##
* *Reflection: Real World Applications
Multiplying Complex Numbers, Day 1 of 4 - Section 3: Multiplication of Complex Numbers

When teaching complex numbers, I often get the question, "What's the point of all this?" It's hard enough to understand why mathematicians would want to say that the square root of -1 is a number, but it's even harder to imagine how this fiction could be *useful*.

However, they are all familiar with vectors from their science classes, so I'll point out to them that complex numbers are vectors. Since scientists often need to work with vectors, complex numbers can be helpful. Also, since every complex number contains 2-dimensional information, they can be used in many cases where two separate numbers would be inconvenient. This is why complex numbers are used in the study of wave motion; a single number can contain information about both amplitude and phase shift. Finally, it's worth noting that the definition of distance (modulus) and argument amount to a translation from rectangular to polar coordinates. For many students, this is their first exposure to such a notion.

*What's the point?*

*Real World Applications: What's the point?*

# Multiplying Complex Numbers, Day 1 of 4

Lesson 8 of 16

## Objective: SWBAT use the unit imaginary number and the field axioms to multiply complex numbers. SWBAT represent and interpret multiplication of complex numbers in the complex number plane.

#### Powers of i

*10 min*

I will begin this lesson with a brief look at the powers of the imaginary unit, *i*. These have already come up once or twice in class, but before we begin multiplying complex numbers, it's important to activate this prior understanding. Beginning with *i* and *i*^2, we'll work quickly through the next few powers of *i*. Along the way, we will constantly reiterate the fact that *i*^2 = -1, and we'll use this fact to simplify one power of *i* after another. Before long, the class should have a pretty long list of powers and should clearly see the repeating pattern of *i*, -1, -*i*, 1. (**MP 8**)

Once this pattern is clear, it's time to dive into multiplication of complex numbers.

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Now, let the class know that today they're going to be considering multiplication with complex numbers. Just as with addition, there is an arithmetic aspect to this operation and there is a geometric aspect. (Do they recall both the arithmetic and geometric aspects of addition?)

In terms of arithmetic, multiplying complex numbers is pretty straightforward. Use the distributive law, remember that *i*^2 = -1, and then combine like terms. But what about the geometric interpretation? Will this create a parallelogram like addition does? The problem set Multiplying Complex Numbers will guide students to the answer.

Before you hand out the problems, however, I'd explain the definitions given on the first page. The relationship between "distance" and "absolute value" should be familiar from the real number line, but the "argument" will be something new. For now, students will just have to trust you that these two quantities will help them to think about the geometric interpretation of multiplication.

Now, pass out the problem set and break the class into small groups. Continually move back and forth between group work and whole-class discussion. As groups begin to finish problem 1, initiate a summary discussion of their solutions. The point is that multiplying by *i* is equivalent to a rotation of 90 degrees. For the summary discussion of #2, the point is that multiplying by a real number is equivalent to scaling the distance. Finally, for the summary of #3, the point is that multiplying by a purely imaginary number is equivalent to scaling and rotating by 90 degrees. (During this first lesson, I would not expect to get much beyond #1.)

Alternatively, you might consider running taking a Socratic approach to this lesson.

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#### Wrapping Up

*5 min*

The end of class will depend on how far we have progressed today. A 3-2-1 Exit Ticket is an appropriate formative assessment on a day like this one, since it gives students a chance to reflect on the *concepts* we've covered, and not just the skills . I expect that many students will still be a little mystified by the geometric interpretation **(MP2)**, and perhaps by the powers of *i*, but I'll have to wait for the exit tickets to know for sure.

#### Resources

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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