About the quaternion algebra,

(accessible to high school students)

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
- 3D re-orientation via Quaternions
- Associativity of Quaternions and Octonions
- Quaternion Analysis
- Quaternion Applications
- Quaternion Polynomials
- Quaternions and conformal mapping
- Transcription of Tait’s An Elementary Treatise on Quaternions