# Euler's Identity

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

e = −1
or, more often
e + 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:

ex = 1 + x + x²2! + x³3! + ...
sin x = xx³3! + x55! − ...
cos x = 1 − x²2! + x44! − ...

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

e = 1 + + i²π²2! + i³π³3! + i4π44! + ...
= 1 + π²2!iπ³3! + π44! + ...
= (1 − π²2! + π44! − ...) + i(π − π³3! + π55! − ...)
= cos π + isin π
= −1 + 0
= −1