In

^{2} can be a root of an integer

_{''n''}) of real numbers is called a ''_{''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 ...

in $\backslash mathbb$, although no negative number does. This shows that the order on $\backslash mathbb$ is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make $\backslash mathbb$ 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 $\backslash mathbb$. Ordered fields that satisfy the same first-order sentences as $\backslash mathbb$ are called nonstandard models of $\backslash mathbb$. 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 $\backslash mathbb$), we know that the same statement must also be true of $\backslash mathbb$.
The

"" ("Real numbers")

, p. 6 The non-negative real numbers can be noted $\backslash mathbb\_$ but one often sees this set noted $\backslash mathbb^+\; \backslash cup\; \backslash .$ In French mathematics, the ''positive real numbers'' and ''negative real numbers'' commonly include

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 their changes (cal ...

, 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 ...

(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 ...

, 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 ...

s. The real numbers include all the rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

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 $\backslash sqrt$ (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 ...

, 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 ...

, 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 ...

, and many more. The set of real numbers is denoted using the symbol R or $\backslash mathbb$ 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 ...

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 ...

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 ...

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 ...

, 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 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 t ...

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 $\backslash mathfrak\; c$ 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 $\backslash aleph\_0$, 'aleph-naught').
The statement that there is no subset of the reals with cardinality strictly greater than $\backslash aleph\_0$ and strictly smaller than $\backslash mathfrak\; c$ 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 ...

(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 theEgyptians
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
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. Composed in Vedic Sanskrit, the texts constitute the ol ...

"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 ...

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 ...

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 ...

, 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 ...

" 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. ...

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 ...

(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). ...

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, 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 ...

(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 ...

(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 ...

(1840) showed that neither ''e'' nor ''e''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. ...

, 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 ...

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 ...

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 ...

, 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 $(\backslash mathbb\; ;\; +\; ;\; \backslash cdot\; ;\; <)$ can be definedaxiomatically
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 ...

, 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 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 t ...

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 ...

.
Axiomatic approach

Let $\backslash mathbb$ denote the set of all real numbers, then: * The set $\backslash mathbb$ is afield
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 ...

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 ...

are defined and have the usual properties.
* The field $\backslash mathbb$ 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 ...

subset ''S'' of $\backslash mathbb$ 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 $\backslash mathbb$ 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 ...

(a.k.a., supremum) in $\backslash mathbb$.
The last property is what differentiates the real numbers from the rational numbers (and from other more exotic ordered fields). For example, $\backslash $ has a rational upper bound (e.g., 1.42), but no ''least'' rational upper bound, because 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 ( ...

(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 $\backslash mathbb\_1$ and $\backslash mathbb\_2$, 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 ...

from $\backslash mathbb\_1$ to $\backslash mathbb$. This uniqueness allows us to think of them as essentially the same mathematical object.
For another axiomatization of $\backslash mathbb$, 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, seeconstruction of the real numbers
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 th ...

.
Properties

Basic properties

* Any non- real number is either ''negative'' or ''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 alinear order
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 ...

of the real numbers along a number line.
* The real numbers make up an infinite set of numbers that cannot be 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
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 ...

. 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 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, each set being a proper subset of the next in the sequence. The 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 measurement
Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ...

s of continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

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
A decimal separator is a symbol used to separate the integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...

; these are often represented like 324.823122147..., where the ellipsis
The ellipsis , , or (as a single glyph
The term glyph is used in typography
File:metal movable type.jpg, 225px, Movable type being assembled on a composing stick using pieces that are stored in the type case shown below it
Typography ...

(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
In mathematics
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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 ...

. 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''Cauchy sequence
In mathematics
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'' 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''square root
In mathematics
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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
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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 ...

, 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
:$e^x\; =\; \backslash sum\_^\; \backslash frac$
converges to a real number for every ''x'', because the sums
:$\backslash sum\_^\; \backslash frac$
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 $\backslash mathbb$ 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 byDavid 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 $\backslash mathbb$. Thus $\backslash mathbb$ 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 $\backslash mathbb$. 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 thecontinuum 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 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 ...

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

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 $\backslash mathbb$ 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 ...

subset of $\backslash mathbb$ 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). Thecontinuum 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 $\backslash aleph\_1$; i.e. the smallest infinite cardinal number after $\backslash aleph\_0$, 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 $\backslash sqrt,$ $\backslash arcsin\; (2/23),$ or $\backslash textstyle\backslash int\_0^1\; x^x\; \backslash ,dx$) 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, $\backslash mathbb$, 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 afield
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 $\backslash mathbb^+$ and $\backslash mathbb^-$, respectively; $\backslash mathbb\_+$ and $\backslash mathbb\_-$ are also used.École Normale Supérieure of Paris"" ("Real numbers")

, p. 6 The non-negative real numbers can be noted $\backslash mathbb\_$ but one often sees this set noted $\backslash mathbb^+\; \backslash cup\; \backslash .$ 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 ...

, and these sets are noted respectively $\backslash mathbb$ and $\backslash mathbb.$ In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted $\backslash mathbb\_*$ and $\backslash mathbb\_*.$
The notation $\backslash mathbb^n$ refers to the Cartesian product of copies of $\backslash mathbb$, 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 $\backslash mathbb^3$ 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 analysisNotes

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