Irrational number

Definition

A real number $x$ is irrational when $x$ is not a rational number. That is, there are no integers $a$ and $b$ with $b \neq 0$ and $x = a / b$. The set of irrational numbers is $\mathbb{R} \setminus \mathbb{Q}$.

Remarks

The number $\sqrt{2}$ is irrational. The numbers $\pi$ and $e$ are also irrational. The decimal expansion of a real number is eventually periodic exactly when the number is rational. Thus the decimal expansion of an irrational number never repeats. The set $\mathbb{Q}$ is countable and $\mathbb{R}$ is not, thus almost all real numbers are irrational.

Explanations

L1 human

An irrational number is a number on the number line that you cannot write as a fraction. The length of the diagonal of a square with side $1$ is such a number: it is $\sqrt{2} = 1.41421\dots$.