Questions tagged [quaternions]

The ring of quaternions is a four dimensional division algebra over the real numbers. They are usually denoted as $\Bbb H$ in honor of the discoverer, William Rowan Hamilton.

The construction of the quaternions was given by Hamilton as follows: take three symbols $\mathrm{i},\mathrm{j},\mathrm{k}$ as imaginary units and define $\mathrm{i}^2=\mathrm{j}^2=\mathrm{k}^2=\mathrm{i}\mathrm{j}\mathrm{k}=-1$. As a result, $\mathrm{i}\mathrm{j}=\mathrm{k}$, and $\mathrm{j}\mathrm{k}=\mathrm{i}$ and $\mathrm{k}\mathrm{i}=\mathrm{j}$. Furthermore, $\mathrm{j}\mathrm{i}=-\mathrm{k}$ and $\mathrm{k}\mathrm{j}=-\mathrm{i}$ and $\mathrm{i}\mathrm{k}=-\mathrm{j}$, so $\mathrm{k}\mathrm{j}\mathrm{i}=1$.

Another construction of the quaternions was given by Carl Friedrich Gauß as follows: take three symbols $\mathrm{i},\mathrm{j},\mathrm{k}$ as imaginary units and define $\mathrm{i}\circ\mathrm{i}=\mathrm{j}\circ\mathrm{j}=\mathrm{k}\circ\mathrm{k}=\mathrm{k} \circ \mathrm{j} \circ \mathrm{i}=-1$. As a result, $\mathrm{i}\circ\mathrm{j}=-\mathrm{k}$, and $\mathrm{j}\circ\mathrm{k}=-\mathrm{i}$ and $\mathrm{k}\circ\mathrm{i}=-\mathrm{j}$. Furthermore, $\mathrm{j}\circ\mathrm{i}=\mathrm{k}$ and $\mathrm{k}\circ\mathrm{j}=\mathrm{i}$ and $\mathrm{i}\circ\mathrm{k}=\mathrm{j}$, so $\mathrm{i}\circ\mathrm{j}\circ\mathrm{k}=1$.

A quaternion is a linear combination and can represented as versor

$q=q_{0} + q_{1} \mathrm{i} + q_{2} \mathrm{j} + q_{3} \mathrm{k} ~ \widehat{=} ~ \left[\begin{matrix} q_{0} \\ q_{1}\\ q_{2}\\ q_{3} \end{matrix}\right]\in \mathbb{R}^{4} $ where $q_{0}, q_{1},q_{2},q_{3}\in \Bbb R$

Multiplication between quaternions is carried out by using the distributive rule and the rules for $\mathrm{i}$, $\mathrm{j}$ and $\mathrm{k}$.

The quaternions turn out to be a noncommutative division ring. In fact, $\Bbb R$ and $\Bbb C$ and $\Bbb H$ are the only associative finite dimensional division rings over $\Bbb R$. They are also the only normed division algebras over $\Bbb R$.