1.2 KiB
+++ title = "MathJax Support" date = 2021-01-03 [taxonomies] categories = ["math"] tags = ["Euler's identity"] [extra] math = true +++
Please add the following lines in the front matter when using MathJax.
[extra]
math = true
Euler's identity
e^{i\pi }+1=0
Geometric interpretation
Any complex number z=x+iy can be represented by the point (x,y) on the complex plane. This point can also be represented in polar coordinates as (r,\theta ), where r is the absolute value of z (distance from the origin), and \theta is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of (r\cos \theta ,r\sin \theta ), implying that z=r(\cos \theta +i\sin \theta ). According to Euler's formula, this is equivalent to saying z=re^{i\theta}.
Euler's identity says that -1=e^{i\pi }. Since e^{i\pi } is re^{i\theta } for r = 1 and \theta =\pi , 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 \pi radians.