Complex numbers in three dimensions

Abstract

A system of commutative hypercomplex numbers of the form w=x+hy+kz are introduced in 3 dimensions, the variables x, y and z being real numbers. The multiplication rules for the complex units h, k are h^2=k, k^2=h, hk=1. The operations of addition and multiplication of the tricomplex numbers introduced in this paper have a simple geometric interpretation based on the modulus d, amplitude \rho, polar angle \theta and azimuthal angle \phi. Exponential and trigonometric forms are obtained for the tricomplex numbers, depending on the variables d, \rho, \theta and \phi. The tricomplex functions defined by series of powers are analytic, and the partial derivatives of the components of the tricomplex functions are closely related. The integrals of tricomplex functions are independent of path in regions where the functions are regular. The fact that the exponential form of the tricomplex numbers contains the cyclic variable \phi leads to the concepts of pole and residue for integrals of tricomplex functions on closed paths. The polynomials of tricomplex variables can be written as products of linear or quadratic factors.