Irrationaalinen

Irrationaalinen is a Finnish mathematical term describing numbers that are not rational. In mathematics, irrationaalinen denotes numbers that cannot be expressed as a ratio of two integers. Equivalently, their decimal expansion is non-terminating and non-repeating.

A rational number is any number that can be written as a/b, where a and b are integers and b is not zero. If such a representation does not exist, the number is irrational. The set of irrational numbers together with the rational numbers constitutes the real numbers.

Examples of irrationaalinen numbers include the square root of 2, pi and e. Some irrationaalinen numbers are algebraic, meaning they are roots of polynomials with integer coefficients (for example, sqrt(2) is algebraic). Others are transcendental, meaning they are not roots of any such polynomial (pi and e are transcendental). The decimal expansions of irrationaalinen numbers never terminate and never settle into a repeating pattern.

Irrationaalinen numbers have several noteworthy properties. They are uncountable as a set, contrasting with the rationals, which are countable. The sum, difference, product, or quotient of irrationaalinen numbers can be rational or irrational, so the set of irrationaalinen numbers is not closed under these operations in general.

Historically, the discovery of irrational numbers is attributed to the ancient Greeks, notably the proof that sqrt(2) cannot be expressed as a ratio of integers, challenging the then-prevailing belief that all quantities were rational. In Finnish usage, irrationaalinen also appears in everyday language to describe irrational behavior or decisions, beyond strict mathematical meaning.