Completeness of the real numbers

Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number. Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction. There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space). The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom. Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields that are ordered and Cauchy complete. When the real numbers are instead constructed using a model, completeness becomes a theorem or collection of theorems. Least-upper-bound property The least-upper-bound property states that every nonempty subset of real numbers having an upper bound must have a least upper bound (or supremum) in the set of real numbers. The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers This set has an upper bound. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be √2, but it does not exist in Q. For any upper bound x ∈ Q, there is another upper bound y ∈ Q with y < x. For instance, take x = 1.5, then x is certainly an upper bound of S, since x is positive and x^2 = 2.25 ≥ 2; that is, no element of S is larger than x.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (15)

Related courses (20)

Related lectures (157)

Related MOOCs (2)

Information, Calcul, Communication: Introduction à la pensée informatique

Dans une première partie, nous étudierons d’abord comment résoudre de manière très concrète un problème au moyen d’un algorithme, ce qui nous amènera dans un second temps à une des grandes questions d

Information, Calcul, Communication: Introduction à la pensée informatique

In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound.

In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set of all rational numbers with its natural order does not have the least upper bound property.

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a

MATH-101(e): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable.

Limit of a Sequence: Reminder and Examples MATH-101(e): Analysis I

Covers the concept of limit of a sequence and introduces the two gendarmes theorem.

Mathematics: Analysis and Algebra Overview MATH-101(e): Analysis I

Provides an overview of analysis and algebra courses, focusing on real numbers, limits, functions, and exams.

Absolute Value and Intervals MATH-101(f): Analysis I

Introduces absolute value, intervals, triangle inequality, and complete numbers in the complex plane.