Truly Hypercomplex Numbers: Unification of Numbers and Vectors

TL;DR

This paper introduces truly hypercomplex numbers of any dimension greater than or equal to three, unifying numbers and vectors through a novel multiplicative law based on spherical coordinates.

Contribution

It defines a new class of hypercomplex numbers that extend complex numbers to higher dimensions using spherical and hyperspherical coordinates, unifying numbers and vectors.

Findings

Defines hypercomplex numbers using spherical coordinates

Numbers form Abelian groups under addition and multiplication

Multiplicative law generally does not distribute over addition

Abstract

Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This paper provides a definitive solution to this problem by defining the truly hypercomplex numbers of dimension N greater than or equal to 3. The secret lies in the definition of the multiplicative law and its properties. This law is based on spherical and hyperspherical coordinates. These numbers which I call spherical and hyperspherical hypercomplex numbers define Abelian groups over addition and multiplication. Nevertheless, the multiplicative law generally does not distribute over addition, thus the set of these numbers equipped with addition and multiplication does not form a mathematical field. However, such numbers are expected to have a tremendous utility in mathematics and in science in general.