A Generalization Of Complex, Dual and Hyperbolic Quaternions: Hybrid Quaternions

Abstract

Hybrid numbers are a new non-commutative number system which a generalization of the complex ($\textbf{i}^2=-1$), dual ($\boldsymbol{\varepsilon}^2=0$), and hyperbolic numbers ($\textbf{h}^2=1$). In this article, firstly we define a new quaternion system called hybrid quaternions by taking the coefficients of real quaternions as hybrid numbers. This novel quaternion system is a combination of complex quaternions (biquaternions), hyperbolic (perplex) quaternions, and dual quaternions, and can be viewed as a generalization of these quaternion systems. We then present the basic properties of hybrid quaternions including fundamental operations, conjugates, inner product, vector product, and norm. Finally, we give a schematic representation of numbers and quaternions.