Body of real numbers

This article presents the main properties of the $ℝ$ body of real numbers. These properties are fundamental to the study of real and complex analysis.

We begin by defining the notion of a body, assumed here to be commutative, and recalling the main rules of calculation common to all bodies, in particular the formula giving the sum of the first n + 1 terms of a geometric series and Newton's binomial formula, which are essential to know.

We then introduce the concept of an ordered set, stressing the notions of upper and lower bounds, which must be mastered in the case of $ℝ$, and we give the definition of a totally ordered body, introducing in passing the notion of absolute value.

After showing certain inadequacies of the $\mathbb{Q}$ body of rational numbers, the body of real numbers is defined as the totally ordered body verifying the equivalent axioms of upper and lower bounds. However, the construction of $ℝ$ –, the principle of which dates back to 1872, whether by Dedekind's method of cuts or Cantor's method of quotient passage through the ring of Cauchy sequences of rational numbers –, has been relegated to an appendix, given its technical nature and its relatively modest interest in the theoretical and practical use of real numbers. The main properties of $ℝ$ are then established, in particular the existence of square roots (and more generally of n^{th roots} for positive numbers), with a reminder of the principle of solving second-degree equations and the Cauchy-Schwarz inequality, followed by the convergence in $ℝ$ of bounded mono-tones and Cauchy sequences of real numbers.

The presentation ends with the approximation of real numbers by rational numbers; the approximation of real numbers is developed:

• on the one hand, by the series of their decimal values, approximated to the nearest 10^–n, which is important for the practical use of real numbers;

• on the other hand, by the sequence of their continued fractions (or continued fractions) which constitute, in a sense that will be specified, the best approximations of real numbers by rational...