Visual proofs of additivity and multiplicativity of complex conjugation?

Question

I am asking if someone knows of visual proofs of (a) additivity of complex conjugation i.e. $\overline{z + z'} = \overline{z} + \overline{z'}$ and (b) multiplicativity of complex conjugation i.e. $\overline{z \cdot z'} = \overline{z} \cdot \overline{z'}$. Obviously I can just write the algebra out and it's clear it works, so I'm looking for a more conceptual i.e. geometric proof.

Answer 1

Well, addition follows readily when you know that, in the complex plane, $\overline z$ is just $z$ but flipped across the real axis (or, visually, flipped vertically). Then by visualizing complex numbers $z = x+iy$ as vectors $(x,y)$ in $\Bbb R^2$, and adding in the usual "tail to tip" fashion, it's easy to see the case for addition, as in the diagram below:

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Multiplication would perhaps be a bit harder to visualize though, and to attempt to do it (at least how I would), we still need a bit of an algebraic framework. I would say the best approach is through the polar form. That is, for $z$ a complex number, we may write $z$ as

$$z = |z| e^{i \theta}$$

$\theta$ will end up being the angle $z$ makes with the positive real axis in this framework. Taking that $\overline z$ is still essentially $z$ but flipped across the real axis, it is easy to justify that

$$\overline z = |z| e^{- \theta}$$

Moreover, if $z_1,z_2$ take the form $z_k = |z_k| e^{i \theta_k}$ for $k=1,2$, then their multiplication looks like

$$z_1 z_2 = |z_1||z_2| e^{i(\theta_1 + \theta_2)}$$

What this defines, in a way, is how $z_2$ "acts" upon $z_1$ (or vice versa). Intuitively, imagine $z_1$ as a vector. Multiplication by $z_2$ stretches $z_1$ by a factor equal to its magnitude, and rotates it further in the complex plane by the angle $z_2$ makes with the positive real axis.

By this, then, for example, multiplication of $z$ by its conjugate $\overline z$ would essentially be like stretching $z$ out by a factor of its own magnitude, and undoing the rotation that brought $z$ off the positive real axis in the first place: a sort of geometric justification for the identity

$$z \overline z = |z|^2$$

So what about $\overline{z \cdot z'}$? Well, what you have done is taken the vector $z$, and stretched it by a factor of $|z'|$, rotated it by the angle $z'$ makes with the positive real axis, and flipped it over the real axis. But notice that, if we instead start by flipping $z$ over the real axis, stretching by $|z'|$, and rotating by the negative of the angle $z'$ makes with the positive real axis (i.e. the angle that $\overline{z'}$ makes with the positive real axis), we get to the same place.

Expressing this in a single diagram like I did above is somewhat asking for trouble, because it would be very cluttered. However, 3Blue1Brown did some amazing visualizations for this over at Khan Academy - though his visualizations focus more on how a multiplication would affect the entirety of the complex plane, rather than any particular vector. (But you also get a sort of grander view of this phenomenon as well and what it can mean, so it's worth it, in my opinion.)