Reaalnumbrite

Reaalnumbrite are the set of all real numbers, usually denoted by R. They comprise every number that can be located on the number line, including integers, fractions, and numbers that cannot be expressed as a ratio of integers, such as the square root of 2 or π. Real numbers provide a complete continuum for representing quantities that vary continuously, such as length, time, and temperature.

Real numbers can be constructed in several equivalent ways. One common approach is to define them as equivalence classes of Cauchy sequences of rational numbers, or as Dedekind cuts of the rational numbers. Every real number can be represented by a decimal expansion, which may be finite or infinite; rational numbers have terminating or eventually repeating decimal expansions, while many irrationals have non-repeating, non-terminating expansions.

Real numbers form a complete ordered field. They are closed under addition, subtraction, and multiplication, and division by nonzero numbers. There is a total order on R that is compatible with these operations, and the field is Archimedean. A key property is completeness: every Cauchy sequence of real numbers converges to a real limit. Geometrically, the real numbers constitute the real line, a connected, unbroken continuum with a rich topological structure.

Some basic and important facts: R is uncountable, having the cardinality of the continuum; between any two distinct real numbers there exists another real number (density). Many fundamental constants, such as π and e, are real numbers. Real numbers underpin real analysis, calculus, geometry, and much of applied mathematics.