On the hypercomplex numbers of all finite dimensions: Beyond quaternions
and octonians
Abstract
In search of a real three-dimensional, normed, associative, division
algebra, Hamilton discovered quaternions that form a non-commutative
division algebra of quadruples. Later works showed that there are only
four real division algebras with 1, 2, 4, or 8 dimensions. This work
overcomes this limitation and introduces generalized hypercomplex
numbers of all dimensions that are extensions of the traditional complex
numbers. The space of these numbers forms non-distributive normed
division algebra that is extendable to all finite dimensions. To obtain
these extensions, we defined a unified multiplication, designated as
scaling and rotative multiplication, fully compatible with the existing
multiplication. Therefore, these numbers and the corresponding algebras
reduce to distributive normed algebras for dimensions 1 and 2. Thus,
this work presents a generalization of $\mathbb{C}$
in higher dimensions along with interesting insights into the geometry
of the vectors in the corresponding spaces.