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Complex Powers (De Moivre)
Complex Powers (De Moivre)
Powers and roots of complex numbers
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Operation
Power zⁿ
nth roots
Real part (a)
Imaginary part (b)
Root degree n
z^(1/n) = r^(1/n)·cis((θ + 2πk)/n), k = 0…n−1
z in polar form
1 ∠ 0°
Modulus |z|
1
The 3 nth roots
w₀ =
1
(1 ∠ 0°)
w₁ =
-0.5 + 0.866i
(1 ∠ 120°)
w₂ =
-0.5 − 0.866i
(1 ∠ -120°)
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Complex Powers (De Moivre)
Argand diagram showing the 3 nth roots of z = 1, spaced evenly around a circle and joined into a regular polygon.
Re
Im
-1
-1
1
1
w₀
w₁
w₂
