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. Every real number can be almost uniquely represented by an infinite decimal expansion.
The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
The set of real numbers is denoted or and is sometimes called "the reals". The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of .
The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root ; these are called algebraic numbers. There are also real numbers which are not, such as ; these are called transcendental numbers.
Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers () are equally spaced.
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.
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. 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. The addition of two real numbers and produce a real number denoted which is the sum of and . The multiplication of two real numbers and produce a real number denoted or which is the product of and . Addition and multiplication are both commutative, which means that and for every real numbers and . Addition and multiplication are both associative, which means that and for every real numbers , and , and that parentheses may be omitted in both cases. Multiplication is distributive over addition, which means that for every real numbers , and . There is a real number called zero and denoted which is an additive identity, which means that for every real number . There is a real number denoted which is a multiplicative identity, which means that for every real number . Every real number has an additive inverse denoted This means that for every real number . Every nonzero real number has a multiplicative inverse denoted or This means that for every nonzero real number . The total order is denoted being that it is a total order means two properties: given two real numbers and , exactly one of or is true; and if and then one has also The order is compatible with addition and multiplication, which means that implies for every real number , and is implied by and
Many other properties can be deduced from the above ones. In particular: for every real number