---
tag: 0GSZ7B0CGR
kind: object
name: Extended real number line
aliases: [extended-reals]
msc: [26A03]
sources:
  - url: https://en.wikipedia.org/wiki/Extended_real_number_line
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Extended_real_number_line
    locator: Order and topological properties
same_as:
  wikidata: Q2039387
  wikipedia: Extended real number line
---

## Description

The extended real number line is the set $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$.
It has two new elements, $-\infty$ and $+\infty$, that are not [[real numbers|08J2M1JYAG]].
Its order extends the order of $\mathbb{R}$, with $-\infty < x < +\infty$ for each real number $x$.

## Remarks

In $\overline{\mathbb{R}}$, each subset has a [[supremum|0BTMQ04556]] and an infimum. For example, $\sup \mathbb{N} = +\infty$ and $\sup \emptyset = -\infty$.
Some operations extend: $x + \infty = +\infty$ for each real $x$. Other expressions, such as $\infty - \infty$, stay undefined.
The set $\overline{\mathbb{R}}$ is not a field.

## Explanations

### L1 human

The extended real line adds two end points to the number line, one at each end.
This lets you write $\lim_{x \to \infty}$ and $\sup$ of unbounded sets with symbols instead of words.