Check if these statements are true for all complex nubmers

Question

I have these two statements and I should check whether they are true in $\mathbb{C}$:

• $\frac{z_1}{\lvert z_1\rvert}=\frac{z_2}{\lvert z_2\rvert}\iff(\exists k \in \mathbb{R}^+) \vec{0z_1}=k\vec{0z_2} $
• $arg(z_1)=arg(z_2)\iff\frac{z_1}{\lvert z_1\rvert}=\frac{z_2}{\lvert z_2\rvert}$

Well, I know these statements are "generally true", but since we are looking at whole complex set, meaning it includes $0$, and argument is not defined for $0$, I am not sure if $\iff$ still holds.

Answer 1

We need to assume $z_1,z_2\neq 0$, otherwise the expression $\frac{z_1}{\lvert z_1\rvert}=\frac{z_2}{\lvert z_2\rvert}$ is not defined.

Then for the first one we need to prove that for $z_1,z_2\neq 0$

$$\frac{z_1}{\lvert z_1\rvert}=\frac{z_2}{\lvert z_2\rvert} \implies z_1=\frac{|z_1|}{\lvert z_2\rvert}z_2=kz_2 $$

and

$$z_1=kz_2 \implies \frac{z_1}{\lvert z_1\rvert}=\frac{kz_2}{\lvert kz_2\rvert} =\frac{z_2}{\lvert z_2\rvert} $$

Can you proceed with the second one?