Irrational numbers are real numbers that cannot be expressed as a simple fraction, meaning they cannot be written as the ratio of two integers. Their decimal representations are non-repeating and non-terminating, which means they go on forever without repeating a pattern. Common examples of irrational numbers include the square root of 2 and the mathematical constant π. These numbers play a crucial role in mathematics, particularly in geometry and calculus. For instance, the length of the diagonal of a square with side length 1 is an irrational number, specifically √2. Understanding irrational numbers helps in grasping more complex mathematical concepts and their applications.