\begin{definition}[Definition:Conjugate Quaternion]
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
The conjugate quaternion of $\mathbf x$ is defined as:
:$\overline {\mathbf x} = a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k$.
\end{definition}