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 $$\mathbb H$$.

Construction from Complex Plane
Let $$\mathbf 1, \mathbf i, \mathbf j, \mathbf k$$ denote the following four elements of the matrix space $$\mathcal M_\C \left({2}\right)$$:


 * $$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} $$ where $$\C$$ is the set of complex numbers.

In Quaternions Defined by Matrices it is shown that these have the appropriate properties as defined above, and that a general element of $$\mathbb H$$ has the form:
 * $$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$$

Also see
In Ring of Quaternions it is shown that $$\mathbb H$$ forms a ring under the operations of conventional matrix addition and matrix multiplication.

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

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

In Complex Numbers Subfield of Quaternions it is shown that $$\C$$ is isomorphic to a subfield of $$\mathbb H$$.

Alternative notation
Some sources use $$V$$ for $$\mathbb H$$.

History
The quaternions were famously conceived by William Rowan Hamilton, 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.