Suppose $z$ is a complex number. Prove that there exists an $r \geq 0$ and a complex number $w$ (with $|w| = 1$) such that $z = rw$. Does $z$ uniquely determine $r$ and $w$?
Let $z = a+bi$. Then $|z| = \sqrt{a^2+b^2}$. So take $r = |z|$. It seems like one can take $w$ to be any complex number such that $|w| = 1$. So I think $z$ uniquely determines $r$ but not $w$.