Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Target Six

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Strolling Along

Age 14 to 18Challenge Level

This resource is part of our Adventures with Complex Numbers collection

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.