Roots of Complex Number/Corollary
Theorem
Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $w$ be one of the complex $n$th roots of $z$.
Then the $n$th roots of $z$ are given by:
- $z^{1 / n} = \set {w \epsilon^k: k \in \set {1, 2, \ldots, n - 1} }$
where $\epsilon$ is a primitive $n$th root of unity.
Proof
By definition of primitive complex $n$th root of unity:
- $\omega = e^{2 m i \pi k}$
for some $m \in \Z: 1 \le m < n$.
Thus:
\(\ds \paren {w \omega^k}^n\) | \(=\) | \(\ds w^n \paren {e^{2 m i \pi k / n} }^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z e^{2 m i \pi k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z \paren {e^{2 i \pi} }^{m k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z \times 1^{m k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z\) |
This demonstrates that $w \omega^k$ is one of the complex $n$th roots of $z$.
All of the complex $n$th roots of unity are represented by powers of $\omega$.
Thus it follows from Roots of Complex Number that:
- $z^{1 / n} = \set {w \omega^k: k \in \set {1, 2, \ldots, n - 1} }$
are the $n$ complex $n$th roots of $z$.
$\blacksquare$
Examples
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:
- $b = \sqrt [8] 8 \paren {\cos \dfrac \pi {16} + i \sin \dfrac \pi {16} }$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity