Question

If $z_1$ and $z_2$ are two complex numbers such that $|z_1|=|z_2|$, then is it necessary that $z_1=z_2$?

Answer 1

Let $z_1=\sqrt{a_1^2+b_1^2}and|z_2|=\sqrt{a_2^2+b_2^2}$
$\therefore |z_1|=|z_2|\Rightarrow \sqrt{a_1^2+b_1^2}=\sqrt{a_2^2+b_2^2}\Rightarrow a_1^2+b_1^2=a_2^2+b_2^2$
It does not imply that $a_1=a_2$ and $b_1=b_2$.
Hence, $z_1$ and $z_2$ need not be equal.