This collection is designed to give an introductory taste of complex numbers, one of the fascinating areas of mathematics that you can discover by studying Further Mathematics at A-level.

Introduction

You might have been taught that we can't solve equations like $x^2 + 6x + 10 = 0$. Try it!

Imagine you are happily getting on with solving some equations. You start with
$$
x^2-6x+5=0.
$$With a little work (perhaps using the quadratic formula), you work out that this this equation is true if $x=5$ or if $x=1$. What about
$$
x^2-6x+9=0?
$$
Well, for this equation the only possibility is $x=3$. You move on to
$$
x^2-6x+10=0.
$$
Oops, hang on a minute... here things get a little weird. If you try the formula here you'll end up taking the square root of a negative number, and we all know that's impossible... Or is it? What if we imagine we can? What mathematical worlds does that open up?

It turns out that with a little imagination and mathematical bravery you can break the rules and find yourself in a whole new mathematical landscape: the complex numbers. This collection gives you an opportunity to explore these ideas yourself, and discover more about the impact and applications of complex numbers in our everyday lives.

We hope you enjoy your adventures with complex numbers and they give you a taste for the exciting mathematics you can discover by choosing Further Mathematics at A-level. If this has whetted your appetite, find out more about studying maths beyond GCSE.

This collection of resources was developed with generous support from the University of Cambridge and the University of Oxford. We would also like to thank the University of Bath and Imperial College, London.