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In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences.[a] Every real number can be almost uniquely represented by an infinite decimal expansion.[b][1]

Conversely, analytic geometry is the association of points on lines (especially axis lines) to real numbers such that geometric displacements are proportional to differences between corresponding numbers.

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The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field.[d] Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence.

The real numbers form an ordered field. Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations, addition and multiplication, and a total order that have the following properties.

These identifications make the set Q {\displaystyle \mathbb {Q} } of the rational numbers an ordered subfield of the real numbers R . {\displaystyle \mathbb {R} .} The Dedekind completeness described below implies that some real numbers, such as 2 , {\displaystyle {\sqrt {2}},} are not rational numbers; they are called irrational numbers.

The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties (axioms). So, the identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with 1 taken as the successor function.

Formally, one has a injective homomorphism of ordered monoids from the natural numbers N {\displaystyle \mathbb {N} } to the integers Z , {\displaystyle \mathbb {Z} ,} an injective homomorphism of ordered rings from Z {\displaystyle \mathbb {Z} } to the rational numbers Q , {\displaystyle \mathbb {Q} ,} and an injective homomorphism of ordered fields from Q {\displaystyle \mathbb {Q} } to the real numbers R . {\displaystyle \mathbb {R} .} The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write

The last two properties are summarized by saying that the real numbers form a real closed field. This implies the real version of the fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.

A key property of real numbers is their decimal representation. A decimal representation consists of a nonnegative integer k and an infinite sequence of decimal digits (nonnegative integers less than 10)

One can use the defining properties of the real numbers to show that a is the least upper bound of the D n . {\displaystyle D_{n}.} So, the resulting sequence of digits is called a decimal representation of a.

A main reason for using real numbers is so that many sequences have limits. More formally, 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):

The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).

The completeness property of the reals is the basis on which calculus, and more generally mathematical analysis, are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.

Additionally, an order can be Dedekind-complete, see Axiomatic approach. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; the description in Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that R {\displaystyle \mathbb {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". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, 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 {\displaystyle \mathbb {R} } . Thus R {\displaystyle \mathbb {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.

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

Every nonnegative real number has a square root in R {\displaystyle \mathbb {R} } , although no negative number does. This shows that the order on R {\displaystyle \mathbb {R} } is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R {\displaystyle \mathbb {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 normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.

The developers of calculus used real numbers without having defined them rigorously. The first rigorous definition was published by 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. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891.

In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.

The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.

Electronic calculators and computers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary definable real numbers, which are inconvenient to manipulate. 17dc91bb1f