Given a complex number $a+bi$, it has a complex conjugate $a-bi$. The product of this complex number with its complex conjugate gives $(a+bi)(a-bi)=a^2+b^2$.
One might imagine flipping the sign of the real part instead of the imaginary part to get a sort of "anticonjugate", resulting in a similar product of $(a+bi)(-a+bi)=-(a^2+b^2)$.
Clearly this "anticonjugate" is the negative of the conjugate. I suspect I've never seen this concept before because of either (1) no finds this anticonjugate useful or (2) the negative of the complex conjugate is considered without giving it a special name.
Is there a different reason why we don't seem to use or consider "anticonjugates"? Or does the above account for this perception?
Best Answer
Complex conjugates are important because $i$ and $-i$ are fundamentally indistinguishable by definition; $i$ is defined to be a number satisfying the equation $i^2 = -1$, but of course $-i$ must satisfy the same equation. So "any fact" which can be stated about complex numbers must remain true if we swap all occurrences of $i$ with $-i$ (though one must take care with "hidden" occurrences). Complex conjugation is therefore a mapping of complex numbers which preserves many algebraic properties.
In contrast, complex anticonjugation as defined in your question does not preserve any useful properties, because $1$ and $-1$ are not fundamentally indistinguishable; $-1$ is not a successor of the number $0$, it is not a multiplicative identity such that $1x \equiv x$, nor does it satisfy any other reasonable definition of the number $1$.