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Irrational numbers are real numbers that cannot be expressed as a simple fraction, meaning they cannot be written as a ratio of two integers. This contrasts with rational numbers, which can be expressed as a fraction where both the numerator and denominator are integers. The set of irrational numbers is infinite and includes numbers such as the square root of 2, pi (π), and Euler's number (e). These numbers have non-repeating, non-terminating decimal expansions, making them inherently different from rational numbers, which either terminate or repeat.

Irrational numbers were first recognized in ancient times, with the Pythagoreans discovering that the diagonal of a square with side length 1 is irrational. This discovery led to the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The irrationality of the square root of 2 was a significant breakthrough in mathematics, as it challenged the Pythagorean belief that all numbers could be expressed as ratios of integers.

Irrational numbers play a crucial role in various areas of mathematics, including geometry, calculus, and number theory. They are used to represent quantities that cannot be measured exactly using rational numbers, such as the circumference of a circle or the area of a triangle. The study of irrational numbers has led to the development of advanced mathematical concepts, such as the theory of real numbers and the concept of transcendental numbers, which are irrational numbers that are not roots of any non-zero polynomial equation with rational coefficients.