In mathematics, 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 realization 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.
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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 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 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 because the precision of such measurements can only be finite.
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 NPcomplete problems, and even #Pcomplete 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 ndimensional space with real coordinates; for example, a value from R^{3} consists of three real numbers and specifies a location in 3dimensional 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.
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]}
In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
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.
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.
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.
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.
Let R denote the set of all real numbers. Then:
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.
The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekindcomplete 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.
For another axiomatization of R, see Tarski's axiomatization of the reals.
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:
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.
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:
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 positive square root of 2.)
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 example, the standard series of the exponential function
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 real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
First, an order can be latticecomplete. It is easy to see that no ordered field can be latticecomplete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.
Additionally, an order can be Dedekindcomplete, 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 Dedekindcompletion 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 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 Dedekindcomplete (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.
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.
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 nonexistence 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.
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, but yield different presentations for the topology  in the order topology as intervals, in the metric topology as epsilonballs. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. 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.
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.
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.
The supremum axiom of the reals refers to subsets of the reals and is therefore a secondorder logical statement. It is not possible to characterize the reals with firstorder logic alone: the LöwenheimSkolem 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 firstorder sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a firstorder 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.
The real numbers can be generalized and extended in several different directions:
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".

A real number is a rational or irrational number. Usually when people say "number" they mean "real number".
Some real numbers are positive. A positive number is bigger than zero. If a positive number is added to another positive number, that number gets bigger. Zero is also a real number. If zero is added to a number, that number does not change.
There are also negative real numbers. If a negative number is added to another number, that number gets smaller.
The real numbers are infinite. There is no smallest or biggest real number. No matter how many real numbers are counted, the last one will never be reached.
The real numbers are dense. That means that if two different real numbers are taken, there will always be another one in between them, no matter how close together the two numbers are.
The real numbers are uncountable. That means that there is no way to put all the real numbers into a sequence. Any sequence of real numbers will miss out a real number, even if the sequence is infinite. This makes the real numbers special. There are more real numbers than there are integers, even though the integers are infinite, too, because the integers are countable.
There are number systems other than the real numbers, such as the complex numbers. Every real number is a complex number, but not every complex number is a real number.
There are different types of real numbers. Depending on the application, only a subset may be needed. Sometimes all the real numbers are not talked about at once. Sometimes only special, smaller sets of them are talked about. These sets have special names. They are:
The number 0 (zero) is special. Sometimes it is taken as part of the subset to be considered, and at other times it is not. It is the Identity element for addition and subtraction. That means that adding or subtracting zero does not change the original number. For multiplication and division, the identity element is 1.
One real number that is not rational is $\backslash sqrt\{2\}$. This number is irrational. If a square is drawn with sides that are one unit long, the length of the line between its opposite corners will be $\backslash sqrt\{2\}$.
The number $\backslash sqrt\{2\}$ is not rational. Here is the proof.
It is not true that $\backslash sqrt\{2\}$ is a rational number. So $\backslash sqrt\{2\}$ is irrational.
