TITLE: Why do we negate the imaginary part when conjugating?
QUESTION [46 upvotes]: For $z=x+iy \in \mathbb C$ we all know the definition for the "conjugate" of $z$, $\bar{z}=x-iy$. Geometrically this is the reflection of $z$ across the $y$ axis.
My question is: couldn't we have defined $\underline{z}=-x+iy$ instead? (this is the reflection across the $x$ axis) Is there anything wrong with it?
I agree that the formula $|z|^2=z \bar{z}$ looks better than $|z|^2=-z \underline{z}$, but are there any more serious problems?

REPLY [2 votes]: The property of a conjugate is that the argument of the conjugate is the additive inverse of the argument of the original.  If $z$ is 30 degrees clockwise from the real axis, then $\bar z$ is 30 degrees counter-clockwise.  The upshot of this is that when you multiply the numbers together, you get a real number, because the arguments cancel out. (As you likely know, multiplication of complex numbers results in addition of arguments.)
Another nice thing about the conjugate is that it basically goes away when we restrict to the real numbers. The conjugate of a real is just that real. So in some cases, $z\bar z$ is an extension of $z^2$ into the complex numbers. For instance distance from the origin is $\sqrt{z^2}$ if $z$ is real, but we generalize it as $\sqrt{z\bar z}$ in the complex numbers. It "specializes back" to $\sqrt{z^2}$.
Nice, simple things like this break if you try to find some other concept for the complex conjugate.