Abstract

Three-complex numbers are introduced for using a geometric vector product in the three-dimensional Euclidean vector space R3 and proving its equivalence with a spherical coordinate product. Based upon the definitions of the geometric power and geometric exponential functions, some Euler-type trigonometric representations of three-complex numbers are derived. Further, a general l23−complex algebraic structure together with its matrix, polynomial and variable basis vector representations are considered. Then, the classes of lp3-complex numbers are introduced. As an application, Euler-type formulas are used to construct directional probability laws on the Euclidean unit sphere in R3.

Highlights

  • While a crucial property of complex numbers x + iy is very briefly described by the equation i2 = −1, the numbers x + iy + jz considered in the present work have the fundamental property i2 = j2 = ij = −1

  • For three-complex numbers or complex numbers in dimension three, xk = xk + iyk + jzk, k = 1, 2 where xk, yk and zk are real numbers and i and j are called imaginary units, we introduce in the usual notation for complex numbers the associative and commutative addition rule x1 + x2 = ( x1 + x2 ) + i (y1 + y2 ) + j(z1 + z2 ) and for 6= (0, 0), k = 1, 2 the multiplication rule x1 · x2 = S(x1, x2 )( x1 x2 −

  • “imaginary” or “alchemical” aspects of this introduction, we present a completely formally correct mathematical introduction to these three-complex numbers in Section 2 and extend this approach in Section 5.1 by introducing the more general notion of a corresponding three-complex algebraic structure

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Summary

Introduction

If zk = xk + iyk , k = 1, 2 denote usual complex numbers the real and imaginary parts of their product z1 z2 = x1 x2 − y1 y2 + i ( x1 y2 + x2 y1 ) may be interpreted as the components of a geometric vector product in the two-dimensional Euclidean vector space R2 , z1 · z2 = The implementation of such an approach in three dimensions is the subject of the present work. The interested reader may find more on “mathematical alchemy” in “Why an unsolved problem in mathematics matters” by Marcus du Sautoy, Fourth Estate, London 2003.) Sections 3 and 4 in between are devoted to an application of the new Euler-type formulas to the construction of directional probability laws and to a geometric view of the present topic, respectively.

Three-Complex Numbers
Directional Probability Laws
Geometric View
A General Three-Complex Algebraic Structure
On l23 -Complex Structures
Classes of l 3p -Complex Numbers
Discussion
Full Text
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