Algebra

The Trinions: The Natural Extension of the Complex and Dual Numbers

Authors: Joerg Moesch
Comments: 27 Pages.

In this paper, the natural three-dimensional extension of the complex and dual numbers is introduced: the trinion algebras Y(k) of the form a+bi+cj with i²=-1, j²=0 and the commutative coupling ij=ji=k, where k = α+βi+γj is an element of the algebra. The family Y(k) is systematically investigated: multiplication, division and inverses are derived. Special attention is given to the phenomena induced by non-associativity, which lead to invertible zero divisors, non-unique division, and rich geometric structures. The eight fundamental algebras with α,β,γ ∈ {0,1} are examined and their zero divisor sets are visualized. Numerical examples for the algebra Y(1) illustrate the observed phenomena.
Category: Algebra