This problem follows on from A Brief Introduction to Complex Numbers
Watch the video below to learn about the Argand diagram.
If you can't see the video, reveal the hidden text which describes the video
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 $z_3$.
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.