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?
Answer
No, the nice thing about conjugation is that it is an automorphism: $\overline{zw} = \bar z\bar w$ and $\overline{z+w}=\bar z+\bar w$.
At heart, what conjugacy shows you is that, while $i$ and $-i$ are different complex numbers, they have exactly the same behavior. That's not surprising, because we define $i$ so that $i^2=-1$, but then $(-i)^2=-1$. How do we distinguish these two square roots? What if we defined the complex numbers as $a+bi+cj$ where $i+j=0$ and $i^2=j^2=-1$? Then which would be the "primary" square root of $-1$? There is really no way to tell. With positive real numbers, we can always pick the positive square root, but we can't define "positive" in the complex numbers. This yields a duality in the complex numbers, represented by conjugation.
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