Proof

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 $$\begin{array}{rcl} |z|&=&\sqrt{\Re(z\cdot z^*)}\\ &=&\sqrt{(\Re(z)+i\Im(z))(\Re(z)-i\Im(z))}\\ &=&\sqrt{\Re(z)^2+i\Im(z)\Re(z)-i\Re(z)\Im(z)-i^2\Im(z)^2}\\ &=&\sqrt{\Re(z)^2+\Im(z)^2}. \end{array}$$
∎

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