentheses open parentheses 6 i close parentheses plus open parentheses short dash 4 i close parentheses open parentheses 2 close parentheses plus open parentheses short dash 4 i close parentheses open parentheses 6 i close parentheses

Going in order, this provides:

16 plus 48 i minus 8 i minus 24 i squared

Recalling that i squared is equal to -1, you can substitute -1 in for i squared, then multiply negative 24 times -1.

table attributes columnalign left end attributes row cell 16 plus 48 i minus 8 i minus 24 left parenthesis negative 1 right parenthesis equals end cell row cell 16 plus 48 i minus 8 i minus left parenthesis negative 24 right parenthesis end cell end table

To simplify your expression, you know that the real parts are like terms, so you can combine 16 minus -24, which equals 40. You also know that your imaginary parts are like terms so you can combine 48i minus 8i, which equals 40i, so your final answer is 40 plus 40i. Note that this is in standard form.

40 plus 4 i

summary
Today you reviewed imaginary numbers, recalling that the square root of a negative number is non-real, or an imaginary number; the imaginary unit i is equal to the square root of -1. You also reviewed the definition of a complex number, which is a value in the form a plus bi, where a is the real part and b times i is the imaginary part of the complex number. You also learned that when multiplying two complex numbers together, you use the FOIL method.

Source: This work is adapted from Sophia author Colleen Atakpu.

Formulas to Know
Imaginary Number

i equals square root of short dash 1 end root
i squared equals short dash 1