We also have
\begin{equation*}
\overline{\overline{z}} = z
\end{equation*}
and, moreover,
\begin{align*}
z + \overline{z} &= (a+bi) + (a-bi) = 2a,\\
z - \overline{z} &= (a+bi) - (a-bi) = 2bi.
\end{align*}
That is,
\begin{equation*}
\fbox{
$\displaystyle z + \overline{z}= 2Re(z)\qquad \text{and}\qquad z + \overline{z}=
2i\,Im(z).$
}
\end{equation*}
The complex conjugate can also be thought of as a reflection in the complex plane about the real axis. In particular, the only numbers that do no change upon this reflection are the real numbers. This implies that
\begin{equation*}
\fbox{
\(z \text{ is a real number if and only if }\,\,\, z = \overline{z}.\)
}
\end{equation*}
If \(z = \overline{z}\text{,}\) it is sometimes said that \(z\) satisfies the reality condition. Recall that absolute value symbol \(|x|=\sqrt{x^2}\) for real numbers \(x\text{.}\) One can think about \(|x|\) as the distance of \(x\) away from zero. In the same spirit, we extend this notation to complex numbers:
\begin{equation*}
|z|=\sqrt{a^2+b^2}\quad\quad (\text{for }z=a+bi)
\end{equation*}
This can also be thought of as the distance of \(z\) away from the origin \(0+0i\text{.}\) The symbol \(|z|\) is called the complex modulus, the norm, or simply the absolute value. Make note of the following useful identity:
\begin{equation*}
z\overline{z} = (a+bi)(a-bi) = a^2-b^2i^2 = a^2-b^2(-1) = a^2+b^2 = |z|^2.
\end{equation*}