An ordered pair of two real numbers with the complex addition and multiplication defined.
Forms both a:
- division algebra if thought of with complex multiplication as the bilinear map of the algebra
- field
Constructs the quaternions from complex numbers, octonions from quaternions, and keeps doubling like this indefinitely.
Kind of extends the complex numbers.
Some facts that make them stand out:
- one of the only three real associative division algebras in addition to the real numbers and complex numbers, according to the classification of associative real division algebras
- the simplest non-commutative division algebra. Contrast for example with complex numbers where multiplication is commutative
Unlike the quaternions, it is non-associative.
