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# Review of Complex Numbers

Lesson 5 of 11

## Objective: SWBAT find the sum, product and quotient and absolute value of a complex number as well as graph a complex number on a coordinate plane?

*40 minutes*

#### Bell work

*15 min*

Over the next 2 lessons students will see how converting complex numbers to trigonometric form can make computations easier. This seems like a change of topic but writing a vector as the sum of unit vectors **i** and **j** is similar to writing complex numbers in trigonometric form.

I begin the day by giving students a worksheet with 3 expressions to simplify this activity allows students to review operations of complex numbers. After about 3 minutes of working alone, I have the students move to groups by counting off by 3's. The groups get together and discuss the problems for a couple of minutes. As they discuss, I review student's work and then give each group one of the problems to share out. If groups ask about a problem I refer the group to pages 159-162 in Larson, "Precalculus with Limits, 2nd ed."

Some questions I ask as I move around the room are:

- What is
*i*? - How do you know when you have a simplified expression?
- What are expression like (3+
*i*) and (3-*i*) called?

I expect students to struggle with the last problem on the worksheet. Most students will not think about removing the complex number from the denominator of the fraction. I ask "when you worked on complex numbers last year did you have any rules you needed to follow?" This will help some students.

After about 3-4 minutes students share the answers the groups found. We go through the examples and review ideas such as *i*^2=-1 and conjugates.

This activity seems to take some time but is necessary to move through the rest of the unit.

#### Resources

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#### Graphing a complex number

*10 min*

Now that we have worked on reviewing operations with complex numbers I develop how to graph a complex number. I use the idea of graphing vectors.

I give students several complex numbers to graph. I use students that remember how to graph to share what they did last year. This allows me to assess whether they understand that the horizontal axis represents real numbers and the vertical axis is the imaginary axis. If a student just plots the point without labeling the axis, I ask "How did you know to plot it this way?" Students sometimes say that the y is imaginary. This opens the discussion about the rectangular coordinate system is for real numbers and we are using another coordinate system. I remind students that labeling is vital for understanding. Once we have the graphs labeled we are able to graph several points.

I now ask "How far is 3+2*i* from 0?" I give students a couple of minutes to discuss this questions. Students need to realize that 0 is at the origin. Students will ask where is zero. My response is "How do you write zero as a complex number?" Once students realize 0 can be written as 0+0*i *students see that 0 is at the origin. Some students graph the 2 points (0 and 3+2i) and see that we just need to find the distance using either the distance formula or Pythagorean Theorem. I explain that this distance is called the absolute value of a complex number. Instead of me just giving students the book definition of the absolute value of a complex number I have the students find the definition. This is not a hard definition and by giving students the first few words students are able to finish the definition.

#### Resources

*expand content*

I now ask "How far is 3+2*i* from 0?" I give students a couple of minutes to discuss this questions. Students need to realize that 0 is at the origin. Students will ask where is zero. My response is "How do you write zero as a complex number?" Once students realize 0 can be written as 0+0*i *students see that 0 is at the origin. Some students graph the 2 points (0 and 3+2i) and see that we just need to find the distance using either the distance formula or Pythagorean Theorem. I explain that this distance is called the absolute value of a complex number. Instead of me just giving students the book definition of the absolute value of a complex number I have the students find the definition. This is not a hard definition and by giving students the first few words students are able to finish the definition.

*expand content*

#### Closure

*5 min*

As we come to an end for today I give students a problem to answer as an exit slip. The students are given 2 complex numbers that differ by a factor. I ask students to determine how the factor changes the graph and the absolute value.

Students should see how the point is reflected across the origin as well as further away. The absolute value is 2 times larger. Many students are surprised that the -2 only changes the absolute value by 2 times. Many think it should be 4 times larger.

#### Resources

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Introduction to Vectors
- LESSON 2: Component Form of Vectors
- LESSON 3: Operation with Vectors
- LESSON 4: Solving Problems with Vectors
- LESSON 5: Review of Complex Numbers
- LESSON 6: Complex Numbers and Trigonometry
- LESSON 7: Operations of Complex Numbers in Trigonometric Form
- LESSON 8: DeMoivre's Theorem
- LESSON 9: Roots of Complex Numbers
- LESSON 10: Review
- LESSON 11: Assessment