Editing Euler's identity (section)

==Explanations==
===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>

===Geometric interpretation===
Any complex number <math>z = x + iy</math> can be represented by the point <math>(x, y)</math> on the [[complex plane]]. This point can also be represented in [[Complex_number#Polar_complex_plane|polar coordinates]] as <math>(r, \theta)</math>, where {{Mvar|r}} is the absolute value of {{Mvar|z}} (distance from the origin), and <math>\theta</math> is the argument of {{Mvar|z}} (angle counterclockwise from the positive ''x''-axis). By the definitions of sine and cosine, this point has cartesian coordinates of <math>(r \cos \theta, r \sin \theta)</math>, implying that <math>z = r(\cos \theta + i \sin \theta)</math>. According to Euler's formula, this is equivalent to saying <math>z = r e^{i\theta}</math>.

Euler's identity says that <math>-1 = e^{i\pi}</math>. Since <math>e^{i\pi}</math> is <math>r e^{i\theta}</math> for {{Mvar|r}} = 1 and <math>\theta = \pi</math>, this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive ''x''-axis is <math>\pi</math> radians.

Additionally, when any complex number {{Mvar|z}} is [[Complex number#Multiplication and division in polar form|multiplied]] by <math>e^{i\theta}</math>, it has the effect of rotating <math>z</math> counterclockwise by an angle of <math>\theta</math> on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point <math>\pi</math> radians around the origin has the same effect as reflecting the point across the origin.  Similarly, setting <math>\theta</math> equal to <math>2\pi</math> yields the related equation <math>e^{2\pi i} = 1,</math> which can be interpreted as saying that rotating any point by one [[turn (angle)|turn]] around the origin returns it to its original position.