Euler's identity is one of the most beautifully compact mathematical formulas. It is:

or, more often

The beauty of the identity lies in the fact that it succinctly expresses a relationship between five of the most fundamental mathematical constants:

`e`, the base of the natural logarithm`i`, the square root of −1, the imaginary unit- π, the ratio of a circle's circumference to its diameter
- 1, the real unit and the multiplicative identity
- 0, the additive identity

This identity isn't too hard to prove, provided that you know elementary trigonometry as well as how to express the following functions as infinite series:

sin

cos

Starting with the first one and letting `x` = `iπ`, we get:

= 1 +

= (1 − π²⁄2! + π

= cos π +

= −1 + 0

= −1