Bicomplex and Hyperbolic Numbers: Conjugates and Modulii

The theory of bicomplex numbers has found extensions and applications in many directions such as Quantum Mechanics, Blind Source Separation etc. The hyperbolic numbers are associated to Space-time geometry as stated by Lorentz transformations of special relativity, etc. Motivated by the wide range of applications, we study the bicomplex numbers and develop the theory of such numbers. Rochon presented a variety of algebraic properties of both bicomplex and hyperbolic numbers. He studied three types of conjugations and four types of modulii for both bicomplex and hyperbolic numbers. We have made a thorough investigation of Rochon?s work. We have established the relations among three types of conjugations. We have also investigated the impact of various kinds of conjugation on Principal ideals and idempotent elements of C2 and D. We have given an alternative definition of j-modulus and we have compared our definition with the corresponding definition given by Rochon. Hyperbolic numbers are particular case of bicomplex numbers. We also introduce notions of conjugates and modulii of hyperbolic numbers.