# Into the Wilderness

##### Stage: 4

To multiply 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.

Pick a complex number $z_1$, eg $4 + 2i$, and a positive real number $z_2$, eg 3.
What is $z_1 z_2$?

What happens as you change $z_1$ and $z_2$?
Can you describe geometrically the effect of multiplying by a positive real number?
What if $z_2$ is negative?

Now explore the effect of multiplying complex numbers by $i$ (you could set $z_2 = i$ in the Geogebra tool).  Can you describe the effect geometrically?

Now try this for multiplication by $-i$.

What about multiplication by $2i$, $-2i$, $\frac{1}{2}i$, $ki$ for real $k$?

What, geometrically, is the effect of multiplying by $1+i$?

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.