\begin{definition}[Definition:Gaussian Integer]
A '''Gaussian integer''' is a complex number whose real and imaginary parts are both integers.
That is, a '''Gaussian integer''' is a number in the form:
:$a + b i: a, b \in \Z$
The set of all '''Gaussian integers''' can be denoted $\Z \sqbrk i$, and hence can be defined as:
:$\Z \sqbrk i = \set {a + b i: a, b \in \Z}$
Some sources use the symbol $J$ for the set $\Z \sqbrk i$.
\end{definition}