## Sunday, December 23, 2007

Part of a series of articles on The mathematical constant, e
Natural logarithm
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 mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation
$e^{i pi} + 1 = 0, ,!$
where
$e,!$ is Euler's number, the base of the natural logarithm,
$i,!$ is the imaginary unit, one of the two complex numbers whose square is negative one (the other is $-i,!$), and
$pi,!$ is pi, the ratio of the circumference of a circle to its diameter.
Euler's identity is also sometimes called Euler's equation.

Nature of the identity
A reader poll conducted by Mathematical Intelligencer named the identity as the most beautiful theorem in mathematics.

Perceptions of the identity
The identity is a special case of Euler's formula from complex analysis, which states that
$e^{ix} = cos x + i sin x ,!$
for any real number x. In particular, if
$x = pi,,!$
then
$e^{i pi} = cos pi + i sin pi.,!$
Since
$cos pi = -1 , !$
and
$sin pi = 0,,!$
it follows that
$e^{i pi} = -1,,!$
which gives the identity
$e^{i pi} +1 = 0.,!$
Note that the arguments to the trigonometric functions (sin and cos) are measured in radians.

Derivation
Euler's identity is a special case of the more general identity that the n-th roots of unity, for n > 1, add up to 0:
$sum_{k=0}^{n-1} e^{2 pi i k/n} = 0 .$
Euler's identity is retrieved from this generalization by putting n = 2.