Irrationaalarvudel

Irrationaalarvudel, or irrational numbers, are real numbers that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. This means their decimal representation is non-terminating and non-repeating. Famous examples include pi (π), the ratio of a circle's circumference to its diameter, and the square root of 2 (√2), the length of the diagonal of a unit square.

The set of irrational numbers is infinite and dense within the real numbers. This means that between any two distinct real numbers, no matter how close, there exists an irrational number. While they cannot be written as exact fractions, their values are well-defined and can be approximated with arbitrary precision.

Irrational numbers arise in various areas of mathematics, including geometry, calculus, and number theory. Their discovery, often attributed to the ancient Greeks with the study of geometric magnitudes, presented a fundamental challenge to early mathematical thought, as it expanded the concept of number beyond simple ratios. Despite their "irrational" nature in terms of representation, they are an indispensable part of the mathematical landscape, essential for describing many natural phenomena and solving complex mathematical problems.