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.}

We have created the GeoGebra interactivity below for you to explore the questions that follow. $z_3$ is the product $z_1z_2$.

Pick a complex number $z_1$, and multiply it by $1+i$. You could set $z_2=1+i$ in the interactivity and vary $z_1$.

Can you describe geometrically what happens? Can you explain why?

Now choose some other complex values for $z_2$, and vary $z_1$.

Can you find any complex numbers where multiplying by them stretches but doesn't rotate?
Can you find any complex numbers where multiplying by them rotates but doesn't stretch?

Can you predict the geometric effect of multiplication by the complex number $a + bi$?

Now that you have explored multiplication of numbers, you might like to start Mapping the Territory.