\begin{definition}[Definition:Conjugate Quaternion/Ordered Pair]
Let $\mathbf x$ be a quaternion defined as an ordered pair $\left({a, b}\right)$  of complex numbers.
The conjugate quaternion of $\mathbf x$ is defined as:
:$\overline {\mathbf x} = \overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$
\end{definition}