# Eulers identity

For other meanings, see Euler function (disambiguation)

In complex analysis, Euler's identity is the equation

$\displaystyle e^{i \pi} = -1 \,\!$ ,

where

$\displaystyle e\,\!$ is the base of the natural logarithm,
$\displaystyle i\,\!$ is the imaginary unit, the complex number whose square is negative one, and
$\displaystyle \pi\,\!$ is Pi, the ratio of the circumference of a circle to its diameter.

The identity is also sometimes expressed equivalently as

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

in order to make explicit the relationship between these five fundamental mathematical constants.

## Derivation

The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. The identity is a special case of Euler's formula from complex analysis, which states that

$\displaystyle e^{ix} = \cos x + i \sin x \,\!$

for any real number $\displaystyle x\,\!$ . If $\displaystyle x = \pi\,\!$ , then

$\displaystyle e^{i \pi} = \cos \pi + i \sin \pi \,\!$

and since, by definition

$\displaystyle \cos \pi = -1\,\!$

and

$\displaystyle \sin \pi = 0\,\!$

it follows that

$\displaystyle e^{i \pi} = -1 \,\!$

## Perceptions of the identity

Benjamin Peirce, the noted 19th century mathematician and Harvard professor, after proving the identity in a lecture, said, "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."1

Richard Feynman called Euler's formula (from which the identity is derived) "the most remarkable formula in mathematics".2 Feynman, as well as many others, found this formula remarkable because it links some very fundamental mathematical constants:

Furthermore, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentiation.