Why do we not name $-a+bi$ in relation to $a+bi$?

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?

Solution 1:

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 $1$.

Solution 2:

There is a tradition in math to avoid (or, at least, minimize) redundancies.

In this case, complex conjugation $\,\overline{a+ib}=a-ib\,$ has been known and used for a long time. It has many applications, from polynomial equations to calculus, abstract algebra, geometry etc.

In contrast, the proposed "anti-conjugate", say we write it as $\,\widetilde{a+ib}=-a+ib\,$, would be a new concept, without any obvious advantage - conceptual or practical. Moreover, it can be easily expressed in terms of the conjugate as $\,\widetilde{a+ib}=-\,\overline{a+ib}=\overline{i\cdot\overline{i \cdot(a+ib)}}\,$. Thus, redundant.

[ EDIT ] $\;$ Side by side summary.

$$ \begin{matrix} & & & \small{\text{conjugate}}\;\overline{\,z\,} & & & \small{\text{anti-conjugate}}\; \widetilde{\,z\,} \\ \small{\text{symmetry}} & & & \small{\text{over real axis}} & & & \small{\text{over imaginary axis}} \\ \small{\text{involution}} & & \small{\text{yes:}} & \overline{\overline{\,z\,}} = z & & \small{\text{yes:}} & \widetilde{\widetilde{\,z\,}} = z \\ \small{\text{distrib over +}} & & \small{\text{yes:}} & \overline{\,z_1+z_2\,} = \overline{\,z_1\,}+\overline{\,z_2\,} & & \small{\text{yes:}} & \widetilde{\,z_1+z_2\,} = \widetilde{\,z_1\,}+\overline{\,z_2\,} \\ \small{\text{distrib over }\times} & & \small{\text{yes:}} & \overline{\,z_1\,\cdot\,z_2\,} = \overline{\,z_1\,}\,\cdot\,\overline{\,z_2\,} & & \color{red}{\small{\text{no:}}} & \widetilde{\,z_1\,\cdot\,z_2\,} \ne \widetilde{\,z_1\,}\,\cdot\,\widetilde{\,z_2\,} \end{matrix} $$