Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Definable real number
Summary
Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge.
Different choices of a formal language or its interpretation give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable.
Constructible number
One way of specifying a real number uses geometric techniques. A real number is a constructible number if there is a method to construct a line segment of length using a compass and straightedge, beginning with a fixed line segment of length 1.
Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube.
A real number is called a real algebraic number if there is a polynomial , with only integer coefficients, so that is a root of , that is, .
Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial has 5 real roots, the third one can be defined as the unique such that and such that there are two distinct numbers less than at which is zero.
All rational numbers are algebraic, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible.
The real algebraic numbers form a subfield of the real numbers.
Official source
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.
Ontological neighbourhood
Related courses (32)
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(ol): Analyse I (online)
Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.
Related lectures (167)
Limit of a Sequence: Reminder and Examples
Covers the concept of limit of a sequence and introduces the two gendarmes theorem.
Optimization Algorithms: Greedy Approach
Explores optimization problems and greedy algorithms for efficient decision-making.
Mathematics: Analysis and Algebra Overview
Provides an overview of analysis and algebra courses, focusing on real numbers, limits, functions, and exams.
Related publications (32)
Related people (5)
Related concepts (12)
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
Mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and pi occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (pi).
Real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. A real closed field is a field F in which any of the following equivalent conditions is true: F is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals.
Related MOOCs (11)
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
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
Analyse I
Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond