Definition:Quaternion

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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:
\(\ds \mathbf i \mathbf j\) \(=\) \(\, \ds -\mathbf j \mathbf i \, \) \(\, \ds = \, \) \(\ds \mathbf k\)
\(\ds \mathbf j \mathbf k\) \(=\) \(\, \ds -\mathbf k \mathbf j \, \) \(\, \ds = \, \) \(\ds \mathbf i\)
\(\ds \mathbf k \mathbf i\) \(=\) \(\, \ds -\mathbf i \mathbf k \, \) \(\, \ds = \, \) \(\ds \mathbf j\)
\(\ds \mathbf i^2 = \mathbf j^2 = \mathbf k^2\) \(=\) \(\, \ds \mathbf i \mathbf j \mathbf k \, \) \(\, \ds = \, \) \(\ds -\mathbf 1\)


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:

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


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 $\tuple {a, b} \in \Bbb H$, where:

$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$

where:

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

and

$\overline {\tuple {a, b} }$ 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 $\H$.

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

  • Results about quaternions can be found here.


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

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