irrationál

Irrational refers to a number that cannot be expressed as a simple fraction, meaning it cannot be written in the form p/q where p and q are integers and q is not zero. This is in contrast to rational numbers, which can be expressed in this form. When irrational numbers are represented as decimals, they have an infinite number of non-repeating digits.

A classic example of an irrational number is pi (π), which is the ratio of a circle's circumference to its diameter. Its decimal representation begins 3.1415926535... and continues infinitely without any repeating pattern. Another well-known irrational number is the square root of 2 (√2). Its decimal expansion starts 1.4142135623... and also goes on forever without repetition.

The concept of irrational numbers was a significant development in mathematics, challenging the understanding that all numbers could be represented as ratios of integers. The existence of irrational numbers highlights the richness and complexity of the number system. Their properties are fundamental to many areas of mathematics, including geometry, calculus, and number theory.