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(D) The roots of unity are given by the solutions to the equation $z^n = 1$ for all positive integers $n \in \mathbb{N}$.These roots are of the form $

Vedclass · Accepted Answer

there are infinitely many roots of unity

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(D) The roots of unity are given by the solutions to the equation $z^n = 1$ for all positive integers $n \in \mathbb{N}$.These roots are of the form $z_k = e^{i \frac{2k\pi}{n}}$ for $k = 0, 1, 2, \dots, n-1$.As $n$ increases,the set of all roots of unity $\bigcup_{n=1}^{\infty} \{e^{i \frac{2k\pi}{n}} : k=0, 1, \dots, n-1\}$ becomes dense on the unit circle $|z|=1$.Since any arc of positive length on the unit circle contains infinitely many points from this dense set,there are infinitely many roots of unity on any such arc.Therefore,the correct option is $D$.

On any given arc of positive length on the unit circle $|z|=1$ in the complex plane,

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