Prove that if $z_1*z_2$=0 then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers

Question

Prove that if $z_1*z_2=0$ then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers by using the following property: $|z_1z_2|=|z_1||z_2|$

Approach:

if $z_1*z_2=0$ then $$|z_1*z_2|=|0|$$ $$|z_1*z_2|=0$$ $$|z_1||z_2|=0$$

This implies that at least one $z_i$ must be 0 so $|z_i|=0$

How does that look?

Answer 1

In polar form, $z_1=r_1.e^{i\theta_1}$ and $z_2=r_2.e^{i\theta_2}$.

Then $z_1.z_2=0\implies r_1r_2=0$ as $e^{i(\theta_1+\theta_2)}\ne0$ which suggests $r_1=0$ or $r_2=0$ i.e $\sqrt{a_i^2+b_i^2}=0$ for $i=1$ or $2$$\implies a_i=b_i=0$ for $i=1$ or $2$