Teaching of real numbers by using the Archimedes–Cantor approach and computer algebra systems

Abstract

Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes–Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes–Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ϕ, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes–Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.