Irrationaalne

Irrationaalne is an Estonian adjective used to describe what is not rational, and in mathematics specifically refers to irrational numbers. An irrational number is a real number that cannot be written as a ratio a/b of two integers with b nonzero. Its decimal expansion is non-terminating and non-repeating.

Examples include the square root of 2, pi and e. By contrast, numbers such as 1/2 or 3 are rational. Some irrational numbers are algebraic, meaning they are roots of polynomial equations with integer coefficients, for example sqrt(2) is a root of x^2 − 2 = 0. Others are transcendental, meaning they are not roots of any such polynomial; pi and e are famous examples of transcendental irrationals. While all transcendental numbers are irrational, not all irrational numbers are transcendental.

Properties of irrational numbers are nuanced: the sum or product of two irrationals can be rational or irrational. For instance, sqrt(2) + (2 − sqrt(2)) equals 2, a rational number, while sqrt(2) · sqrt(2) equals 2, also rational. The set of irrational numbers is uncountable, and real numbers consist of both rationals and irrationals. Rationals are dense in the real numbers, and irrationals are dense as well, meaning every interval contains both.

In everyday language, irrationaalne can describe beliefs or decisions not guided by reason or logic, reflecting a broader sense of irrational behavior beyond mathematics.