Irrational numbers are real numbers that cannot be expressed as the ratio of two integers, meaning they cannot be written as simple fractions

. These numbers are often denoted as R\Q, where the backward slash symbol represents the difference between a set of real numbers and a set of rational numbers

. Some key points about irrational numbers include:

- Irrational numbers are not representable as p/q, where p and q are integers, and q ≠ 0

- Examples of irrational numbers include π (pi), √2, √3, √5, Euler's number (e = 2⋅718281…), and 2.010010001…

- Irrational numbers have decimal expansions that are neither terminating nor recurring

The concept of irrational numbers was introduced in ancient Greece, and the first proof of their existence is usually attributed to a Pythagorean who discovered them while identifying sides of the pentagram

. Irrational numbers are important in various fields, including mathematics, physics, and engineering, as they help describe phenomena that cannot be accurately represented using rational numbers.