University⏱ 4 min

Nth Complex Roots

Roots of any complex number

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w=0+2iz0z1z2

Degree n: 3

w = 0+2i

z3:  ∣z∣1/3=1.260,  arg⁡k=90°+360°k3\sqrt[3]{z}: \; |z|^{1/3} = 1.260, \; \arg_k = \frac{90° + 360°k}{3}

z0: 1.091 + 0.630i (30.0°)

z1: -1.091 + 0.630i (150.0°)

z2: -0.000 + -1.260i (270.0°)

The nth roots of w are evenly spaced on a circle of radius |w|^(1/n). Each root is rotated 360°/n from the previous.