Réelles

Réelles, in mathematics, refers to the set of real numbers, usually denoted by ℝ. They include all rational numbers and all irrational numbers, such as √2 and π. On the real number line, each real number corresponds to a point, and between any two distinct real numbers there are infinitely many others, giving a continuous continuum of values.

Real numbers form a complete ordered field: they support the usual addition, subtraction, multiplication, and division by nonzero numbers, and the natural order is compatible with these operations. They contain the rational numbers as a dense subfield, and every real can be approximated by rationals arbitrarily well.

Construction and uniqueness: Real numbers can be constructed from the rationals by methods such as Dedekind cuts or Cauchy sequences. By a standard theorem, this structure is unique up to isomorphism among complete ordered fields.

Topological and size properties: The real numbers carry the standard metric d(x,y)=|x−y|; the resulting space is complete, archimedean, and connected. The real line is uncountable with cardinality equal to the continuum.

Applications and relations: The reals are foundational to calculus, analysis, probability, and measure theory. They form the real part of the complex numbers, which extend the structure to include a imaginary component.