Complex Roots of Unity/Examples/4th Roots
Example of Complex Roots of Unity
The complex $4$th roots of unity are the elements of the set:
- $U_n = \set {z \in \C: z^4 = 1}$
They are:
| \(\ds e^{0 i \pi / 4}\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds e^{i \pi / 2}\) | \(=\) | \(\ds i\) | ||||||||||||
| \(\ds e^{i \pi}\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds e^{3 i \pi / 2}\) | \(=\) | \(\ds -i\) |
Proof
By definition, the first complex $4$th root of unity $\alpha$ is given by:
| \(\ds \alpha\) | \(=\) | \(\ds e^{2 i \pi / 4}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds e^{i \pi / 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \frac \pi 2 + i \sin \frac \pi 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0 + i \times 1\) | Cosine of $\dfrac \pi 2$, Sine of $\dfrac \pi 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds i\) |
We have that:
- $e^{0 i \pi / 4} = e^0 = 1$
which gives us, as always, the zeroth complex $n$th root of unity for all $n$.
The remaining complex $4$th roots of unity can be expressed as $e^{4 i \pi / 4} = e^{i \pi}$ and $e^{6 i \pi / 4} = e^{3 i \pi / 2}$, but it is simpler to calculate them as follows:
| \(\ds \alpha^2\) | \(=\) | \(\ds i^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -1\) | Definition of Imaginary Unit |
| \(\ds \alpha^3\) | \(=\) | \(\ds \alpha^2 \times \alpha\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1} \times i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Example $1$.
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The $n$th Roots of Unity: $105 \ \text {(a)}$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): fourth root of unity