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
Associativity of Quaternions and Octonions
Quaternions and conformal mapping
Transcription of Tait’s An Elementary Treatise on Quaternions