In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339.... The real numbers include rational numbers, such as 42 and −23/129, and irrational numbers, such as π and the square root of 2, and can be represented as points along an infinitely long number line. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... It has been suggested that this article or section be merged with decimal. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. ...

A more rigorous definition of the real numbers was one of the most important developments of 19th century mathematics. Popular definitions in use today include equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, a more sophisticated version of "decimal representation", and an axiomatic definition of the real numbers as the unique complete Archimedean ordered field. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... The plot of a Cauchy sequence shown in blue, as versus If the space containing the sequence is complete, the ultimate destination of this sequence, that is, the limit, exists. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

The name real numbers arose to distinguish them from what were then called imaginary numbers (and now complex numbers). In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Basic properties

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... A negative number is a number that is less than zero, such as −3. ... A negative number is a number that is less than zero, such as −3. ... Zero redirects here. ...

Real numbers measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823211247... The ellipsis (three dots) indicate that there would still be more digits to come. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... It has been suggested that this article or section be merged with decimal. ... This article is about the punctuation symbol. ...

More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real coefficients has a real root, and that if you add the square root of −1 to the real numbers, obtaining the complex numbers, the result is algebraically closed. In mathematics, an ordered field is a field together with an ordering of its elements. ... The least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ...

Uses

Measurements in the physical sciences are almost always conceived of as approximations to real numbers. While the numbers used for this purpose are generally decimal fractions representing rational numbers, writing them in decimal terms suggests they are an approximation to a theoretical underlying real number. == Headline text ==cant there be some kind of picture somewhere so i can see by picture???? Physical science is a encompassing term for the branches of natural science, and science, that study non-living systems, in contrast to the biological sciences. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to...

A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ... Flowcharts are often used to graphically represent algorithms. ... In mathematics the term countable set is used to describe the size of a set, e. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). ...

Computers can only approximate most real numbers. Most commonly, they can represent a certain subset of the rationals exactly, via either floating point numbers or fixed-point numbers, and these rationals are used as an approximation for other nearby real values. Arbitrary-precision arithmetic is a method to represent arbitrary rational numbers, limited only by available memory, but more commonly one uses a fixed number of bits of precision determined by the size of the processor registers. In addition to these rational values, computer algebra systems are able to treat many (countable) irrational numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their rational approximation. Note that a few programming languages use "real" to describe their main numeric data type, such as AppleScript. This article is about the machine. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... This article is about a form of limited-precision arithmetic in computing. ... On a computer, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique that allows computer programs to perform calculations on integers or rational numbers (including floating-point numbers) with an arbitrary number of digits of precision, typically limited only by the available memory of the host system. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article is about the unit of information. ... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to commonly used values—typically, the values being in the midst of a calculation at a given point in time. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... In programming languages a data type defines a set of values and the allowable operations on those values[1]. For example, in the Java programming language, the int type represents the set of 32-bit integers ranging in value from -2,147,483,648 to 2,147,483,647, and... AppleScript is a scripting language devised by Apple, Inc. ...

Mathematicians use the symbol R (or alternatively, , the letter "R" in blackboard bold, Unicode ℝ) to represent the set of all real numbers. The notation Rⁿ refers to an n-dimensional space with real coordinates; for example, a value from R³ consists of three real numbers and specifies a location in 3-dimensional space. Look up R, r in Wiktionary, the free dictionary. ... An example of blackboard bold letters. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ... 2-dimensional renderings (ie. ...

In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra. As a substantive, the term is used almost strictly in reference to the real numbers, themselves (e.g., The "set of all reals"). In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

History

Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("rule of chords" in Sanskrit), ca. 600 BC, include what may be the first 'use' of irrational numbers^{[citation needed]}. In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... This article or section does not cite its references or sources. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of two. Pythagoras of Samos (Greek: ; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician[1] and founder of the religious movement called Pythagoreanism. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ...

In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Lambert (1761) gave the first flawed proof that π cannot be rational, Legendre (1794) completed the proof, and showed that π is not the square root of a rational number. Ruffini (1799) and Abel (1842) both constructed proofs of Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 – September 25, 1777), was a mathematician, physicist and astronomer. ... Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ... Paolo Ruffini (Valentano, 1765 – Modena, 1822) was an Italian mathematician and philosopher. ... Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Nedstrand, near Finnøy where his father acted as rector. ... The Abel–Ruffini theorem (also known as Abels Impossibility Theorem) states that there is no general solution in radicals to polynomial equations of degree five or higher. ... Graph of a polynomial of degree 5, with 4 critical points. ...

Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e² can be a root of an integer quadratic equation, and then established existence of transcendental numbers, the proof being subsequently displaced by Georg Cantor (1873). Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Hurwitz and Paul Albert Gordan. Galois at the age of fifteen from the pencil of a classmate. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Charles Hermite (pronounced in IPA, ) (December 24, 1822 – January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i. ... | name = David Hilbert | image = Hilbert1912. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... Paul Albert Gordan (April 27, 1837 – December 21, 1912) was a German mathematician. ...

The development of calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. In 1874 he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his method was not his famous diagonal argument, which he published in 1891. For other uses, see Calculus (disambiguation). ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... In mathematics, an uncountable set is a set which is not countable. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics the term countable set is used to describe the size of a set, e. ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ...

Definition

Main article: Construction of real numbers

In mathematics, there are a number of ways of defining the real number system as an ordered field. ...

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. For details and other constructions of real numbers, see construction of real numbers. In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... In mathematics, there are a number of ways of defining the real number system as an ordered field. ...

Axiomatic approach

Let R denote the set of all real numbers. Then: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

• The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
  • if x ≥ y then x + z ≥ y + z;
  • if x ≥ 0 and y ≥ 0 then xy ≥ 0.
• The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.

The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is... The empty set is the set containing no elements. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

For another axiomatization of R, see Tarski's axiomatization of the reals. In 1936, Alfred Tarski axiomatized the real numbers and their arithmetic by means of only 8 axioms. ...

Properties

Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ...

A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε for all n and m that are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other. For other senses of this word, see sequence (disambiguation). ... The plot of a Cauchy sequence shown in blue, as versus If the space containing the sequence is complete, the ultimate destination of this sequence, that is, the limit, exists. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x.

It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true:

Every Cauchy sequence of real numbers is convergent.

That is, the reals are complete.

Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For other uses, see Calculus (disambiguation). ...

For example, the standard series of the exponential function The exponential function is one of the most important functions in mathematics. ...

converges to a real number because for every x the sums

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if the limit is not known in advance.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...

Additionally, an order can be Dedekind-complete, as defined in the section Axioms. 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. In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is...

These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above 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 characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind-complete (and vice versa, of course), this justifies 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. In abstract algebra, an ordered group is a group G equipped with a partial order ≤ which is translation-invariant; in other words, ≤ has the property that, for all a, b, and g in G, if a ≤ b then ag ≤ bg and ga ≤ gb. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... An archimedean field is an ordered field with the archimedean property. ...

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield. | name = David Hilbert | image = Hilbert1912. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ...

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 reals equals that of the set of subsets of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly bigger than the cardinality of N. Since only a countable set of real numbers can be algebraic, 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. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. In mathematics, an uncountable set is a set which is not countable. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In set theory, an infinite set is a set that is not a finite set. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematical logic, a statement S is independent of a theory T if it is impossible to prove S from T and it is impossible to prove not S from T. Many interesting statements in set theory are independent of ZF. It is possible for the statement S is independent... In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. ...

The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected and simply connected), separable metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... Separable can refer to: Separable space in topology Separable sigma algebra in measure theory Separable differential equations This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... 2-dimensional renderings (ie. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... This word should not be confused with homomorphism. ...

Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true: There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and any polynomial... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ...

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1. In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

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 R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R. First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... In mathematical logic, the classic Löwenheim-Skolem theorem states that any infinite model M has a countably infinite submodel N that satisfies exactly the same set of first-order sentences that M satisfies. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural...

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, not even an additive group; 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, not even an additive group. However, it allows division of a non-zero element by zero. It is not ordered anymore.
• The long real line pastes together ℵ₁* + ℵ₁ copies of the real line plus a single point (here ℵ₁* denotes the reversed ordering of ℵ₁) 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 ℵ₁ 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 thus are not Archimedean.
• Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their 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.

In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ... In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced plus infinity and minus infinity). These new elements are not real numbers. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... In mathematics, the projective line is a fundamental example of an algebraic curve. ... In topology, the long line is a topological space analogous to the real line, but much longer. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... For the square matrix section, see square matrix. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In functional analysis, a normal operator on a Hilbert space is a continuous linear operator that commutes with its hermitian adjoint : The main importance of this concept is that the spectral theorem applies to normal operators. ...

"Reals" in set theory

In set theory, specifically descriptive set theory the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals". Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, descriptive set theory is the study of certain classes of well-behaved sets of real numbers, e. ... In mathematics, the Baire space is the set of all infinite sequences of natural numbers. ...

References

• Georg Cantor, 1874, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik, volume 77, pages 258-262.

See also

It has been suggested that this article or section be merged with decimal. ... Look up completeness in Wiktionary, the free dictionary. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true: There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and any polynomial... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... For other uses, see Calculus (disambiguation). ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is... In 1936, Alfred Tarski axiomatized the real numbers and their arithmetic by means of only 8 axioms. ...

External links

• The real numbers: Pythagoras to Stevin
• The real numbers: Stevin to Hilbert
• The real numbers: Attempts to understand