Euler's identity

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  Part of a series of articles on
The mathematical constant e

Natural logarithm · Exponential function

Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay

Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem

People John Napier  · Leonhard Euler

Schanuel's conjecture

In analytical mathematics, Euler's Identity, named for the Swiss-German mathematician Leonhard Euler, is the equality

where

Euler's Identity is also sometimes called Euler's Equation.

Contents

Beauty

Euler's identity is considered by many to be remarkable for its mathematical beauty. These three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

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