19 Facts About Real numbers

Real numbers can be used to measure physical observables such as time, mass, energy; and in one dimension, distance, velocity, acceleration, force, momentum, etc.

The set of real numbers is denoted using the symbol R or and is sometimes called "the reals".

Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced.

The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers.

In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite.

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All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.

The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.

The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound.

Sequence of real numbers is called a Cauchy sequence if for any there exists an integer N such that the distance is less than e for all n and m that are both greater than N This definition, originally provided by Cauchy, formalizes the fact that the xn eventually come and remain arbitrarily close to each other.

Real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

The irrational numbers are dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

Field of real numbers is an extension field of the field of rational numbers, and can therefore be seen as a vector space over.

Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others.

Real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics.

In particular, the real numbers are studied in reverse mathematics and in constructive mathematics.

Hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.

Continuum hypothesis posits that the cardinality of the set of the real numbers is ; i e the smallest infinite cardinal number after, the cardinality of the integers.

Real numbers number is called computable if there exists an algorithm that yields its digits.

Real numbers can be generalized and extended in several different directions:.