TheInfoList

OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a real number is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
that can be used to measure a ''continuous'' one-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a un ...
such as a
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...
, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite
decimal expansion A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, i ...
. The real numbers are fundamental in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...
(and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or $\mathbb$ and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from
imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . ...
s (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. The rest of the real numbers are called irrational numbers, and include algebraic numbers (such as the square root ) and transcendental numbers (such as ). Real numbers can be thought of as all points on an infinitely long
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
called the number line or
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
, where the points corresponding to integers () are equally spaced. Conversely,
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
is the association of points on lines (especially axis lines) to real numbers such that geometric displacements are proportional to differences between corresponding numbers. The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an ...
s and real-valued
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s. A current axiomatic definition is that real numbers form the unique ( up to an isomorphism)
Dedekind-complete 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 has the least-upper-bound property if e ...
ordered field. Other common definitions of real numbers include
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of Cauchy sequences (of rational numbers),
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the ...
s, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

# Properties

## Basic properties

* The real numbers include ''zero'' (), the '' additive identity'': adding to any real number leaves that number unchanged: . * Every real number has an ''
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
'' satisfying . * The real numbers include a ''unit'' (), the ''
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
'': multiplying by any real number leaves that number unchanged: . * Every nonzero real number has a '' multiplicative inverse'' satisfying . * Given any two real numbers and , the results of addition (),
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
(), and multiplication () are also real numbers, as is the result of
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication * Division algorithm, a method for computing the result of mathematical division Military * Division (military), a formation typically consistin ...
() if is not zero. Thus the real numbers are
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under elementary arithmetic operations. * The real numbers form a ''
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
''. * The real numbers are ''
linearly ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
''. For any distinct real numbers and , either or . If and then . (See also
inequality (mathematics) In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different ...
.) * Any nonzero real number is either ''negative'' () or ''positive'' (). * The real numbers are an '' ordered field'' because the order is compatible with addition and multiplication: if then ; if and then . Because the square of any real number is non-negative, and the sum and product of non-negative real numbers is itself non-negative, non-negative real numbers are a ''positive cone'' of . * The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called '' countably infinite''. This establishes that in some sense, there are ''more'' real numbers than there are elements in any countable set. * Any
nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
bounded ''open interval'' (the set of all real numbers between two specified endpoints) can be mapped bijectively by an affine function (
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
and translation of the number line) to any other such interval. Every nonempty open interval contains uncountably infinitely many real numbers. * The real numbers are ''unbounded''. There is no greatest or least real number; the real numbers extend infinitely in both positive and negative directions. * There is a hierarchy of countably infinite subsets of the real numbers, e.g., the integers, the rational numbers, the algebraic numbers and the
computable number In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...
s, each set being a proper subset of the next in the sequence. The complements of each of these sets in the reals (irrational, transcendental, and non-computable real numbers) is uncountably infinite. * Real numbers can be used to express measurements of continuous quantities. They may be expressed by decimal representations, most of them having an infinite sequence of digits to the right of the decimal point; these are often represented like 324.823122147..., where the
ellipsis The ellipsis (, also known informally as dot dot dot) is a series of dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. The plural is ellipses. The term origin ...
indicates that infinitely many digits have been omitted. More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a nonempty set of real numbers has an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an ele ...
, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.

## Completeness

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 space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
s, which is a different sense than the Dedekind completeness of the order in the previous section): A
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
(''x''''n'') of real numbers is called a '' Cauchy sequence'' if for any there exists an integer ''N'' (possibly depending on ε) such that the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...
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 ''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 there exists an integer ''N'' (possibly depending on ε) 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...
, and, more generally
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied i ...
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 :$e^x = \sum_^ \frac$ converges to a real number for every ''x'', because the sums :$\sum_^ \frac$ can be made arbitrarily small (independently of ''M'') by choosing ''N'' sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that $e^x$ 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 (given any element ''z'', is larger). 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 (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 abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...
(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 $\mathbb$ 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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
, 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 $\mathbb$. Thus $\mathbb$ 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 number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals s ...
s, since that construction starts with a proper class that contains every ordered field (the surreals) 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 can be no
one-to-one function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
from the real numbers to the natural numbers. The
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalize ...
of the set of all real numbers is denoted by $\mathfrak c.$ and called the cardinality of the continuum. It is strictly greater than the cardinality of the set of all natural numbers (denoted $\aleph_0$ and called 'aleph-naught'), and equals the cardinality of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is p ...
of the set of the natural numbers. The statement that there is no subset of the reals with cardinality strictly greater than $\aleph_0$ and strictly smaller than $\mathfrak c$ is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(ZFC)—the standard foundation of modern mathematics. In fact, 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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. 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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
), separable and complete metric space of
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
1. The real numbers are
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
but not
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
. 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 $\mathbb$, although no negative number does. This shows that the order on $\mathbb$ is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make $\mathbb$ 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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides w ...
, which is the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, thou ...
on their structure as a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
normalized such that the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analy ...
;1has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g.
Vitali set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many ...
s. 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 $\mathbb$. Ordered fields that satisfy the same first-order sentences as $\mathbb$ are called nonstandard models of $\mathbb$. 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 $\mathbb$), we know that the same statement must also be true of $\mathbb$. The
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
$\mathbb$ of real numbers is an
extension field In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the field $\mathbb$ of rational numbers, and $\mathbb$ can therefore be seen as a vector space over $\mathbb$. Zermelo–Fraenkel set theory with the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
guarantees the existence of a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...
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 In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the ord ...
implies that the real numbers can be
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well- ...
ed if the axiom of choice is assumed: there exists a total order on $\mathbb$ with the property that every nonempty subset of $\mathbb$ has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
does not contain 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 The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the construct ...
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 Simple fractions were used by the Egyptians around 1000 BC; the Vedic "
Shulba Sutras The ''Shulva Sutras'' or ''Śulbasūtras'' ( Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction. Purpose and origins Th ...
" ("The rules of chords") in include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early
Indian mathematicians chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
such as
Manava Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of ''Sulba Sutras.'' The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbasu ...
, 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. The Middle Ages brought about the acceptance of
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...
, negative numbers, integers, and fractional numbers, first by
Indian Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asi ...
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 A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
" and "
magnitude Magnitude may refer to: Mathematics * Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order o ...
" 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 equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quad ...
s, or as coefficients in an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
(often in the form of square roots,
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
s 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 Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated va ...
created the basis for modern
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic nume ...
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" ones. In the 18th and 19th centuries, there was much work on irrational and transcendental numbers.
Lambert Lambert may refer to People *Lambert (name), a given name and surname * Lambert, Bishop of Ostia (c. 1036–1130), became Pope Honorius II * Lambert, Margrave of Tuscany (fl. 929–931), also count and duke of Lucca *Lambert (pianist), stage-name ...
(1761) gave a flawed proof that cannot be rational; Legendre (1794) completed the proof and showed that is not the square root of a rational number.
Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
(1840) showed that neither nor can be a root of an integer
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quad ...
, and then established the existence of transcendental numbers; Cantor (1873) extended and greatly simplified this proof. Hermite (1873) proved that is transcendental, and Lindemann (1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885),
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
(1893), Hurwitz, and Gordan. The developers of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...
used real numbers without having defined them rigorously. The first rigorous definition was published by Cantor in 1871. In 1874, he 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 Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is u ...
was different from his famous
diagonal argument A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: * Cantor's diagonal argument (the earliest) *Cantor's theorem *Russell's paradox *Diagonal lemma ** Gödel's first incompleteness theorem **Tarski ...
published in 1891.

# Formal definitions

The real number system $\left(\mathbb ; + ; \cdot ; <\right)$ can be defined axiomatically up to an isomorphism, which is described hereafter. 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 (say of Hilbert or of Tarski), 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 $\mathbb$ denote the set of all real numbers, then: * The set $\mathbb$ is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
, meaning that
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
and multiplication are defined and have the usual properties. * The field $\mathbb$ is ordered, meaning that there is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
≥ 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 $\mathbb$ with an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an ele ...
in $\mathbb$ has a least upper bound (a.k.a., supremum) in $\mathbb$. The last property is what differentiates the real numbers from the rational numbers (and from other more exotic ordered fields). For example, $\$ has a rational upper bound (e.g., 1.42), but no ''least'' rational upper bound, because $\sqrt$ is not rational. These properties imply the Archimedean property (which is not implied by other definitions of completeness), 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 $\mathbb_1$ and $\mathbb_2$, there exists a unique field isomorphism from $\mathbb_1$ to $\mathbb$. This uniqueness allows us to think of them as essentially the same mathematical object. For another axiomatization of $\mathbb$, 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 (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number—in this case . For details and other constructions of real numbers, see
construction of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complet ...
.

# 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 Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classica ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, quantum mechanics, general relativity and the
standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. ...
are described using mathematical structures, typically smooth manifolds or
Hilbert spaces In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturall ...
, that are based on the real numbers, although actual measurements of physical quantities are of finite
accuracy and precision Accuracy and precision are two measures of '' observational error''. ''Accuracy'' is how close a given set of measurements ( observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each ot ...
. 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 Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in co ...
and in constructive mathematics. The hyperreal numbers as developed by Edwin Hewitt,
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincor ...
and others extend the set of the real numbers by introducing
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
and infinite numbers, allowing for building
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
in a way closer to the original intuitions of Leibniz,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries i ...
, Cauchy and others.
Edward Nelson Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematic ...
's
internal set theory Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, ...
enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory). The continuum hypothesis posits that the cardinality of the set of the real numbers is $\aleph_1$; i.e. the smallest infinite
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
after $\aleph_0$, 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

Electronic calculators 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 Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...
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 Accuracy and precision are two measures of '' observational error''. ''Accuracy'' is how close a given set of measurements ( observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each ot ...
of numerical
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s implemented with approximate arithmetic. Alternately,
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...
s can operate on irrational quantities exactly by manipulating symbolic formulas for them (such as $\sqrt,$ $\arctan 5,$ or $\int_0^1 x^x \,dx$) 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 (the
constant problem In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero. The problem This problem is also referred to as the identity problem or the method of zero estimates. It has no formal statement as su ...
); and arithmetic operations can cause exponential explosion in the size of representation of a single number (for instance, squaring a rational number roughly doubles the number of digits in its numerator and denominator, and squaring a polynomial roughly doubles its number of terms), overwhelming finite computer storage. A real number is called '' computable'' if there exists an algorithm that yields its digits. Because there are only
countably In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
many algorithms, but an uncountable number of reals,
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
real numbers fail to be computable. Moreover, the equality of two computable numbers is an
undecidable problem In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is ...
. Some constructivists accept the existence of only those reals that are computable. The set of
definable number 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, \sqrt, c ...
s is broader, but still only countable.

## Set theory

In set theory, specifically
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ...
, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

# Vocabulary and notation

Mathematicians use mainly the symbol R to represent the set of all real numbers. Alternatively, it may be used $\mathbb$, the letter "R" in
blackboard bold Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pr ...
, which may be encoded in Unicode (and HTML) as . As this set is naturally endowed with the structure of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
, 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 $\mathbb^+$ and $\mathbb^-$, respectively; $\mathbb_+$ and $\mathbb_-$ are also used. École Normale Supérieure of
Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...

"" ("Real numbers")
, p. 6
The non-negative real numbers can be noted $\mathbb_$ but one often sees this set noted $\mathbb^+ \cup \.$ In French mathematics, the ''positive real numbers'' and ''negative real numbers'' commonly include
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...
, and these sets are noted respectively $\mathbb$ and $\mathbb_.$ In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted $\mathbb_^*$ and $\mathbb_^*.$ The notation $\mathbb^n$ refers to the set of the -tuples of elements of $\R$ ( real coordinate space), which can be identified to the Cartesian product of copies of $\mathbb.$ It is an -
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al vector space over the field of the real numbers, often called the coordinate space of dimension ; this space may be identified to the -
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al 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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
. In mathematics, ''real'' is used as an adjective, meaning that the underlying field is the field of the real numbers (or ''the real field''). For example, ''real
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
'', ''real polynomial'' and ''real Lie algebra''. The word is also used as a noun, meaning a real number (as in "the set of all reals").

# Generalizations and extensions

The real numbers can be generalized and extended in several different directions: * The complex numbers contain solutions to all polynomial equations and hence are an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
unlike the real numbers. However, the complex numbers are not an ordered field. * The affinely extended real number system adds two elements and . It is a compact space. It is no longer a field, or even an additive group, but it still has a total order; moreover, it is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
. * The
real projective line In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...
adds only one value . It is also a compact space. Again, it is no longer a field, or even an additive group. However, it allows division of a nonzero element by zero. It has cyclic order described by a separation relation. * The long real line pastes together copies of the real line plus a single point (here denotes the reversed ordering of ) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group. * Ordered fields extending the reals are the hyperreal numbers and the
surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals s ...
s; both of them contain
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
and infinitely large numbers and are therefore
non-Archimedean ordered field In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational function ...
s. *
Self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
s on a Hilbert space (for example, self-adjoint square complex
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
are real and they form a real
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

*
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 li ...
*
Continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
* Definable real numbers *
Positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used ...
* Real analysis

# References

## Sources

* * * * * * * * Vol. 2, 1989. Vol. 3, 1990. * * Translated from the Germa
''Grundlagen der Analysis''
1930. * *