real+numbers

In mathematics, the **real numbers** may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line. These descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realisation that a better definition was needed — 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. These definitions are all described in detail below. The term "real number" is a retronym coined in response to "imaginary number".   Real numbers can be thought of as points on an infinitely long number line. A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to 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 324.823122147… The ellipsis (three dots) indicate that there would still be more digits to come. 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 order 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 resulting field is algebraically closed. In the physical sciences, most of the physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. Note importantly, however, that all actual measurements of physical quantities yield rational numbers due to the (unavoidably) finite precision involved in the measurement. Computers cannot directly operate on real numbers, but only on a finite subset of rational numbers, limited by the number of bits used to store them. However, computer algebra systems are able to treat some irrational numbers exactly by storing their algebraic description (such as "sqrt(2)") rather than their rational approximation. 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, almost all 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. If computers could use unlimited precision real numbers (real computation), then one could solve NP-complete problems, and even #P-complete problems in polynomial time, answering affirmatively the P = NP problem. Unlimited precision real numbers in the physical universe are prohibited by the holographic principle and the Bekenstein bound.[1] 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 **R**//n// refers to an //n//-dimensional space with real coordinates; for example, a value from **R**3 consists of three real numbers and specifies a location in 3-dimensional space. 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"). Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("rule of chords" in, ca. 600 BC, include what may be the first 'use' of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 750–690 BC), who was aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined. [2] Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The Middle Ages saw the acceptance of zero, negative, integral and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects,[3] which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.[4] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.[5] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 //e//2 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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 the real numbers. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Let **R** denote the set of all real numbers. Then: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields **R**1 and **R**2, there exists a unique field isomorphism from **R**1 to **R**2, allowing us to think of them as essentially the same mathematical object. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">For another axiomatization of **R**, see Tarski's axiomatization of the reals. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">A sequence (//xn//) 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 |//xn// − //xm//| 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 //xn// eventually come and remain arbitrarily close to each other. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">A sequence (//xn//) //converges to the limit// //x// if for any ε > 0 there exists an integer //N// (possibly depending on ε) such that the distance |//xn// − //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//. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 : <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">That is, the reals are complete. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 positive square root of 2.) <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">For example, the standard series of the exponential function <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Converges to a real number because for every //x// the sums <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 (i.e., the power set) 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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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 real 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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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**. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The real numbers can be generalized and extended in several different directions: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">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".
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 24pt;">Real number **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Basic properties **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Basic properties **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Uses **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">History **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Definition **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Construction from the rational numbers **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Axiomatic approach **
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">The set **R** is a field, meaning that addition and multiplication are defined and have the usual properties.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">The field **R** is ordered, meaning that there is a total order ≥ such that, for all real numbers //x//, //y// and //z//:
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">if //x// ≥ //y// then //x// + //z// ≥ //y// + //z//;
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">If //x// ≥ 0 and //y// ≥ 0 then //xy// ≥ 0.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">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**.
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Properties **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Completeness **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Every Cauchy sequence of real numbers is convergent. **<span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">"The complete ordered field" **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Advanced properties **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Generalizations and extensions **
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">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.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">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.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">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.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">The long real line pastes together <span style="font-family: "Cambria Math","serif"; font-size: 12pt;">ℵ <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">* + <span style="font-family: "Cambria Math","serif"; font-size: 12pt;">ℵ <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> copies of the real line plus a single point (here <span style="font-family: "Cambria Math","serif"; font-size: 12pt;">ℵ <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">* denotes the reversed ordering of <span style="font-family: "Cambria Math","serif"; font-size: 12pt;">ℵ <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">) 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 <span style="font-family: "Cambria Math","serif"; font-size: 12pt;">ℵ <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> 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.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">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.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">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.
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">"Reals" in set theory **