In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus and in many other branches of mathematics, in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is usually notated as a bold R or the blackboard bold R {\displaystyle \mathbb {R} } . The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of negative numbers. The real numbers include the rational numbers, such as the integer − 5 {\displaystyle -5} and the fraction 4 / 3 {\displaystyle 4/3} . Real numbers that are not rational are irrational. Those real numbers that are roots of polynomials with rational coefficients are algebraic numbers, which include all the rational numbers and also irrational numbers such as √2 = 1.414.... Other real numbers, such as π = 3.1415..., are not roots of polynomials; these are the transcendental numbers. The real numbers can be thought of as the points on a line, called the number line or real line, on which the points corresponding to integers (..., −2, −1, 0, 1, 2, ...) are equally spaced. The informal descriptions above of the real numbers are not sufficient for rigorous reasoning about real numbers. The development of a suitable formal definition was a major achievement of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. One modern axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. 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.
Characterizing properties 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.
Arithmetic 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.
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