irrationalinen

Irrational numbers are a fundamental concept in mathematics, representing numbers that cannot be expressed as a simple fraction of two integers. They are the complement of rational numbers, which can be written as a ratio of two integers. The term "irrational" was first used by René Descartes in the 17th century. Irrational numbers are characterized by their non-repeating, non-terminating decimal expansions. This means that their decimal representation goes on forever without settling into a repeating pattern.

The most well-known examples of irrational numbers are algebraic irrationals, which are roots of polynomials with integer coefficients that cannot be expressed as a ratio of integers. A classic example is the square root of 2, which is approximately 1.41421356... and never ends or repeats. Another example is the golden ratio, approximately 1.6180339887..., which is a solution to the equation x^2 - x - 1 = 0.

Irrational numbers also include transcendental numbers, which are numbers that are not roots of any non-zero polynomial equation with integer coefficients. A famous example is the mathematical constant pi (π), which is approximately 3.14159265... and is fundamental in geometry and trigonometry.

The set of irrational numbers is uncountably infinite, meaning there are more irrational numbers than there are natural numbers. This was proven by Georg Cantor in the 19th century. Irrational numbers play a crucial role in various areas of mathematics, including number theory, calculus, and geometry. They are also essential in the study of real numbers, which form the basis of modern mathematics.