# Quaternions

About the quaternion algebra,

(accessible to high school students)

### Introduction

After seeing the complex numbers, you might wonder whether there were
algebras of even higher dimension that act like the real numbers.
Such algebras do exist. Alexander Hamilton invented the first
in 1843, and named it the quaternions.

The question is, in what way should these algebras behave like the
real numbers?
If an algebra is exactly like the real numbers in every way, then it
*is* the real numbers.

There are many ways of generating such higher-dimensional algebras, depending
on which algebraic properties of the real numbers you want to preserve.
Two such ways are the Cayley-Dickson construction and
the Clifford algebras.

These pages are meant to be understandable to anyone with sufficent mathematical
background to appreciate the issues. Most can be read by students who
have completed a second-year high school algebra course.

Representations of the Quaternions

The Cayley-Dickson construction of
quaternions, octonions, etc.

The Worth of Quaternions in Physics

Quaternions in terms of Components

Quaternions as Matrices

Quaternions in Linear-algebraic terms

Quaternion Geometry

Associativity of Quaternions and Octonions

Quaternion Analysis

Quaternion Applications

Quaternion Polynomials

Quaternions and conformal mapping

Transcription of Tait’s An Elementary Treatise on Quaternions