Operations with Imaginary Numbers

Complex Number Operations

by Steven Giesting

​Objective

​Students will review the definition of complex numbers and learn how to perform operations on complex numbers.

​Definition of Complex Numbers

​Recall that a complex number is made up of a real and an imaginary part.

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Multiple Choice

Find the sum

(-6 + 4i) + (9 + 5i)

• 15 + 9i

• 3 + 9i

• 3 + i

• -15 - i

Multiple Choice

Find the difference

(12 + 5i) - (2 - i)

• 14 + 6i

• 10 + 4i

• 10 + 6i

• 14 + 4i

​Multiplying Complex Numbers

​When multiplying complex numbers, we use the FOIL Method like we did with polynomials.

​Example 1

Simplify the expression: (8+i)(2+7i)

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​First: 8(2) = 16

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​Outer: 8(7i) = 56i

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​Inner: i(2) = 2i

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​Last:

Multiple Choice

Simplify the expression

$$\left(-6-5i\right)\left(1+3i\right)$$  

• $$10-23i$$  

• $$9-13i$$  

• $$-21-23i$$  

• $$9-23i$$  

​Graphing A Complex Number

​We graph or plot complex numbers on the complex number plane.

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​The x-axis is now the real axis and the y-axis is the imaginary axis.

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​A complex number is a point on the graph.

​Absolute Value of Complex Numbers

​Like real numbers, complex numbers have an absolute value whose value is the distance from zero or in this case the origin.

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​Look at the example to the right

Multiple Choice

Find the absolute value of the complex number

$$3-6i$$  

• $$3\sqrt[]{5}$$  

• $$3\sqrt[]{3}$$  

• $$\sqrt[]{45}$$  

• $$\sqrt[]{42}$$