Complex Roots of Unity/Examples/Cube Roots/Proof
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Example of Complex Roots of Unity
The complex cube roots of unity are the elements of the set:
- $U_3 = \set {z \in \C: z^3 = 1}$
They are:
\(\ds \) | \(\) | \(\, \ds e^{0 i \pi / 3} \, \) | \(\, \ds = \, \) | \(\ds 1\) | ||||||||||
\(\ds \omega\) | \(=\) | \(\, \ds e^{2 i \pi / 3} \, \) | \(\, \ds = \, \) | \(\ds -\frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||
\(\ds \omega^2\) | \(=\) | \(\, \ds e^{4 i \pi / 3} \, \) | \(\, \ds = \, \) | \(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\) |
The notation $\omega$ for, specifically, the complex cube roots of unity is conventional.
Proof
\(\ds z^3 - 1\) | \(=\) | \(\ds \paren {z - 1} \paren {z^2 + z + 1}\) | Difference of Two Cubes/Corollary | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds 1\) | |||||||||||
\(\, \ds \text { or } \, \) | \(\ds z^2 + z + 1\) | \(=\) | \(\ds 0\) |
Then:
\(\ds z^2 + z + 1\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \dfrac {-1 \pm \sqrt {1^2 - 4 \times 1 \times 1} } {2 \times 1}\) | Quadratic Formula | ||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 2 \pm i \frac {\sqrt 3} 2\) | simplifying |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: $(3.6)$