Hybrid numbers with the view of multiplicative calculus

Abstract

This paper defines a new number system called multiplicative hybrid numbers. A multiplicative hybrid number is one that satisfies the non-commutative multiplication criteria using any combination of multiplicative complex ($i_{\star }^{2_{\star }}={-}_{\star }{1}_{\star }$), multiplicative dual (${\varepsilon }_{\star }^{2_{\star }}=0_{\star }$), and multiplicative hyperbolic ($h_{\star }^{2_{\star }}=1_{\star }$) numbers. Introducing some algebraic properties, the geometry of multiplicative hybrid numbers is then investigated by examining the effects of an involution on multiplicative hybrid numbers. Additionally, the relationships between involutions and anti-involutions of multiplicative hybrid numbers are given in detail. Taking these properties into account, the geometric interpretation is illustrated by several significant figures.