Extensions of real numbers using coset groups

TL;DR

This paper introduces a systematic method using coset groups to generate and classify extensions of real numbers, such as quaternions and octonions, based on symmetry and group closure properties.

Contribution

It presents a novel coset group-based framework for deriving multiplication rules and classifying real number extensions, linking algebraic structures to symmetry considerations.

Findings

Generated basis and multiplication rules for number systems systematically.

Classified number systems based on group structures of given order.

Derived invariant matrix representations for these number systems.

Abstract

Extensions of real numbers in more than two dimensions, in particular quaternions and octonions are finding applications in physics due to the fact that they naturally capture certain symmetries of physical systems. Here it is shown that the property of closure of coset groups can be used to generate the basis and general multiplication rules for extensions of real numbers in a systematic way. The coset approach has the advantage that multiplication rules follow directly from group closure instead of being postulated. In this approach, constraints on multiplication parameters can be formulated in ways that capture the symmetry features of the coset group. A complete classification of numbers systems is therefore obtained based on possible group structures of a given order. General matrix representations are also obtained through the coset procedure and by construction, the form of these matrices is invariant under matrix addition and multiplication. Since group symmetries are captured naturally into each number system, the coset group approach can add insight into the utility of multidimensional number systems in describing symmetries in nature.