This post summarizes symbols used in complex number theory.
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\begin{description}
\item[Set of complex numbers:]$\mathbb{C}$
\item[Notation 1:]$z=a+ib$
\item[Notation 2:]$i^2=-1$
\item[Real part denotation:]$\Re(z)$ and $\operatorname{Re}(z)$
\item[Imaginary part denotation:]$\Im(z)$ and $\operatorname{Im}(z)$
\end{description}
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The set of complex numbers
\usepackage{amssymb}
$\mathbb{C}$
See here for a complete list of set symbols.
Complex number notation
$a+bi$ $i^2=-1$
Nothing unexpected here, the standard i or sometimes j is used.
Real part
\usepackage{amsmath}
$\Re(z)$ and $\operatorname{Re}(z)$
Imaginary part
\usepackage{amsmath}
$\Im(z)$ and $\operatorname{Im}(z)$