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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