Equality of two complex numbers with respect to argument

Question

If two complex numbers are equal , is it necessary that their arguments are also equal ? Is the vice versa also true ? means that if the arguments of two complex numbers are equal , does it necessarily imply that they’re equal? According to me , the first supposition would be right , but I’m not quite sure about the second one .

Answer 1

If we take complex numbers as $z_1 = a+bi$ and $z_2 = c+di$, $z_1 = z_2$ if and only if $a = c$ and $b = d$. So $\arg(z_1) = \arg(z_2)$, obviously.

However, if $\arg(z_1) = \arg(z_2)$, it doesn't have to imply that $z_1 = z_2$. A simple counter-example is $z_1 = 2z_2$ with $z_1 \ne 0$. Their arguments are equal however $z_1 \ne z_2$ (if you know some about physics, you can think $z_1$ and $z_2$ as two vectors with same direction but different magnitudes. Then of course we can't say that they are equal).