Complex Circles :)!
S ome charming mathematics and a movie recommendation: (Here _ , ^ and * represent subscript, superscript and multiplication respectively.) -------- Theorem: If a complex number z is not equal to 0, and if n is a positive integer, there are exactly n distinct complex numbers z_0, z_1, ... , z_(n-1) (called the nth roots of z), such that: (z_k)^n = z for each k = 0, 1, 2, ... (n-1) Furthermore, these roots are given by the formulas z_k = R*e^(i*phi_k), where R = {abs(z)}^{1/n} and phi_k = (1/n)*arg(z) + (1/n)*(2*pi*k) (k = 0, 1, 2, ... (n-1)) -------- The proof of this theorem should not be too difficult to understand (better still, carry out 😉) for even non math majors. After you've engaged with it, mark off these roots on a circle of radius R = {abs(z)}^{1/n} centered at the origin. Observe how charmingly it falls into place ☺️. After you've done the above, treat yourself to the movie "The Imitation Game". It's a true story based on Alan Turing's contribu...