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,
$|z|=1$
$\Rightarrow z^n=1$
$\Rightarrow z^n=e^{i2R\pi };\kappa \in I$
$\Rightarrow \kappa \in \left(0,n-1\right)$
$\Rightarrow z=e^{\frac{i2\kappa {\pi }^2}{n}}$
$z=1,e^{\frac{i2\pi }{n}},e^{\frac{i4\pi }{n}},\dots ,e^{\frac{i2\left(n-1\right)\pi }{n}}$
All roots of unity lie on arc of circle
$\therefore$ There are infinitely many roots of unity