
This article about complex numbers is a little advanced. See here for a basic introduction to complex numbers.
Many things in mathematics are named after Leonhard Euler, who probably was the most prolific mathematician of all time. In this article we explore a formula carrying his name which reveals a beautiful relationship between the exponential function and trigonometric functions. It also allows us to write complex numbers in an exponential way.
First of all, remember that a complex number has the form
You can associate a complex number

A complex number represented as a point on the plane in Cartesian coordinates.
Now a point

A complex number represented as a point on the plane in polar coordinates.
The relationship between the Cartesian coordinates

Trigonometry tells us the relationship between polar and Cartesian coordinates.
Going back to our complex number
The power of powers
Here comes the really interesting bit.
As we explained in a piece from our Maths in a minute library, the cosine, the sine, and the exponential of a real number
Re-writing z
Now let's see what happens when we work out
(Note that we are allowed to rearrange the terms of the series because it is absolutely convergent, though we won't prove this here.)
From our representations ofWhy do we care?
Euler's formula is beautiful in its own right, but it's also useful. Imagine you want to
multiply two complex numbers

Euler's identity
Finally, we have a quick look look at what's often described as the most beautiful equation in mathematics: Euler's identity:
We can see that the equation is true because
People love this equation because it combines three of the most important numbers in maths —
About this article
Marianne Freiberger is Editor of Plus.
This article is part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

Comments
How to do it with a computer
Here is another take on the topic, plus how to do it with a popular programming language (R): https://blog.ephorie.de/euler-coding-challenge-build-maths-most-beautif…