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.

### Vanishing Roots

If $y=x^2-6x+c$, and we vary $c$, what happens to the roots when $c>9$?

### Opening the Door

What happens when we add together two complex numbers?

### Strolling Along

What happens when we multiply a complex number by a real or an imaginary number?

### Into the Wilderness

Let's go further and see what happens when we multiply two complex numbers together!

### Complex Puzzle

Can you use everything you have learned about complex numbers to crack this puzzle?

### Mapping the Territory

Can you devise a system for making sense of complex multiplication?

### Complex Numbers - Power

Professor Chris Budd looks at how complex numbers play a crucial role in the electricity networks that power our daily lives.

### Complex Numbers - Insight

Dr Holly Krieger explains how she uses complex numbers to understand dynamical systems, including beautiful fractals.

### Complex Numbers - Strength

Professor Ahmer Wadee shows us how complex numbers make structures safe.

### A Complex Mistake?

Discover how Heron of Alexandria missed his chance to explore the unknown mathematical land of complex numbers.

### Maths Goes to the Movies

Got your popcorn? Picked a good seat? Are you sitting comfortably? Then - with a bit of maths - let the credits roll...

### Complex Electricity

Discover how maths helps us keep the lights on.

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.