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 $\mathbb H$.

Also see

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


 * Quaternions Subring of Complex Matrix Space, where it is shown that $\mathbb H$ is a subring of the matrix space $\mathcal M_\C \left({2}\right)$.


 * Quaternions form Skew Field, where is it shown that $\mathbb H$ 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 $\mathbb H$.

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

History
The quaternions were famously conceived by, who was so proud of his flash of insight that he carved:
 * $i^2 = j^2 = k^2 = i j k = -1$

into the stone of Brougham Bridge on October 16, 1843.

Linguistic Note
The word quaternion is derived from the Latin word quaterni, meaning four by four.

The word quaternion is also used for a style of poem in which the theme is divided into four complementary parts.

It's an awkward word -- the fingers keep trying to type it as quaternian which, although it feels more natural, is technically incorrect.