Irrational Number¶

Definition¶

An irrational number is a real number but not a rational number, i.e., it is an element of the set $\mathbb{R}\setminus \mathbb{Q}\text{.}$

Interpretation¶

An irrational number cannot be expressed as a fraction $p/q$ of two integers. Such numbers arise from ordinary geometry, and they are what the real numbers add to the rationals.

$A unit square whose diagonal has length the square root of 2, rotated down onto the number line to mark the point about 1.414, which is not a fraction.$

The diagonal of the unit square has length $\sqrt{2}$ , since $1^2+1^2=2$ $2$ . Rotating it down onto the number line marks the point $1.414\ldots$ : a length that is easy to construct lands at a point that no fraction reaches. The classical proof is by contradiction. If $\sqrt{2}=p/q$ in lowest terms, then $p^2=2q^2$ $2 q^{2}$ , so $p$ is even; writing $p=2k$ gives $q^2=2k^2$ $2 k^{2}$ , so $q$ is even as well, which contradicts lowest terms.

A common slip is to expect the irrationals to be closed under arithmetic. They are not: $\sqrt{2}+(-\sqrt{2})=0$ and $\sqrt{2}\cdot\sqrt{2}=2$ are rational. A rational number plus an irrational number, on the other hand, is always irrational.

Key Results¶

A real number is rational exactly when it has a decimal expansion that terminates or is eventually periodic. Irrational numbers therefore have non-terminating decimal expansions that are not eventually periodic.
Every real number is either rational or irrational, and never both.

Examples¶

$\sqrt{2}$ is an irrational number.
$\pi$ and $\mathrm{e}$ are irrational numbers.
It is still unknown if $\pi+\mathrm{e}$ is irrational or not.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 2

Symbols: pi, rational numbers, real numbers, square root

Last updated: 20 October 2026 03:02