The concept of exponential in real analysis books: a perspective of intuition and rigor point of view

Abstract

In this paper, we aim to examine the approach adopted in Real Analysis textbooks used by Brazilian universities in the construction of the concept of  and to present a more intuitive construction based on the notions of supremum, infimum and the completeness of the real number system. As a theoretical framework, we discuss the relationship between intuition and rigor in the context of teaching Real Analysis. This qualitative study involves documentary analysis of textbooks. We conclude that the most common constructions in Real Analysis textbooks, although rigorous and elegant, typically rely on formal definitions of exponential and logarithmic functions, which are not very intuitive, because they require many previous concepts. The alternative construction we present prioritizes intuition, offering students greater opportunities for visualization and creativity. It allows them to follow a pedagogical and rigorous path to the definition of the exponential function, which is fundamental for future mathematics teachers and mathematicians.