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Abstract
Dispensing with the common property of distributivity and replacing classical trigonometric functions with their l p -counterparts in Euler’s trigonometric representation of complex numbers, classes of l p -complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the l p -absolute value of each l p -complex number invariant under l p -complex numbers multiplication is shown to be a group of elements that have l p -absolute value one but not the symmetry group.
Highlights
IntroductionThe various analytical, geometric and algebraic aspects of a strong mathematical foundation of complex numbers, and their applications to physics and technique, having been a cornerstone in establishing these numbers, can be studied in different levels of exposition in [1,2,3,4,5]
The various analytical, geometric and algebraic aspects of a strong mathematical foundation of complex numbers, and their applications to physics and technique, having been a cornerstone in establishing these numbers, can be studied in different levels of exposition in [1,2,3,4,5].Historically, the theory and applications of mathematics have not always developed in the same rhythm
Various types of functions may appear in dependence on how the single turn of a wire is moved within the magnetic stator
Summary
The various analytical, geometric and algebraic aspects of a strong mathematical foundation of complex numbers, and their applications to physics and technique, having been a cornerstone in establishing these numbers, can be studied in different levels of exposition in [1,2,3,4,5]. For everyday applications the rotor is a thick wound coil and voltage and current are dealt with as the real and imaginary parts of a complex number, respectively, or as the components of a vector that moves within a Euclidean circle in the complex plane with time. The multiplication allows an interpretation in terms of a geometric vector product This circumstance motivates the generalization of the vector realization of the classical complex structure presented in the present paper. In the classical vector realization, C is endowed with the absolute value function, | x + iy| = ( x2 + y2 )1/2 , and geometric vector multiplication with a complex number of absolute value one in particular means moving points from a circle {z ∈ C : |z| = r } without leaving this set, just like orthogonal transformations do.
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