Complex Circles :)!

Some charming mathematics and a movie recommendation:

(Here _ , ^ and * represent subscript, superscript and multiplication respectively.)

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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))
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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 contribution in World War II and is available on YouTube.

Have a lovely weekend ☺️!

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