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## 'Into the Wilderness' printed from http://nrich.maths.org/

*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.