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 , where and 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.

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 , where and are real numbers. When a complex number is written this way, is called the real part of the number, and is called the imaginary part.

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

In this activity, students compare what happens on the real number line and on the complex plane when multiplying 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.

If students have trouble with parentheses when raising to different powers, consider saying:

• “Can you explain how you rewrote the powers of .”

• “How could writing out all the factors help you collect the factors and pair them up?”

Display a blank complex plane for all to see. Invite students to share the points they plotted to represent , , , and . 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 this question 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. (The distance from 0 changes in a similar way. If you multiply the distance from 0 of each of the numbers being multiplied, then the product is that distance from 0. For example, is from 0 and 3 is 3 from 0. So, must by from 0. They may seem different because multiplying complex numbers seems to rotate the position in the complex plane while multiplying real numbers appears to have no rotation. If you consider multiplying by a positive as having 0-degree rotation and multiplying by a negative number as having 180-degree rotation, then multiplying real numbers can also fit this understanding of rotation in the complex plane.)