Reals

The real numbers, denoted R, form the set of all numbers that can be placed on a number line. They include integers, rationals, and irrationals (numbers that are not ratios of integers). The real line provides a standard model for describing continuous quantities in mathematics.

There are several equivalent ways to construct R. One common view is that R is the unique complete ordered field up to isomorphism. Constructions include Dedekind cuts and equivalence classes of Cauchy sequences of rational numbers, both yielding a system in which every Cauchy sequence converges to a real number.

R is an ordered field with the usual addition and multiplication, satisfying the standard algebraic laws and order compatibility. The completeness axiom states that every nonempty set bounded above has a least upper bound (supremum). The Archimedean property ensures there are no infinite or infinitesimal gaps in R; the rationals are dense in R and the irrationals are dense as well.

The real numbers are uncountable; their cardinality is the continuum. With the distance function d(x,y) = |x − y|, R becomes a complete metric space and a connected, locally compact topological space. A subset of R is compact precisely if it is closed and bounded.

Real numbers underpin real analysis, calculus, and many applied disciplines. They provide the standard domain for limits, continuity, differentiation, and integration, forming the natural setting for modeling continuous quantities in science and engineering.