Complex numbers - roots of unity

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Submitted by roamingfree on Fri, 06/25/2021 - 08:36
skills
Know properties of roots of unity
Question
Let $\omega = e^{\frac{2\pi i}{5}}$ be a fifth root of unity. Determine the value of $$ z = 1 + \omega + \omega^2 + \omega^3 $$
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$\frac{1}{4}\left(\sqrt{5}-1 +i\sqrt{2(5+\sqrt{5})}\right)$
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We can either evaluate this directly, or use a bit of cunning. We know, for the $nth$ roots of unity, $n>1$, $\sum_{k=1}^n \omega^k = 0$. It follows that in our case, $ 0 = 1 + \omega + \omega^2 + \omega^3 + \omega^4$, hence $$\begin{align} z &= -\omega^4 \\ &= -e^{\frac{8\pi i}{5}}\\ &= -\left(\cos(\frac{8\pi}{5}) + i\sin(\frac{8\pi}{5})\right) \\ &= \frac{1}{4}\left(\sqrt{5}-1 +i\sqrt{2(5+\sqrt{5})}\right) \end{align} $$