Real number


In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, 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 in many other branches of mathematics, in particular by their role in the classical definitions of limits, continuity and derivatives.
The set of real numbers, sometimes called "the reals", is usually notated as a bold or the blackboard bold.
The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of negative numbers.
The real numbers include the rational numbers, such as the integer and the fraction. Real numbers that are not rational are irrational. Those real numbers that are roots of polynomials with rational coefficients are algebraic numbers, which include all the rational numbers and also irrational numbers such as. Other real numbers, such as, are not roots of polynomials; these are the transcendental numbers.
The real numbers can be thought of as the points on a line, called the number line or real line, on which the points corresponding to integers are equally spaced.
The informal descriptions above of the real numbers are not sufficient for rigorous reasoning about real numbers. The development of a suitable formal definition was a major achievement of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. One modern axiomatic definition is that real numbers form the unique Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences, Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

Characterizing properties

Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence.

Arithmetic

The real numbers form an ordered field. Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations, addition and multiplication, and a total order that have the following properties.
  • The addition of two real numbers and produce a real number denoted which is the sum of and.
  • The multiplication of two real numbers and produce a real number denoted or which is the product of and.
  • Addition and multiplication are both commutative, which means that and for every real numbers and.
  • Addition and multiplication are both associative, which means that and for every real numbers, and, and that parentheses may be omitted in both cases.
  • Multiplication is distributive over addition, which means that for every real numbers, and.
  • There is a real number called zero and denoted which is an additive identity, which means that for every real number.
  • There is a real number denoted which is a multiplicative identity, which means that for every real number.
  • Every real number has an additive inverse denoted This means that for every real number.
  • Every nonzero real number has a multiplicative inverse denoted or This means that for every nonzero real number.
  • The total order is denoted Being a total order means that it has the following two properties:
  1. For any two real numbers and, exactly one of,, is true.
  2. If and then.
  • The order is compatible with addition and multiplication, which means that implies for every real number, and is implied by and
Many other properties can be deduced from the above ones. In particular:
  • for every real number
  • for every nonzero real number

    Auxiliary operations

Several other operations are commonly used, which can be deduced from the above ones.
The total order that is considered above is denoted and read as " is less than ". Three other order relations are also commonly used:
The real numbers and are commonly identified with the natural numbers and. This allows identifying any natural number with the sum of real numbers equal to.
This identification can be pursued by identifying a negative integer with the additive inverse of the real number identified with Similarly a rational number is identified with the division of the real numbers identified with and.
These identifications make the set of the rational numbers an ordered subfield of the real numbers The Dedekind completeness described below implies that some real numbers, such as are not rational numbers; they are called irrational numbers.
The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties. So, the identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with taken as the successor function.
Formally, one has an injective homomorphism of ordered monoids from the natural numbers to the integers an injective homomorphism of ordered rings from to the rational numbers and an injective homomorphism of ordered fields from to the real numbers The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write
These identifications are formally abuses of notation, and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming. In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by the compiler.

Dedekind completeness

Previous properties do not distinguish real numbers from rational numbers. This distinction is provided by Dedekind completeness, which states that every non-empty set of real numbers with an upper bound admits a least upper bound. This means the following. A set of real numbers is bounded above if there is a real number such that for all ; such a is called an upper bound of So, Dedekind completeness means that, if is non-empty and bounded above, it has an upper bound that is less than any other upper bound.
Dedekind completeness implies other sorts of completeness, but also has some important consequences.
The last two properties are summarized by saying that the real numbers form a real closed field. This implies the real version of the fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.

Decimal representation

The most common way of describing a real number is via its decimal representation, a sequence of decimal digits each representing the product of an integer between zero and nine times a power of ten, extending to finitely many positive powers of ten to the left and infinitely many negative powers of ten to the right. For a number whose decimal representation extends places to the left, the standard notation is the juxtaposition of the digits in descending order by power of ten, with non-negative and negative powers of ten separated by a decimal point, representing the infinite series
For example, for the circle constant is zero and etc.
More formally, a decimal representation for a nonnegative real number consists of a nonnegative integer and integers between zero and nine in the infinite sequence
Such a decimal representation specifies the real number as the least upper bound of the decimal fractions that are obtained by truncating the sequence: given a positive integer, the truncation of the sequence at the place is the finite partial sum
The real number defined by the sequence is the least upper bound of the which exists by Dedekind completeness.
Conversely, given a nonnegative real number, one can define a decimal representation of by induction, as follows. Define as decimal representation of the largest integer such that . Then, supposing by induction that the decimal fraction has been defined for one defines as the largest digit such that and one sets
One can use the defining properties of the real numbers to show that is the least upper bound of the So, the resulting sequence of digits is called a decimal representation of.
Another decimal representation can be obtained by replacing with in the preceding construction. These two representations are identical, unless is a decimal fraction of the form In this case, in the first decimal representation, all are zero for and, in the second representation, all 9..
In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9.
The preceding considerations apply directly for every numeral base simply by replacing 10 with and 9 with

Topological completeness

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete :
A sequence of real numbers is called a Cauchy sequence if for any there exists an integer N such that the distance is less than ε 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.
A sequence converges to the limit ''x if its elements eventually come and remain arbitrarily close to x'', that is, if for any there exists an integer N such that the distance is less than ε for n greater than N.
Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.
The set of rational numbers is not complete. For example, the sequence, where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number.
The completeness property of the reals is the basis on which calculus, and more generally mathematical analysis, are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.
For example, the standard series of the exponential function
converges to a real number for every x, because the sums
can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that is well defined for every x.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element.
Additionally, an order can be Dedekind-complete, see. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field and then forms the Dedekind-completion of it in a standard way.
These two notions of completeness ignore the field structure. However, an ordered group defines a uniform structure, and uniform structures have a notion of completeness; the description in § Completeness is a special case. It is not true that is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete, justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences, since it starts with an Archimedean field and forms the uniform completion of it in a standard way.
But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of. Thus is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field and then selects from it the largest Archimedean subfield.

Cardinality

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there exists no one-to-one function from the real numbers to the natural numbers. The cardinality of the set of all real numbers is called the cardinality of the continuum and commonly denoted by It is strictly greater than the cardinality of the set of all natural numbers, denoted and called Aleph-zero or aleph-nought. The cardinality of the continuum equals the cardinality of the power set of the natural numbers, that is, the set of all subsets of the natural numbers.
The statement that there is no cardinality strictly greater than and strictly smaller than is known as the continuum hypothesis. It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice —the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.

Other properties

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
The real numbers form a metric space: the distance between x and y is defined as the absolute value. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible, separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.
Every nonnegative real number has a square root in, although no negative number does. This shows that the order on is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.
The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.
The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as. Ordered fields that satisfy the same first-order sentences as are called nonstandard models of. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model, we know that the same statement must also be true of.
The 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. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.
The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on with the property that every nonempty subset of has a least element in this ordering. Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.
A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.

History

were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" in include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava, who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined.
Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the root of 2">root of a polynomial">root of 2 is irrational.
For Greek mathematicians, numbers were only the natural numbers. Real numbers were called "proportions", being the ratios of two lengths, or equivalently being measures of a length in terms of another length, called unit length. Two lengths are "commensurable", if there is a unit in which they are both measured by integers, that is, in modern terminology, if their ratio is a rational number. Eudoxus of Cnidus provided a definition of the equality of two irrational proportions in a way that is similar to Dedekind cuts, except that he did not use any arithmetic operation other than multiplication of a length by a natural number. This may be viewed as the first definition of the real numbers.
The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation. In Europe, such numbers, not commensurable with the numerical unit, were called irrational or surd.
In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" numbers.
In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert gave a flawed proof that cannot be rational; Legendre completed the proof and showed that is not the square root of a rational number. Liouville showed that neither nor can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor extended and greatly simplified this proof. Hermite proved that e | is transcendental, and Lindemann, showed that is transcendental. Lindemann's proof was much simplified by Weierstrass, Hilbert, Hurwitz, and Gordan.
The concept that many points existed between rational numbers, such as the square root of 2, was well known to the ancient Greeks. The existence of a continuous number line was considered self-evident, but the nature of this continuity, presently called completeness, was not understood. The rigor developed for geometry did not cross over to the concept of numbers until the 1800s.

Modern analysis

The developers of calculus used real numbers and limits without defining them rigorously. In his Cours d'Analyse, Cauchy made calculus rigorous, but he used the real numbers without defining them, and assumed without proof that every Cauchy sequence has a limit and that this limit is a real number.
In 1854 Bernhard Riemann highlighted the limitations of calculus in the method of Fourier series, showing the need for a rigorous definition of the real numbers.
Beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, including Hermann Hankel, Charles Méray, and Eduard Heine, leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as Dedekind cuts, and the other one by Georg Cantor, as equivalence classes of Cauchy sequences. Several problems were left open by these definitions, which contributed to the foundational crisis of mathematics. Firstly both definitions suppose that rational numbers and thus natural numbers are rigorously defined; this was done a few years later with Peano axioms. Secondly, both definitions involve infinite sets, and Cantor's set theory was published several years later. Thirdly, these definitions imply quantification on infinite sets, and this cannot be formalized in the classical logic of first-order predicates. This is one of the reasons for which higher-order logics were developed in the first half of the 20th century.
In 1874 Cantor showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891.

Formal definitions

The real number system can be defined axiomatically up to an isomorphism, which is described hereinafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another approach is to start from some rigorous axiomatization of Euclidean geometry, and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.

Axiomatic approach

Let denote the set of all real numbers. Then:
  • The set is a field, meaning that addition and multiplication are defined and have the usual properties.
  • The field is ordered, meaning that there is a total order ≥ such that for all real numbers x, y and z:
  • * if x ≥ y, then x + z ≥ y + z;
  • * if x ≥ 0 and y ≥ 0, then xy ≥ 0.
  • The order is Dedekind-complete, meaning that every nonempty subset S of with an upper bound in has a least upper bound in.
The last property applies to the real numbers but not to the rational numbers. For example, has a rational upper bound, but no least rational upper bound, because square root| is not rational.
These properties imply the Archimedean property, which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that, and thus, which is a contradiction with the upper-bound property of N.
The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields and, there exists a unique field isomorphism from to. This uniqueness allows us to think of them as essentially the same mathematical object.
For another axiomatization of see Tarski's axiomatization of the reals.

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like converges to a unique real number—in this case. For details and other constructions of real numbers, see Construction of the real numbers.

Applications and connections

Physics

In the physical sciences most physical constants, such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity, and the Standard Model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.
Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.

Logic

The 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 also studied in reverse mathematics and in constructive mathematics.
The 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.
Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are elements of the set of the real numbers.
The 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. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.

Computation

and computers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary definable real numbers, which are inconvenient to manipulate.
Instead, computers typically work with finite-precision approximations called floating-point numbers, a representation similar to scientific notation. The achievable precision is limited by the data storage space allocated for each number, whether as fixed-point, floating-point, or arbitrary-precision numbers, or some other representation. Most scientific computation uses binary floating-point arithmetic, often a 64-bit representation with around 16 decimal digits of precision. Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not. The field of numerical analysis studies the stability and accuracy of numerical algorithms implemented with approximate arithmetic.
Alternately, computer algebra systems can operate on irrational quantities exactly by manipulating symbolic formulas for them rather than their rational or decimal approximation. But exact and symbolic arithmetic also have limitations: for instance, they are computationally more expensive; it is not in general possible to determine whether two symbolic expressions are equal ; and arithmetic operations can cause exponential explosion in the size of representation of a single number, overwhelming finite computer storage.
A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

Set theory

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties that are a technical inconvenience. Elements of Baire space are referred to as "reals".

Vocabulary and notation

The set of all real numbers is notated as or R. An old typewriter convention used double-struck I and R, and an old printing convention used the letters I and R with no kerning.
As it is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.
The sets of positive real numbers and negative real numbers are often noted and, respectively; and are also used. The non-negative real numbers can be noted but one often sees this set noted In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively and In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted and
The notation refers to the set of the -tuples of elements of , which can be identified to the Cartesian product of copies of It is an -dimensional vector space over the field of the real numbers, often called the coordinate space of dimension ; this space may be identified to the -dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a point of the Euclidean space is identified with the tuple of its Cartesian coordinates.
In mathematics real is used as an adjective, meaning that the underlying field is the field of the real numbers. For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number.

Generalizations and extensions

The real numbers can be generalized and extended in several different directions: