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This book arises from original research of the authors on hypercomplex numbers and their applications ([8] and [15]–[23]). Their research concerns extensions to more general number systems of both well-established applications of complex numbers and of functions of a complex variable.
Before introducing the contents of the book, we briefly recall the epistemological relevance of Number in the development of Western Science. In his “Metaphysics of number”, Pythagoras considered reality, at its deepest level, as mathematical in nature. Following Pythagoras, Plato (Timaeus) explained the world by the regular polygons and solids of Euclidean geometry, laying a link between Number, Geometry and Physical World that represents the foundation of Modern Science. Accordingly, Galileo (Il Saggiatore § 6) took geometry as the language of Nature. These ideas that may appear trivial to modern rationalism, still have their own validity. For example, imaginary numbers make sense of algebraic equations which, from a geometrical point of view, could represent problems that admit no solutions. Despite such an introduction, complex numbers are strictly related to Euclidean geometry (Chap. 3) and allow formalizing Euclidean trigonometry (Chap. 4). Moreover, their functions are the means of representing the surface of the Earth on a plane (Chap. 8). In more recent times, another astonishing coincidence has been added to the previous ones: the space-time symmetry of two-dimensional Special Relativity, which, after Minkowski, is called Minkowski geometry, has been formalized [18] by means of hyperbolic numbers, a number system which represents the simplest extension of complex numbers [81]. Finally, N-dimensional Euclidean geometries and number theory have found a unified language by means of “Clifford algebra” [14], [42] and [45], which has allowed a unified formalization of many physical theories.
Preface
Introduction
N-Dimensional Commutative Hypercomplex Numbers
The Geometries Generated by Hypercomplex Numbers 1
Trigonometry in the Minkowski Plane
Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox)
General Two-Dimensional Hypercomplex Numbers
Functions of a Hyperbolic Variable
Hyperbolic Variables on Lorentz Surfaces
Constant Curvature Lorentz Surfaces
Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle)
Appendices
A Commutative Segre’s Quaternions
B Constant Curvature Segre’s Quaternion Spaces
C Matrix Formalization for Commutative Numbers