Short question about Complex Numbers: $\forall z\in\mathbb{C},\exists\theta\in\mathbb{R}:e^{-i\theta}z=-|z|$

Question

Is the following statement true?

$\forall z\in\mathbb{C},\exists\theta\in\mathbb{R}:e^{-i\theta}z=-|z|$

I believe it is because if $z=|z|e^{i\alpha}$ then $\theta=\alpha-\pi$ should work?

Answer 1

Multiplying by $e^{-i\theta}$ represents clockwise rotation about the origin of $\theta$ radians, so essentially your formula says that you can rotate any complex number to get a negative real number, which is evidently true.