Contents

Idea

The quaternions form the largest associative normed division algebra, usually denoted $ℍ$ after William Rowan Hamilton? (since $ℚ$ is taken for the rational numbers).

Normed division algebra structure

Concretely, the structure of $ℍ$ as an $ℝ$-algebra is given by a basis $\left\{1,i,j,k\right\}$ of the underlying vector space of $ℍ$, equipped with a multiplication table where $1$ is the identity element and otherwise uniquely specified by the equations

${i}^{2}={j}^{2}={k}^{2}=ijk=-1,$i^2 = j^2 = k^2 = i j k = -1,

and extended by $ℝ$-linearity to all of $ℍ$. The norm on $ℍ$ is given by

${\parallel \alpha \parallel }^{2}=\alpha \overline{\alpha }${\|\alpha\|}^2 = \alpha \widebar{\alpha}

where given an $ℝ$-linear combination $\alpha =a1+bi+cj+dk$, we define $\overline{\alpha }≔a1-bi-cj-dk$. A simple calculation yields

${\parallel \alpha \parallel }^{2}={a}^{2}+{b}^{2}+{c}^{2}+{d}^{2}${\|\alpha\|}^2 = a^2 + b^2 + c^2 + d^2

whence for $\alpha \ne 0$, the multiplicative inverse is

${\alpha }^{-1}=\frac{1}{{\parallel \alpha \parallel }^{2}}\overline{\alpha }.$\alpha^{-1} = \frac1{{\|\alpha\|}^2} \widebar{\alpha}.

In this way $ℍ$ is a normed division algebra.

References

A survey is in

• T. Y. Lam, Hamilton’s Quaternions (ps)

Revised on November 26, 2012 20:49:18 by Todd Trimble (67.81.93.16)