# Complex Roots of Unity/Examples/4th Roots

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## 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:

\(\displaystyle e^{0 i \pi / 4}\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle e^{i \pi / 2}\) | \(=\) | \(\displaystyle i\) | |||||||||||

\(\displaystyle e^{i \pi}\) | \(=\) | \(\displaystyle -1\) | |||||||||||

\(\displaystyle e^{3 i \pi / 2}\) | \(=\) | \(\displaystyle -i\) |

## Proof

By definition, the first complex $4$th root of unity $\alpha$ is given by:

\(\displaystyle \alpha\) | \(=\) | \(\displaystyle e^{2 i \pi / 4}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e^{i \pi / 2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cos \frac \pi 2 + i \sin \frac \pi 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0 + i \times 1\) | Cosine of $\dfrac \pi 2$, Sine of $\dfrac \pi 2$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle i\) |

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:

\(\displaystyle \alpha^2\) | \(=\) | \(\displaystyle i^2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -1\) | Definition of Imaginary Unit |

\(\displaystyle \alpha^3\) | \(=\) | \(\displaystyle \alpha^2 \times \alpha\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {-1} \times i\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -i\) |

$\blacksquare$

## 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): Entry:**fourth root of unity**