Complex Conjugation Proof

Question

Hello the question is: 1. Prove (z*)^n = (z^n)* where * represents the complex conjugate. This is my proof https://i.stack.imgur.com/Scnp7.jpg . I looked online to verify if my proof was right, but all the ones i come across use induction. I'm assuming my proof is wrong i just cant tell where or why its wrong. If someone could let me know why it is that would be great. Thank you!

Answer 1

I'll admit I don't understand what the idea of your proof is. Maybe you should write out the proof in Mathjax, with running commentary, if you want feedback on that. The way I see it, we first check (direct computation) that $\overline{z_1z_2} = \overline{z_1}\overline{z_2}$ for all $z_1, _2 \in \mathbb{C}$. From this we prove the result by induction in a standard way, indeed.

Answer 2

$$ z^n=|z|^ne^{in\theta}\\ (z^n)^*=|z|^ne^{-in\theta}\\ {z^*}^n=|z|^ne^{-in\theta} $$

Q.E.D