# Lesson 12

Arithmetic with Complex Numbers

## 12.1: Math Talk: Telescoping Sums (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for adding and subtracting integers. These understandings help students develop fluency and will be helpful later in this lesson when students combine like terms to express the sums and products of complex numbers in the form \(a+bi\), where \(a\) and \(b\) are real numbers.

In this activity, students have an opportunity to notice and make use of structure (MP7) because each sum includes several pairs of opposites that sum to 0. Being able to compute efficiently by choosing which parts of an expression to evaluate first will be helpful as students develop fluency with complex number arithmetic.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

*Representation: Internalize Comprehension.*To support working memory, provide students with sticky notes or mini whiteboards.

*Supports accessibility for: Memory; Organization*

### Student Facing

Find the value of these expressions mentally.

\(2 - 2 + 20 - 20 + 200 - 200\)

\(100 - 50 + 10 - 10 + 50 - 100\)

\(3 + 2 + 1 + 0 - 1 - 2 - 3\)

\(1 + 2 + 4 + 8 + 16 + 32 - 16 - 8 - 4 - 2 - 1\)

### Student Response

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### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

*Speaking: MLR8 Discussion Supports.*Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because…” or "I noticed _____ so I….” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

*Design Principle(s): Optimize output (for explanation)*

## 12.2: Adding Complex Numbers (15 minutes)

### Activity

The purpose of this activity is to introduce the idea that sums of complex numbers can be rewritten as a single complex number in the form \(a+bi\), where \(a\) and \(b\) are real numbers. In previous lessons, students have used the complex plane to visualize complex numbers, and now they continue that work using the complex plane to visualize why the sum of two complex numbers is another complex number.

### Launch

Display the image from the task statement for all to see.

Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a brief whole-class discussion. Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image.

Arrange students in groups of 2 and give them a few minutes of work time for the activity, encouraging students to discuss each problem and, if there is disagreement, to work to reach agreement. Follow up with a whole-class discussion.

*Conversing, Writing: MLR5 Co-Craft Questions.*Use this routine to spark students’ curiosity about the sums of complex numbers. Display only the image and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving how parts of the expression are represented on the diagram. This will help students create the language of mathematical questions before feeling pressure to produce solutions.

*Design Principle(s): Maximize meta-awareness; Support sense-making*

### Student Facing

- This diagram represents \((2 + 3i) + (\text- 8 - 8i)\).
- How do you see \(2 + 3i\) represented?
- How do you see \(\text- 8 - 8i\) represented?
- What complex number does \(A\) represent?
- Add “like terms” in the expression \((2 + 3i) + (\text- 8 - 8i)\). What do you get?

- Write these sums and differences in the form \(a+bi\), where \(a\) and \(b\) are real numbers.
- \((\text- 3 + 2i) + (4 - 5i)\) (Check your work by drawing a diagram.)
- \((\text- 37 - 45i) + (11 + 81i)\)
- \((\text- 3 + 2i) - (4 - 5i)\)
- \((\text- 37 - 45i) - (11 + 81i)\)

### Student Response

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### Activity Synthesis

Select students to share their arithmetic results and connect them to arrows in the complex plane. It is important to discuss that doing operations with complex numbers results in another complex number. In order to see this, it is helpful to write the result in the form \(a+bi\), where \(a\) and \(b\) are real numbers. When a complex number is written this way, \(a\) is called the *real part* of the number, and \(bi\) is called the *imaginary part*.

If time allows, ask students, “Is \((2 + 3i) + (\text- 8 - 8i)\) the same as \((\text- 8 - 8i) + (2 + 3i)\)?” (Yes. The diagrams will look different, but, for example, going right 2 then left 8 is the same as going left 8 and right 2.)

*Representation: Internalize Comprehension.*Use color-coding and annotations to highlight connections between representations in a problem. For example, highlight connects between representations by showing the expressions \(2 + 3i\) and using one color to connect it to the arrows in the complex plane. Use a second color to highlight the connection between the expression \(\text-8 - 8i\) and the arrows in the complex plane.

*Supports accessibility for: Visual-spatial processing*

## 12.3: Multiplication on the Complex Plane (15 minutes)

### Activity

In this activity, students compare what happens on the real number line and on the complex plane when we multiply numbers. This reinforces the idea that the complex plane is a way of representing complex numbers, analogous to the real number line, and not like the coordinate plane which represents pairs of numbers.

### Launch

*Engagement: Develop Effort and Persistence.*Connect a new concept to one with which students have experienced success. For example, students are familiar with using integer exponents and drawing points on the real number line. Invite students to use the same process to multiply the repeated factors and represent them on the complex plane, reminding students that \(i^2 = \text-1\).

*Supports accessibility for: Social-emotional skills; Conceptual processing*

### Student Facing

- Draw points to represent 2, 2
^{2}, 2^{3}, and 2^{4}on the real number line. -
- Write \(2i\), \((2i)^2\), \((2i)^3\), and \((2i)^4\) in the form \(a+bi\).
- Plot \(2i\), \((2i)^2\), \((2i)^3\), and \((2i)^4\) on the complex plane.

### Student Response

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### Student Facing

#### Are you ready for more?

- If \(a\) and \(b\) are positive numbers, is it true that \(\sqrt{ab}=\sqrt{a}\sqrt{b}\)? Explain how you know.
- If \(a\) and \(b\) are negative numbers, is it true that \(\sqrt{ab}=\sqrt{a}\sqrt{b}\)? Explain how you know.

### Student Response

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### Anticipated Misconceptions

Students may need to be careful with parentheses when raising \(2i\) to different powers. It may also be helpful for students to write out all the \(2i\) factors, and then collect the \(i\) factors and pair them up.

### Activity Synthesis

Display a blank complex plane for all to see. Invite students to share the points they plotted to represent \(2i\), \((2i)^2\), \((2i)^3\), and \((2i)^4\). Draw each point on the plane, and invite students to agree or disagree with each placement.

Once it is agreed where each point should go, display the following for all to see: “How is multiplication on the complex plane similar to multiplication on the real number line? How are they different?” Give students 1 minute of quiet think time to consider their answers to these questions, then invite them to share with their partner. Follow with a whole-class discussion.

*Speaking, Representing: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. After each student shares the points that represent \(2i\), \((2i)^2\), \((2i)^3\), and \((2i)^4\), provide the class with the following sentence frames to help them respond: "I agree because…” or "I disagree because….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as “The point should be here,” can be restated as a question “Can you explain how you placed that point?” This will help students solidify their understanding plotting points on the complex plane through participating in the discussion.

*Design Principle(s): Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

Display for all to see the prompt "How are complex numbers and real numbers similar? How are they different?"

Ask students to record their thoughts in writing. After 2 minutes of quiet work time, invite students to name a similarity or a difference that they noticed. Write down student responses for all to see. Press students to be specific in their answers (MP6), and help them refine their thoughts if necessary. In particular, if students say that both complex numbers and real numbers can be graphed, ask them to be clear about what they mean; complex numbers can be graphed on the complex plane, but this is not the same kind of graph as the coordinate plane.

## 12.4: Cool-down - Add, Subtract, Multiply (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

When we add a real number with an imaginary number, we get a complex number. We usually write complex numbers as:

\(\displaystyle a + bi\)

where \(a\) and \(b\) are real numbers. We say that \(a\) is the real part and \(bi\) is the imaginary part.

To add (or subtract) two complex numbers, we add (or subtract) the real parts and add (or subtract) the imaginary parts. For example:

\(\displaystyle (2 + 3i )+(4 + 5i) = (2 + 4) + (3i + 5i) = 6 + 8i\)

\(\displaystyle (2 + 3i )-(4 + 5i) = (2 - 4) + (3i - 5i) = \text-2 - 2i\)

In general:

\(\displaystyle (a+bi) + (c+di) = (a+c) + (b+d)i\)

and:

\(\displaystyle (a+bi) - (c+di) = (a-c) + (b - d)i\)

When we raise an imaginary number to a power, we can use the fact that \(i^2=\text-1\) to write the result in the form \(a+bi\). For example, \((4i)^3=4i \boldcdot 4i \boldcdot 4i\). We can group the \(i\) factors together to see how to rewrite this.

\(\begin{align*} 4i \boldcdot 4i \boldcdot 4i &= (4 \boldcdot 4 \boldcdot 4) \boldcdot (i \boldcdot i \boldcdot i) \\ &= 64 \boldcdot (i^2 \boldcdot i) \\ &= 64 \boldcdot \text-1 \boldcdot i \\ &= \text-64i \end{align*} \)

So in this example, \(a\) is 0 and \(b\) is -64.