Euler's identity is one of the most beautifully compact mathematical formulas. It is:
eiπ = −1
or, more often
eiπ + 1 = 0
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:
= 1 + x
= 1 − x²⁄2!
Starting with the first one and letting x = iπ, we get:
eiπ = 1 + iπ + i²π²⁄2! + i³π³⁄3! + i4π4⁄4! + ...
= 1 + iπ − π²⁄2! − iπ³⁄3! + π4⁄4! + ...
= (1 − π²⁄2! + π4⁄4! − ...) + i(π − π³⁄3! + π5⁄5! − ...)
= cos π + isin π
= −1 + 0