TITLE: Complex conjugate of $z$ without knowing $z=x+i y$
QUESTION [6 upvotes]: Is it possible to determine (and if so, how) the complex conjugate $\bar{z}$ of $z$, if you don't already know that $z = x + i y$? 
I think you can use $\log(z)$ to get the angle, and therefore the ratio of $y$ and $x$. But how do you get $|z|$, the radius? How then to get $r$ (so that $x = {\rm Re}(z) = r \cos(\log(z))$ and $y = {\rm Im}(z) = r \sin(\log(z))$)? 
(this is related to How to express in closed form?, namely you can compute Re and Im using the conjugate, but then how do you reduce the conjugate itself fully to elementary functions (if at all))

REPLY [8 votes]: If you knew $z$ and $\bar{z}$ then you could easily recover the real and imaginary parts $x$ and $y$, since:
$x = \frac{z+\bar{z}}{2}$
$y = \frac{z-\bar{z}}{2i}$
So you certainly can't know $z$ and $\bar{z}$ without also knowing $x$ and $y$.