Theorem on properties of conjugation

Theorem 1

Let \(z,w\) be complex numbers. Then

1. \(zz^* = |z|^2\).
2. \((z + w)^* = z^* + w^*\).

3. \((zw)^* = z^* w^*\).
4. \(\left(\frac{z}{w}\right)^*=\frac{z^*}{w^*}\), provided \(w\ne0\).
5. \((z^n)^*=(z^*)^n\) for positive integers \(n\). For \(n=0\) or negative integers \(n\), require \(z\ne0\).