Number systems

Hypercomplex numbers

A hypercomplex number system is a finite-dimensional, unital, distributive (not necessarily associative) algebra over the real numbers.

  • Kantor & Solodovnikov, 1989: Hypercomplex numbers: An elementary introduction to algebras
  • Olariu, 2002: Complex numbers in n dimensions (arxiv)

Normed division algebras

The hypercomplex number systems \(\mathbb{A}\) endowed with a positive-definite, homomorphic quadratic form \(|\cdot|: \mathbb{A} \to \mathbb{R}_{\geq}\) are classified by Hurwitz’s theorem. Up to isomorphism, there are four of them:

  1. The real numbers \(\mathbb{R}\)
  2. The complex numbers \(\mathbb{C}\)
  3. The quaternions \(\mathbb{H}\)
  4. The octonions \(\mathbb{O}\)

Such number systems are also called normed division algebras .

Table 1: Normed division algebras
Property \(\mathbb{R}\) \(\mathbb{C}\) \(\mathbb{H}\) \(\mathbb{O}\)
Ordered ✓      
Commutative ✓ ✓    
Associative ✓ ✓ ✓  
Algebraically closed   ✓ ✓ ✓
  • Lam, 2003: Hamilton’s quaternions (doi, ps )
  • Baez, 2002, in AMS Bulletin: The octonions (doi, arxiv)
  • Bae, Carter, Kim, 2019: Amusing permutation representations of group extensions (arxiv)
    • Sec. 4: Presents the quaternion group using ribbon braids (Fig. 6)