Irrational number

An irrational number is a real number that is not a rational number: it cannot be written as $a/b$ with integers $a,b$ and $b\ne 0$. Its decimal expansion is infinite and not eventually periodic. Both $\sqrt{2}$ and $\pi$ are irrational, but their algebraic behavior differs: $\sqrt{2}$ is an algebraic number, while $\pi$ is a transcendental number.

The irrationality of the square root of two supplies a basic example: a reduced fraction $a/b$ with square two would force both $a$ and $b$ to be even. Irrational numbers are needed to describe lengths such as the diagonal of a unit square, which rational coordinates alone cannot express.

An irrational number is a type of real number that cannot be expressed as a simple fraction or ratio of two integers. This means that if a number is irrational, it cannot be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Irrational numbers have non-repeating, non-terminating decimal expansions. This means their decimal representations go on forever without repeating a pattern.