Irrationnelle

Irrationnelle is the feminine form of the French adjective irrationnel, used to describe something that is not rational, or not guided by reason. In everyday language it can refer to actions, beliefs, or arguments that seem unreasonable or driven by impulse rather than logic. In philosophical or psychological contexts, the term may be used to discuss the role of emotion, intuition, or subconscious factors in decision making.

In mathematics, irrationnelle typically concerns irrational numbers. An irrational number is a real number that cannot be expressed as a quotient of two integers. Classic examples include the square root of 2, pi, and e. Such numbers have non-terminating, non-repeating decimal expansions. They contrast with rational numbers, which can be written as fractions p/q with integers p and q (q nonzero). The set of irrational numbers is uncountable, while the set of rational numbers is countable; together they comprise all real numbers. Some irrationals are algebraic (solutions to polynomial equations with integer coefficients, like sqrt(2)) and others are transcendental (not roots of any such polynomial, like pi and e).

Historically, the concept of irrationality dates back to ancient Greece, with proofs that certain quantities, such as the diagonal of a unit square, cannot be expressed as a ratio of integers. In French, while irrationnelle is the feminine adjective form, the related noun is usually irrationnel or the broader term l’irrationnel when referring to the irrational as a concept or domain.

See also: rational number, irrational number, Euclid, proof by contradiction.