University⏱ 5 min

Binomial Equations in ℂ

Solving zⁿ = a and the regular polygon of roots

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ReImz₀z₁z₂z₃z₄ⁿ√|a| = 1.000

Equation parameters zⁿ = a

z5=1.00z^{5} = 1.00
a=1 (cos⁡0°+isin⁡0°)a = 1\,(\cos 0° + i\sin 0°)

General solution:

zk=15(cos⁡0°+360°k5+isin⁡0°+360°k5)z_k = \sqrt[5]{1}\left(\cos\frac{0° + 360°k}{5} + i\sin\frac{0° + 360°k}{5}\right)

k = 0, 1, …, 4

5 solutions:

z0 = 1.00∠0°
z1 = 0.31 + 0.95i∠72°
z2 = -0.81 + 0.59i∠144°
z3 = -0.81 − 0.59i∠216°
z4 = 0.31 − 0.95i∠288°

The equation zⁿ = a has exactly n solutions. They all lie on a circle of radius ⁿ√|a| and form a regular 5-gon. Adjacent roots are separated by 360°/5 = 72.0°.