 These descriptions of the real numbers are not sufficiently rigorous by the modern standards 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.The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation.