New Developed Pattern on Number System, New Number Field and Their Applications in Physics

TL;DR

This paper introduces a novel hypercomplex number system extending quaternions with matrix representations, explores their algebraic properties, and investigates potential applications in physics.

Contribution

It proposes a new pattern for number systems extending quaternions into matrix forms, creating new hypercomplex number fields with potential physical applications.

Findings

Defines a new matrix-based quaternion extension

Identifies new hypercomplex number fields with special matrices

Explores possible physical interpretations of the new system

Abstract

Based on a brief review on developments of number system, a new developed pattern is proposed. The quaternion is extended to a matrix form aI+bC+cB+dA, in which the unit matrix I and three special matrices C,B,A correspond to number 1 and three units of imaginary number i,j,k, respectively. They form usually a ring. But some fields may be composed of some special 2-rank, even n-rank matrices, for example, three matrices aI+bC, aI+cB, aI+dA and so on. It is a new type of hypercomplex number fields. Finally, the physical applications and possible meaning of the new number system is researched.