The geometries in N -dimensional Euclidean spaces can be described by Clifford algebras that were introduced as extensions of complex numbers. These applications are due to the fact that the Euclidean invariant (the distance between two points) is the same as the one of Clifford numbers. In this paper we consider the more general extension of complex numbers due to their group properties (hypercomplex systems), and we introduce the N -dimensional geometries associated with these systems. For N > 2 these geometries are different from the N -dimensional Euclidean geometries; then their investigation could open new applications. Moreover for the commutative systems the differential calculus does exist and this property allows one to define the functions of hypercomplex variable that can be used for studying some partial differential equations as well as the nonflat N -dimensional spaces. This last property can be relevant in general relativity and in field theories.

A Scheffers theorem states that for commutative hypercomplex numbers the differential calculus does exist and the functions can be introduced in the same way as they are for the complex variable. This property could open new applications of commutative quaternions in comparison with non-commutative Hamilton quaternions.In this article we introduce some quaternionic systems, their algebraic properties and the differential conditions (Generalized Cauchy-Riemann conditions) that their functions must satisfy.Then we show that the functional mapping, studied in the geometry associated with the quaternions, does have the same properties of the conformal mapping performed by the functions of complex variable. We also summarize the expressions of the elementary functions.

No abstract

No abstract

All the commutative hypercomplex number systems can be associated with a geometry. In two dimensions, by analogy with complex numbers, a general system of hypercomplex numbers {z = x + u y; u 2 = α + u β; x, y, α, β ∈ R; u / ∈ R} can be introduced and can be associated with plane Euclidean and pseudo-Euclidean (space-time) geometries. In this paper we show how these systems of hypercomplex numbers allow to generalise some well known theorems of the Euclidean geometry relative to the circle and to extend them to ellipses and to hyperbolas. We also demonstrate in an unusual algebraic way the Hero formula and Pytaghoras theorem, and show that these theorems hold for the generalised Euclidean and pseudo-Euclidean plane geometries.