---
tag: 0NCJ5WXN6Z
kind: object
name: Rational numbers
aliases: [rational-number, rationals, set-of-rational-numbers]
msc: [11A99]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_number
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Rational_number
    locator: Arithmetic
same_as:
  wikidata: Q1244890
  wikipedia: Rational number
---

## Description

A rational number is a number of the form $a / b$, where $a$ and $b$ are [[integers|0C4JS9RF5M]] and $b \neq 0$.
We write $\mathbb{Q}$ for the set of rational numbers.
Two fractions $a / b$ and $c / d$ are the same rational number when $a d = b c$.
With addition and multiplication, $\mathbb{Q}$ is a [[field|0N2WPCC00T]], and with its usual order it is an [[ordered field|09RFKMBQ63]].
The set $\mathbb{Q}$ contains $\mathbb{Z}$, because each integer $a$ is the fraction $a / 1$.

## Remarks

Each rational number has exactly one [[representation in lowest terms|0R64A2CVN4]] $a / b$ with $b > 0$.
The set $\mathbb{Q}$ is countable.
Not every real number is rational: $\sqrt{2}$ is [[irrational|0DZGSDHD92]].

## Explanations

### L1 human

The rational numbers are the fractions, such as $1/2$, $-3/4$ and $7 = 7/1$.
Many fractions name the same number: $1/2$, $2/4$ and $3/6$ are all equal.
In $\mathbb{Q}$, you can divide by any number except $0$.