On Real Numbers

Abstract

Ordinary real numbers permit subdivision to arbitrary depth. This paper investigates a real-number syntax with finite resolution depth, while retaining ordinary real numbers as a language of theoretically infinite depth. Rather than truncating real numbers or replacing them with finite decimals, the proposed quantum real-number system changes the rules of operation at a nonzero terminal depth h. Classical operations are retained away from terminal depth. At the terminal level, higher powers of h are treated according to their relational role through collapse, retention, or carrying. Relational computation precedes terminal reduction, so a contribution that can return to a distinguishable scale is not eliminated prematurely. As initial tests, the paper develops finite-depth differences of polynomials, an exact finite-difference product rule, a chain relation with an operation-generated outer step, and symmetric second differences, without taking h to zero. It also discusses the relation to the historical problem of infinitesimals. This is a preliminary framework, not a completed algebraic field theory. Open problems include a rigorous definition of the terminal operation Q_h, criteria for collapse and carrying, compatibility with arithmetic laws and division, orientation information, finite-depth integration, and conversion between ordinary and quantum real-number readings.