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Complex Squares

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A Brief Introduction to Complex Numbers

In this problem, we define complex numbers and invite you to explore what happens when you add and multiply them.

A Brief Introduction to the Argand Diagram

Stage: 4 and 5 Challenge Level: Challenge Level:1

Why do this problem?

This problem is a suitable follow-up to A Brief Introduction to Complex Numbers to encourage students to explore the Argand diagram and get a feel for what happens geometrically when we multiply complex numbers.

Possible approach

Students will need computers to access the online GeoGebra worksheet, or to recreate the worksheet for themselves with the free open-source GeoGebra software or an alternative graphing package.

Show students the video to introduce the Argand diagram and the complex numbers functions in GeoGebra.


Then invite students to explore the questions posed:

Fix $z_1$, and move $z_2$ until $z_3$ is on the x-axis. What can you say about the trajectory of $z_2$ as you move it to keep $z_3$ on the x-axis?

Repeat the above for other values of $z_1$:
  • In each case, can you make predictions about where $z_2$ needs to be for $z_3$ to be on the x-axis?
  • Can you predict where $z_2$ needs to be when you want $z_3$ to be at a given point on the x-axis?
Once students have had a chance to explore with GeoGebra, bring the class together to share what they have found:
"Can you use algebra to explain why the values of $z_2$ you found for each $z_1$ give real values for $z_3$?"

Students can also explore what happens when they try to keep $z_3$ on the y-axis.

Key questions

What can be said about the complex number $z_3=x+iy$ if it lies on the x-axis?
If $z_1$ is at the point $a+ib$, what can be said about the point $z_2$ in order for $z_3$ to be on the x-axis?

Possible extension

Once students have grasped the Argand diagram, they could try Complex Squares.

Possible support

Ask students to pick a point for $z_1$ such as $2+i$, move $z_2$ around, and keep a record of the points where $z_3$ is on the x-axis. What do they notice about the coordinates of $z_2$ and $z_3$?

Suggest they repeat for some other points and share their results.