##
* *Reflection: Connection to Prior Knowledge
Dividing Complex Numbers - Section 2: Division of Complex Numbers

Emphasize the geometric interpretation! It's a fact that when the imaginary unit was first proposed, it was derided as just that: imaginary. It might look good on paper, but it was regarded as essentially meaningless by all of the best mathematicians.

Until, that is, it was given a geometric interpretation. Once the imaginary unit was understood geometrically, mathematicians were willing to accept it. Why should our students be any different? If anything, they have an even greater need for these abstract notions to be given a concrete context!

Be sure to recall for your students the parallelogram rule for adding complex numbers. This same rule can be applied to the inverse operation: subtraction. Likewise, once we have a geometric interpretation of multiplication, the same interpretation can be used for the inverse operation. The factors and the product correspond to three points in the plane, A, B, and C. The difference between the operations is simply that when multiplying you are given A & B and must find C, but when dividing you are given A & C and must find B.

This geometric interpretation gives meaning to the operation. Once your students can attach some meaning to it, then you can teach them how to carry it out more efficiently with algebra.

*Thinking Geometrically*

*Connection to Prior Knowledge: Thinking Geometrically*

# Dividing Complex Numbers

Lesson 13 of 16

## Objective: SWBAT use the properties of complex conjugates to divide two complex numbers. They will also understand and explain the corresponding graphical representation.

## Big Idea: Division of complex numbers is best understood in its relation to multiplication and transformations of the complex plane.

*45 minutes*

#### Lost in the Shuffle

*10 min*

*N.B. This lesson addresses a content standard typically reserved for Year 4 courses. For more advanced classes, I would use this as an extension lesson.*

Set the stage for division of complex numbers.

Problem #1: What do we call two numbers like (4 + 9i) & (4 - 9i)? What is their product? A real number! (That is important.)

Problem #2: Is it possible to factor the difference of two squares (4 - 9)? Is it possible to factor the sum of two squares (4 + 9)?

Problem #3: Let's multiply (7-8i)(2+11i). Notice what happens to the -88i^2 term; it gets combined with the other real term! Looking at our final number, 102 + 61i, we've lost some infomation. The part of that 102 which was originally real is mixed up with the part that was orignially imaginary!

*expand content*

#### Division of Complex Numbers

*20 min*

By now we've established the basic problem: information is lost in the multiplication of complex numbers. At this point, I'd put the following problem on the board:

(3 + 2i)(*z*) = (22 -7i)

This is a division problem, but I've expressed it in such a way that students are led to think of it in terms of the missing factor. And so I'll say, "Your job is to find the missing factor."

Students will be given 5 minutes of individual time (**MP 1**) followed by about 15 minutes of group time. Hopefully that will be enough for a handful of students to find both a graphical and algebraic solution to the problem.

If students are stuck, I might try the following suggestions:

1) Make use of the graphical representation. You know how multiplication works graphically, so you can think about it in those terms.

2) Express *z* as a ratio or fraction and then simplify. How? Find a way to make the denominator a real number. How? Recall that the product of two complex conjugates is always a real number.

*Please see the video for some more details on this section.*

*expand content*

#### Student Solutions

*15 min*

At this point, it is time for students to present their solutions to the division problem to their peers. During the work period, I would have pre-selected the students I intend to call on because their work exhibited certain insights, ways of thinking, or even instructive errors. These students would be asked to use the **document camera** to show their classmates their work and to carefully explain their thought process. The rest of the class would be encouraged to ask questions until they were confident that they understood the solution being presented. (**MP 3**) For more details on this, please see my **strategy video on Student Presentations**.

I am looking for two key insights here. The first is that the graphical representation of multiplication of complex numbers can fairly easily be "undone" to identify the missing factor. First, the argument of the missing factor is the difference of the arguments of the product and the given factor. Second, the magnitude of the missing factor is the quotient of the magnitudes of the product and the given factor. With both the argument and the magnitude known, the missing factor is known.

The second key insight is that the missing factor can be found algebraically by simplifying the ratio of the product and the given factor. When multiplied by its complex conjugate, the denominator can *always* be converted into a real number. Once the denominator is a real number, the quotient is fairly easy to obtain.

This algebraic method will ultimately be the most useful for my students, but it's vitally important that they understand the graphical method, as well.

All of this being done, tonight's homework will be four problems of complex division. Since there are so few, I'll just write them on the board for the class to copy down. They are the following:

(2 + i)(z) = (5 + 5i)

(3 - i)(z) = (17 + i)

(29 + 11i) / (5 + 7i) = z

(-6 + 18i) / (2 + 2i) = z

Alternatively, you might just assign *one* of these problems with the added instructions to solve it both graphically and algebraically.

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Quadratic Equations, Day 1 of 2
- LESSON 2: Quadratic Equations, Day 2 of 2
- LESSON 3: Inconceivable! The Origins of Imaginary Numbers
- LESSON 4: Complex Solutions to Quadratic Equations
- LESSON 5: Complex Addition
- LESSON 6: The Parallelogram Rule
- LESSON 7: Complex Arithmetic and Vectors
- LESSON 8: Multiplying Complex Numbers, Day 1 of 4
- LESSON 9: Multiplying Complex Numbers, Day 2 of 4
- LESSON 10: Muliplying Complex Numbers, Day 3 of 4
- LESSON 11: Multiplying Complex Numbers, Day 4 of 4
- LESSON 12: Practice & Review
- LESSON 13: Dividing Complex Numbers
- LESSON 14: Quadratic Functions Revisited, Day 1
- LESSON 15: Quadratic Functions Revisited, Day 2
- LESSON 16: Complex Numbers Test