Euler identity

In mathematics, Euler’s identity[note 1] (also known as Euler’s equation) is the equality

{\displaystyle e^{i\pi }+1=0}{\displaystyle e^{i\pi }+1=0}
where
e is Euler’s number, the base of natural logarithms,
i is the imaginary unit, which by definition satisfies i2 = −1, and
π 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}{\displaystyle e^{ix}=\cos x+i\sin x} when evaluated for x = π. 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.

Contents
1 Mathematical beauty
2 Explanations
2.1 Imaginary exponents
2.2 Geometric interpretation
3 Generalizations
4 History
5 See also