Introduction to Hybrid Numbers

Abstract

In this study, we define a new non-commutative number system called hybrid numbers. This number system can be accepted as a generalization of the complex $$\left( {\mathbf {i}}^{2}=-1\right) $$ , hyperbolic $$\left( {\mathbf {h}} ^{2}=1\right) $$ and dual number $$\left( \varvec{\varepsilon }^{2}=0\right) $$ systems. A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation $$\mathbf { ih=-hi=i}+\varvec{\varepsilon }.$$ Because these numbers are a composition of dual, complex and hyperbolic numbers, we think that it would be better to call them hybrid numbers instead of the generalized complex numbers. In this paper, we give some algebraic and geometric properties of this number set with some classifications. In addition, we examined the roots of a hybrid number according to its type and character.