Real

Real numbers, denoted by R, are the set of all numbers that can be placed on a continuous number line. They include rational numbers, such as 1/2 and -4, and irrational numbers, such as √2 and π.

R forms a complete ordered field: you can add, subtract, multiply, and divide (except by zero) and compare any two elements; the order is total; and every nonempty set that is bounded above has a least upper bound, or supremum. Every real number has a decimal expansion, and every infinite decimal corresponds to a real number.

Real numbers provide the rigorous basis for calculus and real analysis, enabling limits, continuity, differentiation, and integration. They also underpin physical measurement and most applied science, serving as the standard model for quantities that vary continuously.

Historically, the existence and properties of the real numbers were formalized in the 19th century through constructions such as Dedekind cuts and Cauchy sequences, which gave precise meanings to completeness and irrational numbers. Earlier work showed that certain equations require irrationals, highlighting gaps in the rational numbers alone.

Key properties include density (between any two real numbers lies another real), uncountability (there are more real numbers than integers), and the usual metric d(x,y)=|x−y|, which makes R a complete metric space. The real numbers extend the rationals and form a subfield of the complex numbers, linking real analysis with broader areas of mathematics.

See also real analysis, Dedekind cut, Cauchy sequence, decimal expansion, complex numbers.