But what's happening geometrically when we multiply complex numbers?
We have created the GeoGebra interactivity below for you to explore.
$z_3$ is the product $z_1z_2$.
You can move the complex number $z_1$ around the circle, and you can move $z_2$ anywhere on the grid. Use the slider $a$ to change the radius of the circle.
Choose a radius, and a position for $z_2$.
Move $z_1$ around the circle.
What do you notice?
Now explore what happens when you fix $z_1$ and move $z_2$. You may find it helpful to "Show Lines" joining each point to the origin.
For which values of $z_2$ does the line through $z_1$ and $z_3$ pass through the origin?
What is special about the positions of $z_1$ for which $z_2$ and $z_3$ are equidistant from the origin?
Now that you have explored multiplication of numbers, you might like to try the Complex Puzzle.