Roots of Complex Number/Corollary/Examples
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Examples of Roots of Complex Number: Corollary
Complex Cube Roots
Let $z \in \C$ be a complex number.
Let $z \ne 0$.
Let $w$ be one of the (complex) cube roots of $z$.
Then the complete set of (complex) cube roots of $z$ is:
- $\set {w, w \omega, w \omega^2}$
where:
- $\omega = e^{2 \pi / 3} = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$
Fourth Roots of $2 - 2 i$
The complex $4$th roots of $2 - 2 i$ are given by:
- $\paren {2 - 2 i}^{1/4} = \set {b, bi, -b, -bi}$
where:
- $b = \sqrt [8] 8 \paren {\cos \dfrac \pi {16} + i \sin \dfrac \pi {16} }$