Many mathematicians talk of there being an inherent beauty within the subject, especially with certain numbers, sequences or formulae that hold special properties.

For instance, the Golden Ratio and the Fibonacci Sequence both turn up many times in nature.

What defines a beautiful formula? This can have many answers, but it will suffice here to present a formula that combines the 5 most important numbers in elementary maths [e, π, i, 1, 0] and the two most important operations [+, =].

Here we present a derivation to get to the following "beautiful" formula:

eiπ + 1 = 0

Before we derive this formula, let's get some basic definitions out of the way.

e:   The base of the natural logarithm, approximately
i:   The square root of -1
π:   The ratio of a circle's circumference to its diameter, approximately

The first step in this formula involves theorems about complex numbers, of the form z = x + yi, where x and y are both real numbers. This is the Cartesian form for complex numbers, (x,y).

They can be re-written in polar form, (r,θ), where r is the modulus of z, sqrt(x2 + y2) and θ the angle represented by tan-1(x/y).

The diagram illustrates this, and how we arrive at the following definition of polar form:

(r,θ) = r(cosθ + isinθ)

This can be re-written into another form for complex numbers, the exponential form, using the following identities (the derivation of these will not be shown):

cos(x) = (eix + e-ix)/2
sin(x) = (eix - e-ix)/2i

We can now re-write the polar form of a complex number as the following:

(r,θ) = r(cosθ + isinθ) = reiθ
(If you don't see how this is the case, then plug in the identities for sin and cos in exponential form and then using fairly simple algebra you will reduce it to the form above)

All that remains now is to express the complex number -1 + 0i in polar form and then convert it to exponential form:

Cartesian: -1 + 0i
r = sqrt((-1)2 + (0)2) = 1
θ = tan-1(0) = π
 Polar: (1,π) = cosπ + isinπ
 Exponential: (1,π) = eiπ

 Therefore
eiπ = cosπ + isinπ

We now evaluate the two trigonometric functions at π, which gives -1 and 0 respectively (back to the Cartesian form), and rearrange:

eiπ = -1 + 0i
eiπ + 1 = 0