realnumber

Real numbers are the values that can be found on the real number line. They include rational numbers, such as 3/4 and -2, and irrational numbers, such as sqrt(2) and pi. Real numbers support the usual arithmetic operations and form an ordered field that extends the rational numbers. They provide a precise continuum for measuring quantities and for expressing limits, sequences, and continuous functions.

Key properties of the real numbers include completeness, order, and density. The order is total: any two real numbers can be compared. Completeness means every nonempty set of real numbers that is bounded above has a least upper bound (supremum). This property underpins many results in analysis. The Archimedean property states that real numbers are not infinitely large or small relative to the integers: for any real x there exists an integer n with n > x. The rational numbers are dense in the real numbers: between any two real numbers there is a rational number.

Real numbers can be constructed in several equivalent ways, such as by completing the rational numbers with respect to the usual metric, using Dedekind cuts, or via Cauchy sequences. Decimal expansions offer a practical representation: rationals have terminating or repeating decimals, while irrationals have nonrepeating, nonterminating decimals.

Applications of real numbers are foundational to real analysis, calculus, and physics. The real line carries a natural metric d(x, y) = |x − y|, making it a complete, connected, one-dimensional topological space. Notable real numbers include rational numbers like 1/2 and irrational numbers like π and e.