# Definition:Hurwitz Quaternion

## Definition

A Hurwitz quaternion is a quaternion whose components are all either:

integers

or:

integers plus a half, that is, halves of odd integers.

The set $H$ of all Hurwitz quaternions can therefore be defined as:

$H = \left\{{a + b \mathbf i + c \mathbf j + d \mathbf k \in \Bbb H: \left({a, b, c, d \in \Z}\right) \text { or } \left({a, b, c, d \in \Z + \dfrac 1 2}\right)}\right\}$

## Source of Name

This entry was named for Adolf Hurwitz.