\begin{definition}[Definition:Hurwitz Quaternion]
A '''Hurwitz quaternion''' is a quaternion whose components are all either:
:integers
or:
:integers plus a half, that is, halves of odd integers.
The set $H$ of all '''Hurwitz quaternions''' can therefore be defined as:
:$H = \set {a + b \mathbf i + c \mathbf j + d \mathbf k \in \H: \paren {a, b, c, d \in \Z} \text { or } \paren {a, b, c, d \in \Z + \dfrac 1 2} }$
\end{definition}