On Complex Numbers in Higher Dimensions

Abstract

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

Keywords

• geometric vector product
• geometric vector power
• geometric exponential function
• vector division
• spherical geometric coordinate product
• spherical geometric coordinate vector ratio