*This article about complex numbers is a little advanced. Take a look behind yesterday's door 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 where and are real numbers and is the square root of .

You can associate a complex number with the point in the plane that has Cartesian coordinates .

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

Now a point in the plane can also be described by its *polar coordinates* . Here is the distance from to and is the angle formed by the line connecting to and the positive -axis (measured anti-clockwise).

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

The relationship between the Cartesian coordinates and the polar coordinates can be worked out using a little trigonometry. It is

Trigonometry tells us the relationship between polar and Cartesian coordinates.

Going back to our complex number , we now see that it can also be written as

Here comes the really interesting bit. We can also prove, using *power series*, that

So, in summary, *Euler's formula* is that a complex number

can also be written as

where

and

It is associated to the point in the plane that has Cartesian coordinates and polar coordinates .

Euler's formula is beautiful in its own right, but it's also useful. It makes multiplying two complex numbers much easier. And of course it also leads to 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 — , , , , and — with three of the most important mathematical operations— addition, multiplication, and exponentiation. It almost feels like a miracle that these numbers and operations combine in such a beautifully elegant way.

*You can read more about Euler's formula in our longer article. Or you can return to the Plus advent calendar 2021.*

*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.*