\begin{definition}[Definition:Quaternion]
A '''quaternion''' is a number in the form:
: $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
where:
: $a, b, c, d$ are real numbers
: $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way:
{{begin-eqn}}     
{{eqn | l = \mathbf i \mathbf j
      | m = -\mathbf j \mathbf i
      | mo= =
      | r = \mathbf k
}}
{{eqn | l = \mathbf j \mathbf k
      | m = -\mathbf k \mathbf j
      | mo= =
      | r = \mathbf i
}}
{{eqn | l = \mathbf k \mathbf i
      | m = -\mathbf i \mathbf k
      | mo= =
      | r = \mathbf j
}}
{{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2
      | m = \mathbf i \mathbf j \mathbf k
      | mo= =
      | r = -\mathbf 1
}}
{{end-eqn}}
The set of all '''quaternions''' is usually denoted $\H$.
\end{definition}