irracionálního

Irracionální refers to something that cannot be expressed as a simple fraction of two integers. In mathematics, irrational numbers are real numbers that are not rational. This means their decimal representation neither terminates nor repeats. Famous examples include pi (π), which is approximately 3.14159, and the square root of 2 (√2), approximately 1.41421.

The concept of irrational numbers emerged in ancient Greece with the discovery of the Pythagorean theorem. It was found that the diagonal of a square with side length 1 has a length of √2. This value could not be represented as a ratio of two whole numbers, challenging the prevailing belief that all magnitudes could be expressed in terms of rational numbers.

Irrational numbers are distinct from rational numbers, which can be written as p/q, where p and q are integers and q is not zero. For instance, 1/2, 0.75 (which is 3/4), and repeating decimals like 0.333... (which is 1/3) are all rational.

The set of irrational numbers, along with the set of rational numbers, forms the set of real numbers. The existence and properties of irrational numbers are fundamental to calculus, analysis, and many other branches of mathematics. Their decimal expansions are infinite and non-repeating, making them essential for representing quantities that cannot be precisely captured by simple fractions.