The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight in the world of maths.

## Complex numbers

This teacher package brings together all *Plus* content on complex numbers. We've split the articles and videos into three groups:

- What are complex numbers and what are they used for?
- Complex numbers, chaos and fractals
- Beyond complex numbers

If you are a teacher or a student then visit this great collection of resources involving complex numbers on our sister site NRICH.

### What are complex numbers and what are they used for?

Maths in a minute: Complex numbers — A quick and concise introduction to complex numbers, how to add and multiply them, and how to represent them geometrically

Maths in a minute: Euler's identity — Here's a quick introduction to the beauty queen amongst mathematical formulas, which is based on complex numbers.

101 uses of a quadratic equation: Part II — An introduction to complex numbers, a bit of history, and interesting applications involving mobile phones.

Maths goes to the movies — Got your popcorn? Picked a good seat? Are you sitting comfortably? Then let the credits roll. Find out how complex numbers help create your favourite movies. This article includes a link to a more detailed article on the same topic.

Complex electricity — This article explores how complex numbers keep the lights on, even if the entire country decides to boil the kettle at the same time. It includes a link to a more detailed article on the same topic.

Missing a complex opportunity — Find out how Heron of Alexandria missed his chance to explore the unknown mathematical land of complex numbers.

Complex numbers: Strength — In 1940 a minor gale hit the Tacoma Narrows Bridge, the bridge started swinging and, eventually, dramatically collapsed. In this video the engineer Ahmer Wadee explains how complex numbers can help prevent such disasters.

Complex numbers: Power — So many people boiled their kettles during half time of the England v Germany match of the 1990 World Cup they nearly brought down the UK power network. In this video the mathematician Chris Budd explains how such problems can be described, and prevented, with complex numbers.

Complex numbers: Insight — In pure mathematics complex numbers are a great tool to explore dynamical systems. In this video the mathematician Holly Krieger introduces dynamical systems and explains how complex numbers come into the picture.

### Complex numbers, chaos and fractals

Chaos and fractals can arise from the simplest of mathematical objects: the humble quadratic equation and the mild-mannered Möbius transformation. To get the full benefit of their beautifully intricate behaviour, you have to look at them as complex functions. These articles tell you how.

What is the Mandelbrot set? — A hands-on introduction to the Mandelbrot set with great images. A short introduction to complex numbers is included.

Unveiling the Mandelbrot set — This longer version of the above article also looks at beautiful cousins of the Mandelbrot set, called Julia sets.

A fat chance of chaos — A news item reporting on a break-through in the field of complex dynamics: a proof that Julia sets can be fat.

Pandora's 3D box — A new 3D version of the Mandelbrot set.

Non-Euclidean geometry and Indra's pearls — An intuitive and largely equation-free introduction to hyperbolic geometry and the fractals created by tilings of hyperbolic space. Great movies and images. The article briefly discusses the role of complex numbers and Möbius transformations — a good starting point for a student project.

### Beyond complex numbers

Curious quaternions — This article gives a comprehensive introduction to complex numbers and then goes on to explore quaternions, a generalisation of complex numbers.

Ubiquitous octonions — Complex numbers were the first step, then there were the above-mentioned quaternions, and next up were the octonions. This article gives an introduction, wonders if there's anything beyond octonions, and looks at possible applications, for example in string theory.

Don't forget to visit this great collection of resources involving complex numbers on our sister site NRICH.