Editing Euler's identity (section)

===Imaginary exponents===
{{main|Euler's formula}}
{{See also|Exponentiation#Complex_exponents_with_a_positive_real_base|l1=Complex exponents with a positive real base}}
[[File:ExpIPi.gif|thumb|right|In this animation {{mvar|N}} takes various increasing values from 1 to 100. The computation of {{math|(1 + {{sfrac|''iπ''|''N''}})<sup>''N''</sup>}} is displayed as the combined effect of {{mvar|N}} repeated multiplications in the [[complex plane]], with the final point being the actual value of {{math|(1 + {{sfrac|''iπ''|''N''}})<sup>''N''</sup>}}. It can be seen that as {{mvar|N}} gets larger {{math|(1 + {{sfrac|''iπ''|''N''}})<sup>''N''</sup>}} approaches a limit of −1.]]
Euler's identity asserts that <math>e^{i\pi}</math> is equal to −1. The expression <math>e^{i\pi}</math> is a special case of the expression <math>e^z</math>, where {{math|''z''}} is any [[complex number]]. In general, <math>e^z</math> is defined for complex {{math|''z''}} by extending one of the [[characterizations of the exponential function|definitions of the exponential function]] from real exponents to complex exponents. For example, one common definition is:

:<math>e^z = \lim_{n\to\infty} \left(1+\frac z n \right)^n.</math>

Euler's identity therefore states that the limit, as {{math|''n''}} approaches infinity, of <math>(1 + \tfrac {i\pi}{n})^n</math> is equal to −1. This limit is illustrated in the animation to the right.

[[File:Euler's formula.svg|thumb|right|Euler's formula for a general angle]]
Euler's identity is a [[special case]] of [[Euler's formula]], which states that for any [[real number]] {{math|''x''}},

: <math>e^{ix} = \cos x + i\sin x</math>

where the inputs of the [[trigonometry|trigonometric functions]] sine and cosine are given in [[radian]]s.

In particular, when {{math|''x'' {{=}} ''π''}},

: <math>e^{i \pi} = \cos \pi + i\sin \pi.</math>

Since

:<math>\cos \pi = -1</math>

and

:<math>\sin \pi = 0,</math>

it follows that

: <math>e^{i \pi} = -1 + 0 i,</math>

which yields Euler's identity:

: <math>e^{i \pi} +1 = 0.</math>