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2 Tutorials that teach Multiplying Complex Numbers
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Multiplying Complex Numbers

Multiplying Complex Numbers

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In this lesson, students will learn how to multiply complex numbers with the imaginary unit i.

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Tutorial
This tutorial covers how to multiply complex numbers, through the definition and discussion of:
  1. Imaginary and Complex Numbers
  2. Multiplying Imaginary and Complex Numbers
  3. Multiplying Complex Numbers using FOIL


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.

i equals square root of negative 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^2 is equal to -1.

KEY FORMULA
i squared equals negative 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.

File:1587-nonreal1.PNG

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


2. Multiplying Imaginary and Complex Numbers

Suppose you want to multiply an imaginary number by a complex number, as in the example:

2 i left parenthesis 5 plus 3 i right parenthesis

You would begin by multiplying these numbers together using distribution, which provides:

File:1588-multcomp1.PNG

10 i plus 6 i squared equals

Next, you can simplify by remembering that i^2 is equal to -1, so you can rewrite your expression as:

10 i plus 6 left parenthesis negative 1 right parenthesis equals

Multiplying 6 times -1 gives you the complex number below, which you would rewrite in standard form with the real part first, -6, and the imaginary part second, 10i, to provide your final answer:

table attributes columnalign left end attributes row cell 10 i minus 6 equals end cell row cell negative 6 plus 10 i end cell end table


3. 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.

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

Suppose you want to multiply the following complex numbers. Using FOIL, you start by multiplying according to the steps for distributing the factors.

File:1589-multcomp2.PNG

Going in order, this provides:

16 plus 48 i minus 8 i minus 24 i squared

Recalling that i^2 is equal to -1, you can substitute -1 in for i^2, 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.

File:1590-multcomp3.PNG

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.

File:1591-multcomp4.PNG

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 learned that when multiplying an imaginary and a complex number together, you use distribution. 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.

TERMS TO KNOW
  • KEY FORMULA

    i^2 = -1