Abstract
We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.
Highlights
The history of complex numbers dates all the way back to the 16th century and begins with a period of ‘empirical’ discoveries and the derivation of surprising individual formulae, from which people observed the possibility of working successfully with imaginary numbers without being able to provide a satisfactory explanation. Real masters expanded their usage of complex numbers in various computations, and some of them provided the first descriptions of these numbers
The continued search for a firm mathematical foundation, and with it a philosophical interpretation, lasted until the nineteenth century and can be followed in detail in [1], where the interested reader will find all the famous names involved until 1831, and the remark that “in 1854 Richard Dedekind . . . judged the situation differently”. Dedekind said at their Göttingen habilitation presentation in the presence of Gauss, ‘Until now, no theory of complex numbers has been accessible that is absolutely free from criticism . . . or at least none has been published’
Solving a quadratic equation within different realizations of an abstract complex algebraic structure allows the derivation of numerous new solutions beyond those known for the classical complex numbers
Summary
The history of complex numbers dates all the way back to the 16th century and begins with a period of ‘empirical’ discoveries and the derivation of surprising individual formulae, from which people observed the possibility of working successfully with imaginary numbers without being able to provide a satisfactory explanation. Solving a quadratic equation within different realizations of an abstract complex algebraic structure allows the derivation of numerous new solutions beyond those known for the classical complex numbers In the rest of this section, the analytical definition of the vector p-product for p < 0, in Definition 1, will by expressed equivalently in a geometric way For this purpose, we define the lp-type polar coordinate transformation Polp : (0, ∞) × (0, 2π)∗ → R2 for the present case p < 0 as (x, y)T = Polp[r, φ] with x = r cosp(φ), y = r sinp(φ). Consideration of the limiting situation p → 0 as well as cases of more general functionals is left open, here
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