numbersreal

Numbersreal is a term used to describe the real numbers as a mathematical system for quantities that can vary continuously. In this usage, numbersreal refers to the set R of all real numbers, including both rational and irrational values, arranged along the real number line and equipped with the standard operations of addition and multiplication.

Properties: numbersreal forms a complete ordered field, extending the rational numbers. It is uncountable and densely ordered, meaning between any two real numbers there lies another real number. It satisfies the Archimedean property, and it has the least upper bound (completeness) axiom, ensuring that every nonempty set bounded above has a supremum within numbersreal.

Representations and constructions: real numbers can be represented by infinite decimal expansions, by equivalence classes of Cauchy sequences of rationals, or by Dedekind cuts. Any real number can be approximated to arbitrary precision by rational numbers.

Examples: familiar numbers include pi, e, sqrt(2), and all rational numbers.

Applications: numbersreal underpins calculus, analysis, and most physical measurements, providing the rigorous basis for limits, continuity, integration, and differentiation.

History: the rigorous development of the real numbers culminated in the 19th century through contributions by Cantor, Dedekind, and Weierstrass, among others, who formalized notions of completeness and representation.

See also: Real numbers, Real analysis, Decimal expansion, Continuity.