A new proposal to the extension of complex numbers

TL;DR

This paper introduces a novel extension of complex numbers by defining a new domain where negative and imaginary probabilities can exist, involving a unique unit based on an unsolvable equation in the complex domain.

Contribution

It proposes a new extended number system with a unique unit derived from an unsolvable equation, expanding the complex number framework to include new concepts like negative and imaginary probabilities.

Findings

Defined a new number space with a special unit satisfying |z|^2= i

Established properties such as positive-definiteness and linearity in the new space

Introduced a new multiplication and mapping operation to preserve algebraic properties

Abstract

We propose the extension of the complex numbers to be the new domain where new concepts, like negative and imaginary probabilities, can be defined. The unit of the new space is defined as the solution of the unsolvable equation in the complex domain: $|z|^2= z^* z = i$. The existence of the unsolvable equation in a closed domain as complex's lead to the definition of a new type of multiplication, for not violate the fundamental theorem of algebra. The definition of the new space also requests the inclusion of a new mapping operation, so the absolute value of the new extended number being real and positive. We study the properties of the vector space like positive-definiteness, linearity, and conjugated symmetry.