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 R or the blackboard bold \(\R\).

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 \(-5\) and the fraction \(4/3\). 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 √2 = 1.414.... Other real numbers, such as π = 3.1415..., 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 (..., −2, −1, 0, 1, 2, ...) 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 (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

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.

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 a and b produces a real number denoted \(a+b,\) which is the sum of a and b.
• The multiplication of two real numbers a and b produces a real number denoted \(ab,\) \(a\cdot b\) or \(a\times b,\) which is the product of a and b.
• Addition and multiplication are both commutative, which means that \(a+b=b+a\) and \(ab=ba\) for all real numbers a and b.
• Addition and multiplication are both associative, which means that \((a+b)+c=a+(b+c)\) and \((ab)c=a(bc)\) for all real numbers a, b and c.
• Multiplication is distributive over addition, which means that \(a(b+c)=ab+ac\) and \((a+b)c=ac+bc\) for all real numbers a, b and c.
• There is a real number called zero and denoted 0 which is an additive identity, which means that \(a+0=0+a=a\) for every real number a.
• There is a real number denoted 1 which is a multiplicative identity, which means that \(a\times 1 =1\times a=a\) for every real number a.
• Every real number a has an additive inverse denoted \(-a.\) This means that \(a+(-a)=(-a)+a=0\) for every real number a.
• Every nonzero real number a has a multiplicative inverse denoted \(a^{-1}\) or \(\tfrac 1a.\) This means that \(aa^{-1}=a^{-1}a=1\) for every nonzero real number a.
• The total order is denoted \(a

1. For any two real numbers a and b, exactly one of \(a
2. If \(a

• The order is compatible with addition and multiplication, which means that \(a

Many other properties can be deduced from the above properties. In particular:

• \(0\cdot a=0\) for every real number a
• \(0<1\)
• \(0

Several other operations are commonly used, which can be deduced from the above properties.

• Subtraction: the subtraction of two real numbers a and b results in the sum of a and the additive inverse −b of b; that is, \[a-b=a+(-b).\]
• Division: the division of a real number a by a nonzero real number b is denoted \(\frac ab,\) or \(a/b\) and defined as the multiplication of a with the multiplicative inverse of b; that is, \[\frac ab=ab^{-1}.\]
• Absolute value: the absolute value of a real number a, denoted \(|a|,\) measures its distance from zero, and is defined as \[|a|=\max(a,-a).\]

The total order that is considered above is denoted \(a

• Greater than: \(a>b,\) read as "a is greater than b", is defined as \(a>b\) if and only if \(b
• Less than or equal to: \(a\le b,\) read as "a is less than or equal to b" or "a is not greater than b", is defined as \((a
• Greater than or equal to: \(a\ge b,\) read as "a is greater than or equal to b" or "a is not less than b", is defined as \((b

The real numbers 0 and 1 are commonly identified with the natural numbers 0 and 1. This allows identifying any natural number n with the sum of n real numbers equal to 1.

This identification can be pursued by identifying a negative integer \(-n\) (where \(n\) is a natural number) with the additive inverse \(-n\) of the real number identified with \(n.\) Similarly a rational number \(p/q\) (where p and q are integers and \(q\ne 0\)) is identified with the division of the real numbers identified with p and q.

These identifications make the set \(\Q\) of the rational numbers an ordered subfield of the real numbers \(\R.\) The Dedekind completeness described below implies that some real numbers, such as \(\sqrt 2,\) 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 (axioms). 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 1 taken as the successor function.

Formally, one has an injective homomorphism of ordered monoids from the natural numbers \(\N\) to the integers \(\Z,\) an injective homomorphism of ordered rings from \(\Z\) to the rational numbers \(\Q,\) and an injective homomorphism of ordered fields from \(\Q\) to the real numbers \(\R.\) The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write

\(\N\subset \Q \subset \R.\)

These identifications are formally abuses of notation (since, formally, a rational number is an equivalence class of pairs of integers, and a real number is an equivalence class of Cauchy series), 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.

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 \(S\) is bounded above if there is a real number \(u\) such that \(s\le u\) for all \(s\in S\); such a \(u\) is called an upper bound of \(S.\) So, Dedekind completeness means that, if S 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 (see below), but also has some important consequences.

• Archimedean property: for every real number x, there is an integer n such that \(x
• Equivalently, if x is a positive real number, there is a positive integer n such that \(0 <\tfrac 1n
• Every positive real number x has a positive square root, that is, there exists a positive real number \(r\) such that \(r^2=x.\)
• Every univariate polynomial of odd degree with real coefficients has at least one real root (if the leading coefficient is positive, take the least upper bound of real numbers for which the value of the polynomial is negative).

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.

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 x whose decimal representation extends k places to the left, the standard notation is the juxtaposition of the digits \(b_k b_{k-1}\cdots b_0. a_{1} a_{2}\cdots,\) in descending order by power of ten, with non-negative and negative powers of ten separated by a decimal point, representing the infinite series

\(x =b_k 10^k+b_{k-1} 10^{k-1}+ \cdots + b_0 +\frac{a_1}{10} +\frac{a_2}{10^2}+\cdots.\)

For example, for the circle constant \(\pi = 3.14159\cdots,\) k is zero and \(b_0=3,\) \(a_1=1,\) \(a_2=4,\) etc.

More formally, a decimal representation for a nonnegative real number x consists of a nonnegative integer k and integers between zero and nine in the infinite sequence

\(b_k, b_{k-1},\ldots, b_0, a_{1}, a_{2},\ldots.\)

(If \(k>0,\) then by convention \(b_k\neq 0.\))

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 n, the truncation of the sequence at the place n is the finite partial sum

\(\begin{aligned} D_n &=b_k 10^k+b_{k-1} 10^{k-1}+ \cdots + b_0 +\frac{a_1}{10}+\cdots+\frac{a_n}{10^n}\\ &=\sum_{i=0}^k b_i 10^i + \sum_{j=1}^n a_j10^{-j} \end{aligned}\)

The real number x defined by the sequence is the least upper bound of the \(D_n,\) which exists by Dedekind completeness.

Conversely, given a nonnegative real number x, one can define a decimal representation of x by induction, as follows:

Condensed: the full section is in Wikipedia.

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):

A sequence \((x_n)\) of real numbers is called a Cauchy sequence if for any \(\varepsilon > 0\) there exists an integer \(N\) (possibly depending on \(\varepsilon\)) such that the distance \(|x_n-x_m|\) is less than \(\varepsilon\) for all \(n\) and \(m\) that are both greater than \(N\). This definition, originally provided by Cauchy, formalizes the fact that the \(x_n\) eventually come and remain arbitrarily close to each other.

A sequence \((x_n)\) converges to the limit \(x\) if its elements eventually come and remain arbitrarily close to \(x\), that is, if for any \(\varepsilon > 0\) there exists an integer \(N\) (possibly depending on \(\varepsilon\)) such that the distance \(|x_n-x|\) is less than \(\varepsilon\) 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 (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), 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 (in the real numbers, in contrast, it converges to the positive square root of 2).

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

\(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\)

\(\sum_{n=N}^{M} \frac{x^n}{n!}\)

Condensed: the full section is in Wikipedia.

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 (given any element \(z\), \(z+1\) is larger).

Additionally, an order can be Dedekind-complete, see § Axiomatic approach. 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 (the rationals) 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 (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that \(\mathbb{R}\) 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 (and vice versa), 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 (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) 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 \(\mathbb{R}\). Thus \(\mathbb{R}\) 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 (the surreals) and then selects from it the largest Archimedean subfield.

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers {1, 2, 3, 4, ...} 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 \(\mathfrak c.\) It is strictly greater than the cardinality of the set of all natural numbers, denoted \(\aleph_0\) 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 \(\aleph_0\) and strictly smaller than \(\mathfrak c\) is known as the continuum hypothesis (CH). The axiom system most commonly used in mathematics, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), is insufficient to decide whether CH holds: assuming that ZFC is consistent, CH can be neither proved nor disproved within ZFC, since some models of ZFC satisfy CH, while others violate it.

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 \(|x-y|\). 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 (hence connected and simply connected), 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 \(\mathbb{R}\), although no negative number does. This shows that the order on \(\mathbb{R}\) is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make \(\mathbb{R}\) 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 \([0;1]\) 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 \(\mathbb{R}\). Ordered fields that satisfy the same first-order sentences as \(\mathbb{R}\) are called nonstandard models of \(\mathbb{R}\). This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in \(\mathbb{R}\)), we know that the same statement must also be true of \(\mathbb{R}\).

Condensed: the full section is in Wikipedia.

Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" ("The rules of chords") in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava (c. 750-690 BC), 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 square 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 (c. 390−340 BC) provided a definition of the equality of two irrational proportions in a way that is similar to Dedekind cuts (introduced more than 2,000 years later), except that he did not use any arithmetic operation other than multiplication of a length by a natural number (see Eudoxus of Cnidus). 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 (the latter being made possible by the development of algebra). 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 (c. 850-930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots, and fourth roots). In Europe, such numbers, not commensurable with the numerical unit, were called irrational or surd ("deaf").

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 (1761) gave a flawed proof that π cannot be rational; Legendre (1794) completed the proof and showed that π is not the square root of a rational number. Liouville (1840) showed that neither e nor e can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor (1873) extended and greatly simplified this proof. Hermite (1873) proved that e is transcendental, and Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), Hilbert (1893), Hurwitz, and Gordan.

Condensed: the full section is in Wikipedia.

The developers of calculus used real numbers and limits without defining them rigorously. In his Cours d'Analyse (1821), 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 (Dedekind cuts and sets of the elements of a Cauchy sequence), 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.