Which of the following best defines irrational numbers?

Irrational numbers are best defined as numbers that cannot be expressed as the ratio of two integers. This means that an irrational number cannot be written in the form ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b ) is not zero. A classic example of an irrational number is ( \sqrt{2} ), which cannot be simplified to a fraction of integers.

Irrational numbers typically have non-repeating, non-terminating decimal representations; they continue infinitely without forming a repeating pattern. This characteristic further distinguishes them from rational numbers, which either terminate or repeat when expressed as decimals. The understanding of irrational numbers plays a crucial role in mathematics, especially in areas dealing with real numbers and number theory.