Real number

Definition and basic properties

Informally, a real number is any value representing a position on a continuous line. Formally the reals form an ordered field that is complete in the sense that every nonempty set bounded above has a least upper bound (supremum). This completeness property distinguishes the real numbers from simpler systems such as the rational numbers, and it is essential for limits, continuity and the foundations of calculus. The order permits comparison: numbers greater than zero are positive and those less than zero are negative. The real line is also Archimedean: for any real there exists an integer larger than it.

Classes and examples

• Rational numbers are those expressible as a ratio of two integers and include finite or repeating decimal expansions; see rational numbers.
• Irrational numbers cannot be written as a ratio of integers and have nonterminating, nonrepeating decimal expansions; common examples are √2, π and e, discussed at irrational numbers.
• Algebraic and transcendental: some real numbers are roots of polynomial equations with integer coefficients (algebraic); others, like π and e, are transcendental.

Density, order and arithmetic

The real numbers are dense: between any two distinct real numbers there exists another real number, and in fact infinitely many. They are ordered so arithmetic comparisons and standard operations (addition, subtraction, multiplication and division by nonzero numbers) behave in the familiar way. Because zero is a real number, it serves as the additive identity. The arithmetic and order properties together make the reals a complete ordered field, a structure often used as the standard model for continuous quantities.

Constructions and formalization

The intuitive idea of all points on a line was made rigorous in the 19th century by several equivalent constructions. Two widely used approaches are Dedekind cuts and the completion of the rationals by equivalence classes of Cauchy sequences. Both constructions produce a set with the same algebraic and order properties and with completeness. These formal descriptions explain why limits of Cauchy sequences of rationals need not be rational, and show how the irrationals naturally fill in the "gaps" left by rationals.

Cardinality and measure

The set of real numbers is uncountable: there is no way to arrange all real numbers into a single sequence that includes each one exactly once. Cantor's diagonal argument provides a simple demonstration of this fact and shows that there are strictly more real numbers than integers or rationals. Although rationals are dense in the reals, they form a countable subset and therefore have Lebesgue measure zero; most points on the line are irrational in a measure-theoretic sense. For discussions of sequences and countability see sequence and material on uncountability and uncountability.

Topology and analysis

Topologically, the real line is a connected, complete metric space. Open and closed intervals, convergence of sequences, continuity of functions, and integration are defined in terms of the real topology. Completeness ensures that Cauchy sequences converge in ℝ, while compactness of closed bounded intervals underlies many fundamental theorems, such as the extreme value theorem and the Heine–Borel property in the one-dimensional setting.

Relations to other number systems

Number systems are nested: the integers and rationals are subsets of the real numbers, and the real numbers are a subset of the complex numbers. Thus every real number is also a complex number (with zero imaginary part); for more on extensions beyond the reals, see complex numbers. The reals are also the basis for various function spaces and for algebraic structures used throughout mathematics and physics.

Uses and significance

• Real numbers are the standard model for continuous measurement in physics, engineering and applied sciences, though practical work uses finite approximations.
• Calculus and most classical analysis rely on the completeness and order of the reals to define derivatives, integrals and solutions of differential equations.
• Probability, statistics and modeling typically use real-valued variables and parameters to represent outcomes and quantities.

Historical notes

The distinction between rational and irrational quantities was recognized in antiquity (for example through incommensurable magnitudes). Over centuries decimal notation and continued fractions provided practical ways to represent reals. The rigorous, axiomatic treatment of the reals emerged in the 19th century with formal constructions such as Dedekind cuts and completion by Cauchy sequences; subsequent work by Cantor and others clarified cardinality and set-theoretic aspects.

Readers seeking further introductory material can follow links on the real number concept, explore detailed expositions of real-line intuition and consult resources on positivity and sign at positive numbers and zero. For negative values see negative real numbers.