section_path: Fundamentals/Complex numbers and Euler's identity
section_name: Complex numbers and Euler's identity
paragraph_text: Complex numbers and Euler's identity
  
  Any complex number, say z, can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or r) is used to represent z 's distance from the origin of the complex plane, and the other (angle or φ) the counter-clockwise rotation from the positive real line: z = r ⋅ (cos φ + i sin φ), {\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi),} where i is the imaginary unit satisfying i 2 = − 1 {\displaystyle i^{2}=-1}. The frequent appearance of π in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula: e i φ = cos φ + i sin φ, {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi,} where the constant e is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of e and points on the unit circle centred at the origin of the complex plane. Setting φ = π {\displaystyle \varphi =\pi } in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants: e i π + 1 = 0. {\displaystyle e^{i\pi }+1=0.}
  
  There are n different complex numbers z satisfying z n = 1 {\displaystyle z^{n}=1}, and these are called the " n -th roots of unity " and are given by the formula: e 2 π i k / n (k = 0, 1, 2, …, n − 1). {\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots,n-1).}