N-Dimensional dual complex numbers

Abstract

It is well-known that the quadratic algebrasQa,b = {z|z =x +qy,q2 =a +qb ,a, b, x, y e ℝ,q ∉ ℝ }, also expressible as ℝ[x]/(x2 -bx -a), are, up to isomorphism, equivalent to just three algebras, corresponding to elliptic, parabolic and hyperbolic. These three types are usually represented byQ−1,0,Q0,0,Q1,0 and called complex numbers, dual complex numbers and hyperbolic complex numbers, respectively. Each in turn describes a Euclidian, Galilean and Minkowskian plane. The hyperbolic complex numbers thus provide a 2-dimensional spacetime for special relativity physics (see e.g. [6]) and the dual complex numbers a 2-dimensional spacetime for Newtonian physics (see e.g. [17]). The present authors considered extensions of the hyperbolic complex numbers ton dimensions in [8], and here, in somewhat parallel fashion, some elements of algebra (in Section 1) and analysis (in Section 2) will be presented forn-dimensional dual complex numbers.