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Multiply Complex Numbers

Author: Sophia

what's covered
In this lesson, you will learn how to multiply two complex numbers. Specifically, this lesson will cover:

Table of Contents

1. Complex Numbers

A complex number is a number in the form a plus b i, containing both a real part and an imaginary part. The imaginary part is followed by i, which is the imaginary unit, square root of short dash 1 end root.

Recall the following formulas for imaginary numbers:

formula to know
Imaginary Number
table attributes columnalign left end attributes row cell i equals square root of short dash 1 end root end cell row cell i squared equals short dash 1 end cell end table

When multiplying complex numbers, we follow a similar process when multiplying binomial factors we may be familiar with when studying quadratics. The multiplication process is often referred to as FOIL, which distributes terms into the factors being multiplied. Let's take a moment to review FOIL with real numbers before looking at examples of complex number multiplication.


2. FOIL Review

FOIL stands for First, Outside, Inside, Last, and refers to terms that are multiplied together to form individual addends to the product.

EXAMPLE

Multiply open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses.

open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses Multiply first terms: x times x equals x squared
x squared Multiply outside terms: x times short dash 3 equals short dash 3 x
x squared minus 3 x Multiply inside terms: 2 times x equals 2 x
x squared minus 3 x plus 2 x Multiply last terms: 2 times short dash 3 equals short dash 6
x squared minus 3 x plus 2 x minus 6 Combine like terms
x squared minus x minus 6 Our solution

When multiplying two complex numbers, we will be following the same procedure but will need to make an additional consideration when the imaginary unit is squared.


3. Multiplying Complex Numbers

When multiplying complex numbers, we'll want to consider the imaginary unit, i.

EXAMPLE

Multiply the complex numbers open parentheses 2 plus 3 i close parentheses open parentheses 4 plus 2 i close parentheses.

open parentheses 2 plus 3 i close parentheses open parentheses 4 plus 2 i close parentheses Multiply first terms: 2 times 4 equals 8
8 Multiply outside terms: 2 times 2 i equals 4 i
8 plus 4 i Multiply inside terms: 3 i times 4 equals 12 i
8 plus 4 i plus 12 i Multiply last terms: 3 i times 2 i equals 6 i squared
8 plus 4 i plus 12 i plus 6 i squared Combine like terms
8 plus 16 i plus 6 i squared Simplify i squared

The final step here is to simplify the last term, containing the imaginary unit squared. Recall that the imaginary unit is square root of short dash 1 end root. When this is squared, it becomes the real number -1.

To simplify i squared terms, we can remove i squared completely, but reverse the sign of its coefficient. For example, plus 6 i squared simplifies to -6. This is a real number that can be combined with other like terms.

8 plus 16 i plus 6 i squared Rewrite plus 6 i squared as -6
8 plus 16 i minus 6 Combine like terms
2 plus 16 i Our solution

big idea
To multiply complex numbers, we use the FOIL process to multiply the terms in the two complex numbers. During this process, we simplify i squared to -1, which is a real number.

summary
Complex numbers consist of a real part and an imaginary part. The square root of negative 1 is imaginary because no real number squared results in a negative number. Because complex numbers are binomials, use FOIL when Complex numbers consist of a real part and an imaginary part. The square root of negative 1 is imaginary because no real number squared results in a negative number. Because complex numbers are binomials, use FOIL when multiplying complex numbers. Simplify expressions with an i squared term by substituting negative 1 for i squared, multiplying by the coefficient, and writing as a real number with the opposite sign.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Formulas to Know
Imaginary Number

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