\begin{definition}
Let $z = x + iy = Re^{i \theta}$, then the \emph{complex conjugate} of $z$ is defined:
$$z^* = x-iy=Re^{-i \theta}$$
\end{definition}

Some properties of the complex conjugate:

\begin{claim}
The product of a complex number and its conjugate is the magnitude squared.
$$ z \cdot z^* = r^2 = \left| z \right|^2 $$
\end{claim}

\begin{proof}
\begin{align*}
z \cdot z^* &= (a + ib)(a - ib) \quad \mbox{Using rectangular representation}\\
z \cdot z^* &= a^2 +aib - aib -i^2b^2 \\
z \cdot z^* &= a^2 +b^2 \\
z \cdot z^* &= \sqrt{a^2 +b^2}^2 \\
z \cdot z^* &= R^2
\end{align*}
\end{proof}

\begin{proof}
\begin{align*}
z \cdot z^* &= Re^{i \theta} \cdot Re^{-i \theta} \quad \mbox{using Polar representation} \\
z \cdot z^* &= R^2e^{i(\theta -\theta)} \\
z \cdot z^* &= R^2e^0 \\
z \cdot z^* &= R^2
\end{align*}
\end{proof}

\begin{claim}
The sum of a complex number and its conjugate is twice the complex number's real part.
$$ z + z^* = 2\mbox{Re}(z) $$
\end{claim}

\begin{proof}
\begin{align*}
z + z^* &= (a + ib) + (a - ib) \\
z + z^* &= 2a
\end{align*}
\end{proof}

\begin{claim}
The difference of a complex number and its conjugate is twice the complex number's imaginary part.
$$ z - z^* = 2i\mbox{Im}(z) $$
\end{claim}

\begin{proof}
\begin{align*}
z - z^* &= (a + ib) - (a - ib) \\
z - z^* &= 2ib 
\end{align*}
\end{proof}

\begin{claim}
The division of a complex number by its conjugate results in a doubling of the angle $\theta$
$$ z / z^* = e^{i2 \theta}  $$
\end{claim}

\begin{proof}
\begin{align*}
z / z^* &=   e^{i\theta} / e^{-i\theta} \\
z / z^* &=   e^{i\theta} \cdot e^{i\theta} \\
z / z^* &=   e^{i(\theta + \theta)} = e^{i2 \theta} 
\end{align*}
\end{proof}

Complex Conjugate