How to define a real number through the concept of Dedekind cut?

Abstract

The aim of this article is to provide a logical building of the real number system starting from the rational number system. The real number system plays most vital role in the study of calculus and real analysis in undergraduate classes, in fact, real numbers play an important role in school mathematics also. This article is mainly intended for undergraduate students. Assuming that the rational number system is the smallest ordered field we talk of the existence of gaps in them and try to fill them up by the notion of Dedekind cut and thus land up in the system of real numbers. The system of real numbers is shown to be a complete ordered field. We then define a real number as an element of a complete ordered field after proving that any two complete ordered fields are order isomorphic.