The Foundations of Real Numbers: An Axiomatic Approach and Historical Perspective

Abstract

This paper provides a comprehensive study of real numbers (), tracing their evolution from ancient philosophical dilemmas to modern axiomatic definitions. It delves into the historical motivations that necessitated a rigorous foundation for numbers, particularly the Pythagorean crisis stemming from the discovery of irrational quantities. The report then systematically explicates the axiomatic framework of real numbers, detailing the Field Axioms that define their algebraic structure, the Order Axioms that establish their linear arrangement, and most critically, the Completeness Axiom, which ensures the continuity of the real number line and distinguishes  from other ordered fields like the rational numbers. A significant portion is dedicated to Dedekind's theory and Dedekind cuts as a primary method for constructing real numbers, illustrating how this formal approach fills the "gaps" inherent in the rational number system. The topological properties of real numbers, including their nature as a metric space and the concepts of limits and continuity, are also explored, highlighting their fundamental role in mathematical analysis. Finally, the report addresses pedagogical considerations for undergraduate students, offering insights into effective teaching strategies and common misconceptions encountered in the study of real analysis.