Question

If for the complex numbers $z_1,z_2,.....,z_n$, $|z_1|=|z_2|=.....=|z_n|=1$. Then prove that $|\overline{z_1+z_2+.....+z_n}|=|\frac{1}{z_1}+\frac{1}{z_2}+......+\frac{1}{z_n}|$.

Answer 1

Given, for the complex numbers $z_1,z_2,.....,z_n$, $|z_1|=|z_2|=.....=|z_n|=1$.

As it is given $|z_1|=1$ then gives $z_1.\overline{z_1}=1$ or $\overline{z_1}=\frac{1}{z_1}$. Same things happens for the rest complex number.

Now ,

$|\overline{z_1+z_2+.....+z_n}|$

$=|\overline{z_1}+\overline{z_2}+.....+\overline{z_n}|$

$=|\frac{1}{z_1}+\frac{1}{z_2}+.....+\frac{1}{z_n}|$. [ Using (1) and consequence of (1)]