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Strolling Along

Stage: 4 and 5 Challenge Level: Challenge Level:1
This activity follows on from Opening the Door.

We can multiply a complex number by a real number or an imaginary number.

We just need to remember that $i^2=-1$. So, for example, $3i \times 4i=12i^2=-12$.

We have created the GeoGebra applet below for you to explore the questions that follow. You can move the complex number $z_1$, and $z_2$, which can either be real or imaginary, and the applet will show you their product, $z_3$.

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 a variety of complex numbers by $i$ (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 $2i$, $3i$, $-2i$, $-4i$, $\frac{1}{2}i$...?
Can you describe geometrically the effect of multiplying by any multiple of $i$? 

Now that you've started to explore multiplication of complex numbers, you might like to step Into the Wilderness.