n-Dimensional hyperbolic complex numbers

Direct product rings have received relatively little attention, perhaps because they are sometimes labeled “trivial” [8, p.6]. Nevertheless, the 2-dimensional direct product ring of the reals, when expressed in the “hyperbolic basis”, is analogous in many ways to the system of complex numbers and also has a physical interpretation. This prompted an exploratory foray into the world ofn-dimensional direct product rings of the reals to see how much can be extended from the 2-dimensional case (see, e.g. [3,4,5]). Section 1 provides algebraic notation, up to the point of defining polar coordinates. Section 2 uses analysis to explore differentiability and conformality.