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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a real number is a value of a continuous
quantity Quantity 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 in terms of a unit of measu ...
that can represent a distance along a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
(or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this context was introduced in the 17th century by
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

René Descartes
, who distinguished between real and
imaginary Imaginary may refer to: * Imaginary (sociology), a concept in sociology * The Imaginary (psychoanalysis), a concept by Jacques Lacan * Imaginary number, a concept in mathematics * Imaginary time, a concept in physics * Imagination, a mental faculty ...
roots A root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They most often ...
of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s. The real numbers include all the
rational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
s, such as the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
−5 and the
fraction A fraction (from Latin ', "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-fifths ...
4/3, and all the
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, such as \sqrt (1.41421356..., the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
, an irrational
algebraic number An algebraic number is any complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
). Included within the irrationals are the real
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, such as

(3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
,
mass Mass is the quantity Quantity 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 ...
,
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

energy
,
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
, and many more. The set of real numbers is denoted using the symbol R or \mathbb and is sometimes called "the reals". Real numbers can be thought of as points on an infinitely long
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
called the
number line In elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) include ...

number line
or
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
, such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one. The real line can be thought of as a part of the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, and the real numbers can be thought of as a part of the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique
Dedekind-complete In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
up to Two mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is a ...
an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
, whereas popular constructive definitions of real numbers include declaring them as
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es of
Cauchy sequence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s (of rational numbers),
Dedekind cut In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, or infinite
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
s, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent. The set of all real numbers is
uncountable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, in the sense that while both the set of all
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s and the set of all real numbers are
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s, there can be no
one-to-one function In , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ...
from the real numbers to the natural numbers. In fact, the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the set of all real numbers, denoted by \mathfrak c and called the
cardinality of the continuum In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, ...
, is strictly greater than the cardinality of the set of all natural numbers (denoted \aleph_0, 'aleph-naught'). 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(CH). It is neither provable nor refutable using the axioms of
Zermelo–Fraenkel set theory In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...
including the
axiom of choice In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

axiom of choice
(ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.


History

Simple fractions were used by the
Egyptians Egyptians ( arz, المصريين, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group of people originating from the country of Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a spanning t ...
around 1000 BC; the
Vedic FIle:Atharva-Veda samhita page 471 illustration.png, upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the ''Atharvaveda''. The Vedas (, , ) are a large body of religious texts originating in ancient India. Com ...
"
Shulba Sutras The ''Shulba Sutras'' or ''Śulbasūtras'' (Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch o ...
" ("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 The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...
such as
Manava Manava (c. 750 BC – 690 BC) is an author of the text of ''.'' The Manava is not the oldest (the one by is older), nor is it one of the most important, there being at least three Sulbasutras which are considered more important. Histori ...
, who were aware that the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
s of certain numbers, such as 2 and 61, could not be exactly determined. Around 500 BC, the Greek mathematicians led by
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

Pythagoras
realized the need for irrational numbers, in particular the irrationality of the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
. The
Middle Ages In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...
brought about the acceptance of
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
,
negative numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, integers, and fractional numbers, first by
Indian Indian or Indians refers to people or things related to India, or to the indigenous people of the Americas, or Aboriginal Australians until the 19th century. People South Asia * Indian people, people of Indian nationality, or people who come ...
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 A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
" 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 of ...
" 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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

quadratic equation
s, or as
coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s in an
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
(often in the form of square roots,
cube root In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

cube root
s and fourth roots). In the 16th century,
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
created the basis for modern
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
notation, and insisted that there is no difference between rational and irrational numbers in this regard. In the 17th century,
Descartes
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.
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French language, French; 26 or 28 August 1728 – 25 September 1777) was a Switzerland, Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics ...
(1761) gave the first flawed proof that cannot be rational;
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named a ...
(1794) completed the proof, and showed that is not the square root of a rational number.
Paolo Ruffini Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

Paolo Ruffini
(1799) and
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

Niels Henrik Abel
(1842) both constructed proofs of the
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general algebraic equation, polynomial equations of quintic equation, degree five or higher with arbitrary coef ...
: that the general
quintic In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
(1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of
Galois theory In , Galois theory, originally introduced by , provides a connection between and . This connection, the , allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the ...
.
Joseph 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 L ...

Joseph Liouville
(1840) showed that neither ''e'' nor ''e''2 can be a root of an integer
quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

quadratic equation
, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof.
Charles Hermite Charles Hermite () FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently ...
(1873) first proved that ''e'' is transcendental, and
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
(1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gr ...
(1893), and has finally been made elementary by
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

Adolf Hurwitz
and
Paul Gordan __NOTOC__ Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German The history of the Jews in Germany goes back at least to the year 321, and continued through the Early Middle Ages (5th to 10th centuries CE) and High M ...
. The development of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
in the 18th century used the entire set of real numbers without having defined them rigorously. The first rigorous definition was published by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
in 1871. In 1874, he showed that the set of all real numbers is
uncountably infinite In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

uncountably infinite
, but the set of all algebraic numbers is
countably infinite In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Contrary to widely held beliefs, his first method was not his famous
diagonal argumentDiagonal argument in mathematics may refer to: *Cantor's diagonal argument (the earliest) *Cantor's theorem *Halting problem *Diagonal lemma See also

* Diagonalization (disambiguation) {{mathdab ...
, which he published in 1891. For more, see
Cantor's first uncountability proof Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studie ...
.


Definition

The real number system (\mathbb ; + ; \cdot ; <) can be defined
axiomatically An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
up to an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 sequence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 grassl ...
, meaning that
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

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 (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
≥ 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
non-empty#REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...

non-empty
subset ''S'' of \mathbb with an
upper bound In mathematics, particularly in order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
in \mathbb has a
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
<math>\sqrt</math>
is not rational. These properties imply the
Archimedean property In abstract algebra and analysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics ( ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 Complete metric space, 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; ...) Limit of a sequence, converges to a unique real number—in this case . For details and other constructions of real numbers, see construction of the real numbers.


Properties


Basic properties

* Any non-0 (number), ''zero'' real number is either negative number, ''negative'' or positive number, ''positive''. * The sum and the product of two non-negative real numbers is again a non-negative real number, i.e., they are closed under these operations, and form a ''positive cone'', thereby giving rise to a linear order of the real numbers along a number line. * The real numbers make up an infinite set of numbers that cannot be injective function, injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called countable set, countably infinite. This establishes that in some sense, there are ''more'' real numbers than there are elements in any countable set. * 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 numbers, each set being a proper subset of the next in the sequence. The complement (set theory), complements of all these sets (irrational, transcendental, and non-computable real numbers) in the reals are all uncountably infinite sets. * Real numbers can be used to express measurements of Continuous function, continuous quantities. They may be expressed by
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
s, 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#In mathematical notation, ellipsis (three dots) indicates that there would still be more digits to come. This hints at the fact that we can precisely denote only a few, selected real numbers with finitely many symbols. More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound axiom, least upper bound property. The first says that real numbers comprise a Field (mathematics), field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a non-empty set of real numbers has an Upper and lower bounds, upper bound, then it has a real
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 limit (mathematics), limits. More formally, the reals are completeness (topology), 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'' if for any there exists an integer ''N'' (possibly depending on ε) such that the distance is less than ε for all ''n'' and ''m'' that are both greater than ''N''. This definition, originally provided by Augustin Louis Cauchy, 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 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

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_^ \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 complete lattice, 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 this case, the additive group of the field) defines a uniform space, uniform structure, and uniform structures have a notion of completeness (topology), completeness; the description in #Completeness, § 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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gr ...
, 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 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.


Advanced properties

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the continuum, cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of \mathbb. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the
continuum hypothesis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The continuum hypothesis can neither be proved nor be disproved; it is logical independence, independent from the axiomatic set theory, axioms of set theory. As a topological space, the real numbers are separable space, 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 (hence connected space, connected and simply connected), separable space, separable and complete space, complete metric space of Hausdorff dimension 1. The real numbers are local compactness, locally compact but not compact space, compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable total order, order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 (mathematics), 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. 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 grassl ...
\mathbb of real numbers is an extension field of the field \mathbb of rational numbers, and \mathbb can therefore be seen as a vector space over \mathbb.
Zermelo–Fraenkel set theory In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...
with the
axiom of choice In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

axiom of choice
guarantees the existence of a basis (linear algebra), 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 \mathbb with the property that every
non-empty#REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...

non-empty
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 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 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 number, computable or uncomputable; either Algorithmically random sequence, algorithmically random or not; and either Algorithmically random sequence#Stronger than Martin-Löf randomness, arithmetically random or not.


Applications and connections to other areas


Real numbers and 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 Constructivism (mathematics), 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 (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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
posits that the cardinality of the set of the real numbers is \aleph_1; i.e. the smallest infinite cardinal number after \aleph_0, the cardinality of the integers. Paul Cohen (mathematician), 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.


In 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.


In computation

With some Symbolic computation, exceptions, most calculators do not operate on real numbers. Instead, they work with finite-precision approximations called floating-point numbers. In fact, most Computational science, scientific computation uses floating-point arithmetic. Real numbers satisfy the Field (mathematics)#Definition and illustration, usual rules of arithmetic, but Floating-point arithmetic#Accuracy problems, floating-point numbers do not. Computers cannot directly store arbitrary real numbers with infinitely many digits. The achievable precision is limited by the number of bits allocated to store a number, whether as floating-point numbers or arbitrary-precision arithmetic, arbitrary-precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas for them (such as \sqrt, \arcsin (2/23), or \textstyle\int_0^1 x^x \,dx) rather than their rational or decimal approximation. It is not in general possible to determine whether two such expressions are equal (the constant problem). A real number is called ''computable number, computable'' if there exists an algorithm that yields its digits. Because there are only countably infinite, 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 constructivism (mathematics), constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.


"Reals" in set theory

In set theory, specifically descriptive set theory, the Baire space (set theory), 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 the symbol R, or, alternatively, \mathbb, the R, letter "R" in blackboard bold (encoded in Unicode as ), to represent the set of all real numbers. 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 grassl ...
, 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
"" ("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 A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
, 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 Cartesian product of copies of \mathbb, which is an -dimensional vector space over the field of the real numbers; this vector space may be identified to the -dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. For example, a value from \mathbb^3 consists of a tuple of three real numbers and specifies the coordinates of a Point (geometry), point in 3‑dimensional space. 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 (mathematics), matrix'', ''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 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. * The real projective line 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 non-zero element by zero. It has cyclic order described by a separation relation. * The Long line (topology), 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 numbers; both of them contain infinitesimal and infinitely large numbers and are therefore non-Archimedean ordered fields. * Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrix (math), matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvector, eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.


See also

* Completeness of the real numbers * Continued fraction * Definable real number, Definable real numbers * Positive real numbers * Real analysis


Notes


References


Citations


Sources

* Georg Cantor, Cantor, Georg (1874). "". ', volume 77, pp. 258–62. * Solomon Feferman, Feferman, Solomon (1989). ''The Number Systems: Foundations of Algebra and Analysis'', AMS Chelsea, . * Katz, Robert (1964). ''Axiomatic Analysis'', D.C. Heath and Company. * Edmund Landau, Landau, Edmund (2001). ''Foundations of Analysis''. American Mathematical Society,. * Howie, John M. ''Real Analysis''. Springer, 2005, . * .


External links

* {{Authority control Real numbers, Real algebraic geometry Elementary mathematics