(related to Proposition: Absolute Value of Complex Conjugate)

- Let $z\in\mathbb C$ be a complex number.
- By definition of the complex conjugate $z^*$, the real part are equal $\Re(z)=\Re/.$
- By the definition of the absolute value of complex numbers, it follows $$|z|=\sqrt{\Re(z\cdot z^*)}=|z^*|.$$
- Moreover, by definition of the multiplication of complex numbers $$\begin{array}{rcl}
|z|&=&\sqrt{\Re(z\cdot z^*)}\\
&=&\sqrt{(\Re(z)+i\Im(z))(\Re(z)-i\Im(z))}\\
&=&\sqrt{\Re(z)^2+i\Im(z)\Re(z)-i\Re(z)\Im(z)-i^2\Im(z)^2}\\
&=&\sqrt{\Re(z)^2+\Im(z)^2}.
\end{array}$$∎