On the hypercomplex numbers and normed division algebra of all
dimensions: A unified multiplication
Abstract
Mathematics is the mother of all the sciences, engineering and
technology, and a normed division algebra of all finite dimensions is
the mathematical holy grail. 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, and
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.