reelle

Réelle is the French term for real, and in mathematics it most commonly denotes the real numbers, the set of numbers corresponding to points on the real line. This set, usually written as R or ℝ, includes both rational numbers (such as 1/2) and irrational numbers (such as √2). Real numbers form the standard ambient field for real-valued quantities in analysis, geometry, and applied mathematics.

Mathematically, the real numbers form a complete ordered field. They contain the rational numbers as a dense subset, and they satisfy the least upper bound (completeness) property: every nonempty subset that is bounded above has a least upper bound in R. Equipped with the metric d(x, y) = |x − y|, they form the real line, a connected, uncountable, metric space. Real analysis rests on these properties, with the notion of limits, continuity, differentiation, and integration defined relative to real numbers.

Real numbers can be constructed rigorously from the rationals via methods such as Dedekind cuts or Cauchy sequences, ensuring no “gaps” remain in the continuum. They provide the standard setting for modeling continuous quantities in mathematics and physics.

In French usage, réelle (feminine) or réel (masculine) functions as an adjective meaning real as opposed to imaginary or virtual. Common phrases include valeur réelle (real value), nombre réel (real number), monde réel (real world), and en temps réel (in real time). The term adapts to gender and number (réelle, réelles) as required.