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A Brief Introduction to the Argand Diagram

Age 14 to 18
Challenge Level

This problem follows on from A Brief Introduction to Complex Numbers

Watch the video below to learn about the Argand diagram.

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Complex numbers can be represented on an Argand diagram. The real part of a number corresponds to the $x$ coordinate and the imaginary part of a number corresponds to the $y$ coordinate. So the number $z=x+iy$ would be represented by the point $(x,y)$.
The Argand diagram below shows the numbers $z_1=2+2i$ and $z_2=3-i$.


You can explore the Argand diagram using GeoGebra, a free-to-download graphing package.
We have created an online GeoGebra worksheet for you to explore the questions below.

Let $z_1$ and $z_2$ be complex numbers represented on an Argand diagram, and let $z_3$ be their product.

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 $z_1$, keeping a record of the values of $z_2$ and $z_3$.

  • 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?

Can you use algebra to explain why the values of $z_2$ you found for each $z_1$ give real values for $z_3$?

Now carry out the same process but this time aiming to keep $z_3$ on the y-axis.

You may want to have a go at Complex Squares next.