---
tag: 08J2M1JYAG
kind: object
name: Real numbers
aliases: [real-number, reals, set-of-real-numbers, real-line]
msc: [26A03]
sources:
  - url: https://en.wikipedia.org/wiki/Real_number
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Real_number
    locator: Characterizing properties
same_as:
  wikidata: Q12916
  wikipedia: Real number
---

## Description

The real numbers are the numbers that measure continuous quantities, such as lengths. We picture them as the points of a line.
We write $\mathbb{R}$ for the set of real numbers.
With addition, multiplication and its order, $\mathbb{R}$ is an [[ordered field|0QSPNASE50]].
This ordered field is complete: each nonempty subset of $\mathbb{R}$ that is bounded above has a [[least upper bound|0AXWET80YW]].
Up to a unique isomorphism, $\mathbb{R}$ is the only [[complete ordered field|0A9E6X0S3E]].

## Remarks

The set $\mathbb{R}$ contains the [[rational numbers|0NCJ5WXN6Z]] and also [[irrational numbers|080T2TXVQV]] such as $\sqrt{2}$ and $\pi$.
Two common constructions of $\mathbb{R}$ from $\mathbb{Q}$ use [[Dedekind cuts|04QNZZ4HHY]] or [[Cauchy sequences|0D8HE681SC]].
The set $\mathbb{R}$ is uncountable.

## Explanations

### L1 human

The real numbers fill the number line without gaps.
The rational numbers leave gaps, for example at $\sqrt{2}$, and the real numbers fill these gaps.
Each real number has a [[decimal expansion|06XY4X9430]], such as $3.14159\dots$ for $\pi$.