Lesson Objectives

- Demonstrate an understanding of a complex number: a + bi
- Learn how to plot complex numbers on the complex plane

## How to Plot Complex Numbers on the Complex Plane (Argand Diagram)

In this lesson, we want to talk about plotting complex numbers on the complex plane. Previously, we learned about the imaginary unit i. We generally define the imaginary unit i as:$$i=\sqrt{-1}$$or$$i^2=-1$$ When we combine our imaginary unit i with real numbers in the format of: a + bi, we obtain what is known as a complex number. We previously talked about complex numbers and how to perform various operations with complex numbers. Let’s recall that for any complex number written in standard form:$$a + bi$$a » the real part of the complex number

b » the imaginary part of the complex number

b is the real number that is multiplying the imaginary unit i, and just to be clear, some textbooks will refer to bi as the imaginary part. For the purposes of our lesson, we will just stick to stating that b is the imaginary part. We should also remember that the real numbers are a subset of the complex numbers. This means that every real number can be written as a complex number.$$3=3 + 0i$$$$-14=-14 + 0i$$Now we will learn how to plot a complex number on the complex plane. If you understand how to plot ordered pairs, this process is just as easy.

Example #1: Plot the given complex number.$$4 + 3i$$How can we plot this on the complex plane? We move from the origin 4 units right on the real axis since 4 is the real part. Next, we move 3 units up on the imaginary axis since 3 is the imaginary part. Label the point as 4 + 3i Example #2: Plot the given complex number.$$-9 - 6i$$How can we plot this on the complex plane? We move from the origin 9 units left on the real axis since -9 is the real part. Next, we move 6 units down on the imaginary axis since -6 is the imaginary part. Label the point as -9 - 6i

b » the imaginary part of the complex number

b is the real number that is multiplying the imaginary unit i, and just to be clear, some textbooks will refer to bi as the imaginary part. For the purposes of our lesson, we will just stick to stating that b is the imaginary part. We should also remember that the real numbers are a subset of the complex numbers. This means that every real number can be written as a complex number.$$3=3 + 0i$$$$-14=-14 + 0i$$Now we will learn how to plot a complex number on the complex plane. If you understand how to plot ordered pairs, this process is just as easy.

### Plotting Complex Numbers

First and foremost, our complex plane looks like the same coordinate plane we worked with in our real number system. The difference here is that our horizontal axis is labeled as the real axis and the vertical axis is labeled as the imaginary axis. Since we use the form: a + bi, where a is the real part and b is the imaginary part, you will also see the horizontal axis sometimes labeled as a, and the vertical axis labeled as b. This will vary, but you need to understand what’s going on if you come across different labeling.Example #1: Plot the given complex number.$$4 + 3i$$How can we plot this on the complex plane? We move from the origin 4 units right on the real axis since 4 is the real part. Next, we move 3 units up on the imaginary axis since 3 is the imaginary part. Label the point as 4 + 3i Example #2: Plot the given complex number.$$-9 - 6i$$How can we plot this on the complex plane? We move from the origin 9 units left on the real axis since -9 is the real part. Next, we move 6 units down on the imaginary axis since -6 is the imaginary part. Label the point as -9 - 6i

#### Skills Check:

Example #1

Find the graphed complex number.

Please choose the best answer.

A

$$9 - 8i$$

B

$$9$$

C

$$8 + 9i$$

D

$$9 + 8i$$

E

$$8i$$

Example #2

Find the graphed complex number.

Please choose the best answer.

A

$$6 - 3i$$

B

$$3 + 6i$$

C

$$-3 + 6i$$

D

$$-3 - 6i$$

E

$$-6 + 3i$$

Example #3

Find the graphed complex number.

Please choose the best answer.

A

$$8 + 7i$$

B

$$-8 + 7i$$

C

$$7 + 8i$$

D

$$-7 + 8i$$

E

$$7 - 8i$$

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