Into the Wilderness

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.