Algebraic Properties of Bihyperbolic Numbers

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.