Irrational number - Knowlpedia

An irrational number is a real number outside $\mathbb Q$ , the set of rational numbers. Equivalently, it cannot be expressed as $p/q$ with integers $p,q$ and $q\ne0$ .

Example

The positive square root of $2$ is irrational. If $\sqrt2=p/q$ in lowest terms with $q>0$ , then $p^2=2q^2$ . The parity of squares implies $p$ is even, and substitution then implies $q$ is even, contradicting lowest terms. Irrationality alone does not give a specified quantitative rate at which rational numbers can approximate a number.