Irrational Numbers

Intuition and core definition

An irrational number is a real number that cannot be written as a ratio $a/b$ of integers with $b\ne0$. Its decimal expansion neither terminates nor repeats a fixed block. Familiar examples include $\sqrt2$, $\pi$, and $e$; an irrational number is still a precise point on the real number line.

Notation, language, and conditions

$\mathbb Q$ denotes rational numbers and $\mathbb R\setminus\mathbb Q$ the irrationals. The symbol $\approx$ reports an approximation, so $\pi\approx3.14159$, whereas $=$ claims exact equality. Not every radical is irrational: $\sqrt{49}=7$ is rational; for a positive integer $n$, $\sqrt n$ is irrational exactly when $n$ is not a perfect square after simplification.

Why this idea matters

Irrational numbers fill locations that no ratio of integers can name, while still supporting approximation, ordering, and exact symbolic calculation.

A dependable method

1. Simplify radicals by extracting any perfect-square factors.
2. If the result is an integer or fraction, classify it as rational.
3. For a displayed decimal, look for termination or a proven repeating block; finite observation alone cannot prove nonrepetition.
4. Keep exact forms such as $\sqrt2$ or $\pi$ during algebra.
5. Use a decimal approximation only when locating, measuring, or rounding is requested.

Worked example

Common mistakes