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 $\mathbb H$.

Quaternion Addition
The sum of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:


 * $\mathbf x_1 + \mathbf x_2 := \left({a_1 + a_2}\right) \mathbf 1 + \left({b_1 + b_2}\right) \mathbf i + \left({c_1 + c_2}\right) \mathbf j + \left({d_1 + d_2}\right) \mathbf k$

Quaternion Multiplication
The product of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:

This is proved to be consistent with the definition of quaternions in Quaternion Multiplication.

Construction from Complex Matrices
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.

In Matrix Form of Quaternion it is shown that a general element $\mathbf x$ of $\mathbb H$ has the form:
 * $\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$

Construction from Complex Pairs
A quaternion can be defined as an ordered pair $\left({x, y}\right)$ where $x, y \in \C$ are complex numbers, on which the operations of addition and multiplication are defined as follows:

Quaternion Addition of Complex Pairs
Let $x_1, x_2, y_1, y_2$ be complex numbers.

Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:


 * $\left({x_1, y_1}\right) + \left({x_2, y_2}\right):= \left({x_1 + x_2, y_1 + y_2}\right)$

Quaternion Multiplication of Complex Pairs
Let $x_1, x_2, y_1, y_2$ be complex numbers.

Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:


 * $\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_2 \overline {y_1}, \overline {x_1} y_2 + y_1 x_2}\right)$

where $\overline {x_1}$ and $\overline {y_1}$ are the complex conjugates of $x_1$ and $y_1$ respectively.

From Quaternions Defined by Ordered Pairs this definition can be seen to be equivalent to the main definition above.

Quaternion Algebra Over a Field
A quaternion algebra can be defined over any field as follows:

Let $\mathbb K$ be a field, and $a$, $b \in \mathbb K$.

Define the quaternion algebra $\left\langle{ a,b }\right\rangle_\mathbb K$ to be the $\mathbb K$-vector space with basis $\{1, i, j, k\}$ subject to:
 * $i^2 = a$
 * $j^2 = b$
 * $ij = k = -ji$

Formally this could be achieved as a multiplicative presentation of a suitable group, or as a linear subspace of a finite extension of $\mathbb K$.

Taking $\mathbb K = \R$ and $a = b = -1$ we see than this generalises Hamilton's quaternions above.

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.

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.