ABSTRACT
Important theorems of calculus often rely on the properties of real numbers. In the course of the nineteenth century it became clear that these needed to be proved as well, and for that it was necessary to make a precise definition of real numbers. In any area of mathematics, statements need to be proved, and this always involves the use of previously established results. This approach, the deductive method, has as its foundation a set of axioms, from which other assertions can be derived. The natural numbers are not suitable for the development of calculus. For example, we cannot do the usual arithmetic operations of subtraction and division. In order to remedy this deficiency, mathematicians have been using rational numbers since the earliest times. The chapter closes with a brief discussion of several issues regarding the deductive approach that we have used. On a positive side, once a result is proved, there is no dilemma that it is true.
