irrationality

Irrationality, in mathematics, refers to real numbers that cannot be expressed as a ratio of two integers. An irrational number has a non-terminating, non-repeating decimal expansion. Classic examples include the square root of 2, pi, and Euler's number e. The property is typically established by a contradiction, such as showing that sqrt(2) cannot equal a/b for integers a and b with b nonzero. Irrational numbers are abundant: between any two rational numbers there are infinitely many irrationals, and the set of real numbers is uncountable.

Beyond numbers, the term also appears in everyday and academic language to describe actions, beliefs, or judgments that seem unsupported by sound reasons. In psychology and behavioral economics, rationality is analyzed as the coherence of beliefs and preferences and adherence to consistent decision rules. Irrationality can arise from cognitive biases, misperceptions, or emotional factors that lead choices away from those predicted by standard models like expected utility theory. In philosophy, rationality concerns the norms governing reasoning and action, including the justification of beliefs and the alignment of means with ends. Discussions often distinguish normative (how one ought to reason) from descriptive (how people actually reason) notions of rationality, with irrationality marking deviations from normative standards.

In cultural usage, irrationality can describe phenomena that appear to resist logical explanation. While mathematically irrational numbers have precise definitions, perceived irrationality in behavior or culture is typically a label for complexity, bias, or incomplete information rather than a strict mathematical property.