The real numbers are the complete ordered field β the number system that corresponds to every point on the continuous number line, with no gaps. Every rational number is real, and so is every irrational number (like β2, Ο, and e). The key property distinguishing the reals from the rationals is completeness: every bounded increasing sequence of real numbers converges to a real limit, every Cauchy sequence has a limit in β, every non-empty set bounded above has a least upper bound. This completeness is the foundation of calculus, analysis, and all continuous mathematics.
The rational numbers (fractions like 1/2, 3/7, 22/7) are dense on the number line β between any two rationals there is another rational. But they leave infinitely many gaps: β2 is not rational, Ο is not rational, e is not rational. These gaps are real, in both an everyday and mathematical sense β there are genuine points on the number line that no fraction can name exactly. The real numbers fill in all these gaps, producing a complete, gapless number line.
The entire edifice of calculus β limits, derivatives, integrals β requires the real numbers specifically because of their completeness. If you tried to do calculus over just the rationals, you would immediately run into problems: the sequence 3, 3.1, 3.14, 3.141, 3.1415, ... gets closer and closer to Ο, but Ο is not rational, so this sequence of rationals has no rational limit. Calculus would be full of sequences "wanting" to converge but having nowhere to go. The real numbers are precisely the number system where every convergent sequence actually converges.
Before the 1870s, mathematicians used real numbers freely but without a rigorous definition β they relied on geometric intuition about "the continuous number line." Two independent constructions in 1872 placed the reals on a firm logical foundation:
Dedekind cuts (Richard Dedekind, 1872): A real number is defined as a "cut" of the rationals β a partition of β into two non-empty sets (L, R) where every element of L is less than every element of R and L has no greatest element. For example, β2 corresponds to the cut where L = {q β β : q < 0 or qΒ² < 2} and R = {q β β : q > 0 and qΒ² > 2}. This gives a clean, rigorous definition of every real number purely in terms of rationals.
Cauchy sequences (Georg Cantor, 1872; building on Cauchy's earlier work): A real number is an equivalence class of Cauchy sequences of rationals β sequences (qβ) where |qβ β qβ| β 0 as m,n β β β where two sequences are equivalent if their difference converges to 0. The completion of the rationals under this equivalence relation is precisely β.
Rather than constructing β from β, it can also be characterised axiomatically as the unique complete ordered field. The key axiom is the least upper bound property (or supremum property): every non-empty subset of β that is bounded above has a least upper bound (supremum) in β. This single axiom, combined with the field and order axioms, completely characterises β up to isomorphism. The following are all equivalent to it:
Cantor's diagonal argument (1891) proves that β is uncountable β strictly larger than the set of natural numbers (see the Continuum Hypothesis entry). There is no surjection from β to β, no way to list all real numbers in a sequence. The cardinality of β is denoted 2^β΅β or π (the "cardinality of the continuum"). Whether there exists a set strictly between β and β in cardinality is the Continuum Hypothesis β proven independent of ZFC by GΓΆdel and Cohen.
β carries a natural topology β the standard topology generated by open intervals (a, b) β which makes it a connected, locally compact, second-countable Hausdorff space. These topological properties are deeply intertwined with its order and field structure: β is the unique (up to homeomorphism) connected, locally compact, second-countable topological field. The topology of β underlies all of classical analysis: continuity, differentiability, and integrability are all defined relative to this topology.
Henri Lebesgue's 1902 construction of the Lebesgue measure on β β assigning a "length" to subsets of β in a way that extends the obvious length of intervals while being compatible with countable additivity β is one of the foundational achievements of 20th-century analysis. Most subsets of β encountered in practice are Lebesgue measurable; there exist non-measurable sets (the Vitali construction, 1905), but only if the axiom of choice is assumed β a connection to the Banach-Tarski paradox (see that entry). Lebesgue integration, built on this measure theory, is strictly more powerful than Riemann integration and is the standard integral of modern analysis.
The real number system is Archimedean: for any positive real x, there exists a natural number n such that n > x (there are no infinitely large real numbers, and no infinitely small positive ones). Non-Archimedean number systems β hyperreal numbers (Abraham Robinson's non-standard analysis, 1960s) and surreal numbers (John Conway, 1970s) β extend β to include genuine infinitesimals (positive numbers smaller than every positive real) and genuine infinities. Robinson's non-standard analysis rigorously vindicates Leibniz's original infinitesimal intuitions and provides an alternative foundation for calculus β producing exactly the same theorems as standard analysis, but often with more intuitive proofs.