## complex number

Here's a quick introduction to the beauty queen amongst mathematical formulas.

This year's Abel Prize has been awarded to the Belgian mathematician Pierre Deligne for "seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields".

Solving equations often involves taking square roots of numbers and if you're not careful you might accidentally take a square root of something that's negative. That isn't allowed of course, but if you hold your breath and just carry on, then you might eventually square the illegal entity again and end up with a negative number that's a perfectly valid solution to your equation.

If you're bored with your holiday snaps, then why not turn them into fractals? A new result by US mathematicians shows that you can turn any reasonable 2D shape into a fractal, and the fractals involved are very special too. They are intimately related to the famous Mandelbrot set.

**Chris Sangwin**takes us through excerpts of Euler's algebra text book and finds that modern teaching could have something to learn from Euler's methods.

**Joan Lasenby**tells us about the mathematics and engineering behind them.

**Robert L. Devaney**explores the maths behind these beauties and shows that they're loaded with mathematical meaning.

**John Baez**declares himself fascinated by exceptions in mathematics. This interest has led him to study the octonions, and, through them, to find out more about the origins of complex numbers and quaternions. In the second of two articles, he talks about the characters of the different dimensions, beauty and utility in mathematics, and just why he likes dimension 8 so much.