Into the Wilderness

Age 14 to 18 Challenge Level:

This resource is part of our Adventures with Complex Numbers collection

This activity follows on from Strolling Along.

To multiply two complex numbers we can expand brackets, taking care to remember that $i^2 = -1$.

For example:

$$\eqalign{\left(4+2i\right)\left(\frac{7}{5}- 3i\right) &= \frac{28}{5} + \frac{14}{5}i - 12i - 6i^2 \\ &=
\left(\frac{28}{5} + 6\right) + \left(\frac{14}{5} - 12\right)i \\ &= \frac{58}{5} - \frac{46}{5}i.}$$

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.