I want to rigorously verify that
$$\overline{\left(\frac{z_1}{z_2}\right)}={\left(\frac{\overline{z_1}}{\overline{z_2}}\right)}$$
Here's my attempt at thinking:
$$\overline{\left(\frac{z_1}{z_2}\right)}=\overline{\left({z_1}\cdot \frac{1}{z_2}\right)}={\left(\overline{z_1}\cdot \overline{\left(\frac{1}{z_2}\right)}\right)}$$
Now, it appears that we are in a "vicious circle" because we don't know if $$\overline{\left(\frac{1}{z_2}\right)}=\frac{1}{\overline{z_2}}$$
One approach might be to prove that
$$\overline{\left(\frac{1}{z_2}\right)}=\overline{{z_2}^{-1}}=\left(\overline{z_2}\right)^{-1}=\frac{1}{\overline{z_2}}$$
But is there a better way?
Best Answer
You are right when you reduce the original problem to the problem of proving that$$\overline{\left ( \frac 1z \right )}=\frac 1 {\overline z}.\tag1$$Now, in order to prove this, all you have to do is to notice that$$\overline{\left ( \frac 1z \right )}.\overline z=\overline{{\left ( \frac 1z \right)}\times z}=\overline1=1.$$Therefore, $(1)$ holds.