Editing Euler's identity (section)

===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.