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$\forall n \in\mathbb N$, determine which $n$-th root of unity is closest to $\frac12$.

I'm really struggling with where to even begin with this question. Any help would be appreciated

Thomas Andrews
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    What have you tried? Do you know what the $n$th roots of unity are in the complex plane? Also, I'm assuming the question is about primitive roots of unity, since otherwise, it's pretty trivial. – Rushabh Mehta Feb 25 '20 at 20:09
  • What is the nearest point to $1/2$ on the circle $|z|=1?$ – Thomas Andrews Feb 25 '20 at 20:18

1 Answers1

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Well, the answer(s) are given geometrically in the comments.

An analytic approach:

The $n^\textrm{th}$-roots of unity are $\left\{1,\omega, \omega^2, \dots \omega^{n-1} \right\},$ where $\omega = \exp \frac{2\pi i k}{n}.$

We want to minimize $$|z-1/2|^2=\left(e^{2\pi i k/n} -1/2\right)\left(e^{-2\pi i k/n} -1/2\right) = \frac{5}{4} - \cos \frac{2k \pi}{n}.$$

Clearly, $k=0$ gives the minimum at $z=1$. If we disallow $z=1$, then $k=1$ or $k=n-1$ are the next closest along the circle, and also next closest to $1/2$.

mjw
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