Question

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

• A. there need not be any root of unity
• B. there lies exactly one root of unity
• C. there are more than one but finitely many roots of unity
• D. there are infinitely many roots of unity

Answer 1

Answer: (D)

We have,
$|z|=1$
$\Rightarrow z^{n}=1$
$\Rightarrow z^{n}=e^{i2R \pi} ;\kappa \in I $
$\Rightarrow \kappa \in\left(0, n-1\right)$
$\Rightarrow z=e^{\frac{i2\kappa\pi^2}{n}} $
$z=1, e^{\frac{i 2\pi}{n}}, e^{\frac{i 4\pi}{n}}, \ldots, e^{\frac{i2\left(n-1\right)\pi}{n}}$
All roots of unity lie on arc of circle
$\therefore $ There are infinitely many roots of unity