Complex numbers: Why do we love them?

Imagine you are minding your own business, solving some equations at school or work. You start with $x^{2} = 1$ , and pretty quickly realise this this equation is true if $x = 1$ or if $x = - 1$ . What about $x^{2} = 0 ?$ Well, for this equation the only possibility is $x = 0$ .

You move on to $x^{2} = - 1.$ Now hang on a minute... here things get a little weird. Taking the square root of a negative number is not allowed... Or is it?

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 package gives you a brief introduction to this new world, and explores the impact it has on our everyday lives and people who use complex numbers in their work everyday.

If you are looking for teaching and learning resources involving complex numbers, you might also want to have a look at the package Adventures with complex numbers, which we developed in collaboration with our sister site NRICH. And for more articles on complex number on Plus, see our teacher package.

Videos

Complex numbers: Strength — In 1940 a minor gale hit the Tacoma Narrows Bridge, the bridge started swinging and, eventually, dramatically collapsed. 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. 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. Holly Krieger introduces dynamical systems and explains how complex numbers come into the picture.

Articles

Maths in a minute: Complex numbers — Here's a brief introduction to complex numbers, including their geometrical interpretation as points on the plane.

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.

Complex electricity — Chris Budd explains how complex numbers keep the lights on, even if the entire country decides to boil the kettle at the same time.

What is the Mandelbrot set? — The Mandelbrot is one of the most famous fractals of them all. Find out what it is and what it has to do with complex numbers.

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