Euler's identity

Part of a series of articles on the
mathematical constant e
Properties
Natural logarithm Exponential function
Applications
compound interest Euler's identity Euler's formula half-lives exponential growth and decay
Defining e
proof that e is irrational representations of e Lindemann–Weierstrass theorem
People
John Napier Leonhard Euler
Related topics
Schanuel's conjecture
v t e

In mathematics, Euler's identity^{[note 1]} (also known as Euler's equation) is the equality

$${\displaystyle e^{i\pi }+1=0}$$

${\displaystyle e}$ is Euler's number, the base of natural logarithms,

${\displaystyle i}$ is the imaginary unit, which by definition satisfies ${\displaystyle i^{2}=-1}$, and

${\displaystyle \pi }$ is pi, the ratio of the circumference of a circle to its diameter.

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula ${\displaystyle e^{ix}=\cos x+i\sin x}$ when evaluated for ${\displaystyle x=\pi }$. Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof^[3][4] that π is transcendental, which implies the impossibility of squaring the circle.

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