# Definition:Quaternion

## Contents

## 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:

\(\displaystyle \mathbf i \mathbf j = -\mathbf j \mathbf i\) | \(=\) | \(\displaystyle \mathbf k\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \mathbf j \mathbf k = -\mathbf k \mathbf j\) | \(=\) | \(\displaystyle \mathbf i\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \mathbf k \mathbf i = -\mathbf i \mathbf k\) | \(=\) | \(\displaystyle \mathbf j\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k\) | \(=\) | \(\displaystyle -\mathbf 1\) | $\quad$ | $\quad$ |

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

### 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:

- $\mathbf x_1 + \mathbf x_2 := \paren {a_1 + a_2} \mathbf 1 + \paren {b_1 + b_2} \mathbf i + \paren {c_1 + c_2} \mathbf j + \paren {d_1 + d_2} \mathbf k$

### 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:

\(\displaystyle \mathbf x_1 \mathbf x_2\) | \(:=\) | \(\displaystyle \left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) \mathbf 1\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) \mathbf i\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) \mathbf j\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) \mathbf k\) | $\quad$ | $\quad$ |

## 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.

## Construction from Cayley-Dickson Construction

The set of quaternions $\Bbb H$ can be defined by the Cayley-Dickson construction from the set of complex numbers $\C$.

From Complex Numbers form Algebra, $\C$ forms a nicely normed $*$-algebra.

Let $a, b \in \C$.

Then $\left({a, b}\right) \in \Bbb H$, where:

- $\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$
- $\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$

where:

- $\overline a$ is the complex conjugate of $a$

and

- $\overline {\left({a, b}\right)}$ is the conjugation operation on $\Bbb H$.

It is clear by direct comparison with the Construction from Complex Pairs that this construction genuinely does generate the Quaternions.

## Quaternion Algebra over a Field

An algebra of quaternions 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 that this generalises Hamilton's quaternions.

## Also denoted as

Some sources use $V$ for $\H$.

## 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 $\map {\mathcal M_\C} 2$.

- 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 $\H$.

## Historical Note

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 $16$th October $1843$.

However, it is also worth noting that Carl Friedrich Gauss independently came up with the same concept in $1819$.

## 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.

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.2$: Some examples of rings: Ring Example $9$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem - John C. Baez:
*The Octonions*(2002)*: 1 Introduction*