This paper introduces the meta-real number system, a novel framework designed to unify various mathematical concepts, with a particular emphasis on the behavior of the number zero. Traditional real number systems prohibit division by zero; however, various extensions in topology and abstract algebra have explored alternative approaches. Inspired mainly by the real projective line, the meta-real number system incorporates ideas from affine spaces, projective spaces and multi-projective spaces.The paper establishes a set of axioms governing arithmetic operations within this system, including series addition and subtraction, parallel addition and subtraction, multiplication, and division. It differentiates between two absorbing elements—unsigned zero and unsigned infinity—each linked to distinct geometric interpretations. A key feature of this system is its ability to handle values and distances using both numerator-based and denominator-based measures, offering new insights into geometric transformations.The document also examines the role of the involutory function ρ as a transformation, which swaps zeros and asymptotes in rational expressions. The paper concludes by presenting examples of the system's applications, such as four alternative definitions of the classical derivative, each paired with its own corresponding tangent curve. These definitions give rise to four distinct expansions of an infinitely differentiable real-valued function at a point (a).This work proposes that the meta-real number system can unify diverse mathematical concepts under a single framework and suggests future directions for mathematical research.
Louis Thériault (Sun,) studied this question.