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 A\mathbb{A} endowed with a positive-definite, homomorphic quadratic form ∣⋅∣:A→R≥|\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 R\mathbb{R}
  2. The complex numbers C\mathbb{C}
  3. The quaternions H\mathbb{H}
  4. The octonions O\mathbb{O}

Such number systems are also called normed division algebras .

Table 1: Normed division algebras
Property R\mathbb{R} C\mathbb{C} H\mathbb{H} O\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)