Historical Analysis of Continuum for Teaching Continuity of Real Numbers

This study analyzes the historical development process of continuum and discusses the pedagogical implications of continuity of real numbers based on this. As a result of historical analysis, the following was confirmed. First, the process of constructing real numbers was a very difficult process that took a long time surrounding understanding and justification of infinity. Second, reductio ad absurdum was a premise condition to revealing that completeness is equivalent to the various ways of constructing real numbers. Teaching implications for this are as follows. First, the method of constructing real numbers from rational numbers can be divided into various levels from intuitive understanding to mathematical justification, and pedagogical treatment appropriate to students' understanding level should be provided. Second, it is not enough to distinguish the properties of rational numbers from the continuity of real numbers in the number line model, so the alternatives need to be studied. These findings provide significant implications for teaching and learning about the continuity of real numbers.