n-Dimensional dual complex numbers

It is well-known that the quadratic algebrasQ _a,b = {z|z =x +qy,q ² =a +qb ,a, b, x, y ε ℝ,q ∉ ℝ }, also expressible as ℝx]/(x ² -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,Q _0,0,Q _1,0 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.