Quaternion

From AoPSWiki

The quaternions are a division ring (that is, a ring in which each element has a multiplicative inverse; alternatively, a noncommutative field) which generalize the complex numbers.

Formally, the quaternions are the set , where are any real numbers and the behavior of is "as you would expect," with the properties:

• 
• , and

Note in particular that multiplication of quaternions is not commutative. However, multiplication on certain subsets does behave well: the set act exactly like the complex numbers.

This article is a stub. Help us out by expanding it.