Definition:Quaternion

Definition
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:

The set of all quaternions is usually denoted $\H$.

Also denoted as
Some sources use $V$ for $\H$.

Some sources develop a more abstract presentation for this structure, and use symbols such as $\lambda_0$, $\lambda_1$, $\lambda_2$ and $\lambda_3$ instead of $\mathbf 1$, $\mathbf i$, $\mathbf j$ and $\mathbf k$.

Also see

 * Ring of Quaternions is Ring, where it is shown that $\H$ forms a ring under the operations of conventional matrix addition and matrix multiplication.


 * Quaternions Subring of Complex Matrix Space, where it is shown that $\H$ is a subring of the matrix space $\map {\MM_\C} 2$.


 * Quaternions form Skew Field, where is it shown that $\H$ actually forms a skew field under the operations of conventional matrix addition and matrix multiplication.


 * Complex Numbers form Subfield of Quaternions, where it is shown that $\C$ is isomorphic to a subfield of $\H$.


 * Definition:Quaternion Group