Algebraic Properties of Bihyperbolic Numbers

• Published in: Advances in applied Clifford algebras Vol. 30; no. 1
• Main Authors: Bilgin, Merve, Ersoy, Soley
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-02-2020; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Banach spaces; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Real numbers; Theoretical; Idempotent representations of bihyperbolic numbers; Moduli of bihyperbolic numbers; 14K10; 13A18; Bihyperbolic numbers; Hyperbolic numbers
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-019-1036-2

In this paper, we study the four-dimensional real algebra of bihyperbolic numbers. Under consideration of the spectral representation of the bihyperbolic numbers, we give a partial order of bihyperbolic numbers which allows us to obtain some relations in the ordered vector space of bihyperbolic numbers. Moreover, we state that the set of bihyperbolic numbers form a real Banach algebra with a new defined norm. We introduce conjugates, three hyperbolic valued moduli, real moduli, and multiplicative inverse of the bihyperbolic numbers. We give the concept of the absolute value of a bihyperbolic number which generalizes that of real numbers. Also, we represent the polar form of invertible bihyperbolic numbers.