# Definition:Extended Real Number Line

## Definition

### Definition 1

The **extended real number line** $\overline \R$ is defined as:

- $\overline \R := \R \cup \set {+\infty, -\infty}$

that is, the set of real numbers together with two auxiliary symbols:

- $+\infty$,
**positive infinity** - $-\infty$,
**negative infinity**

such that:

- $\forall x \in \R: x < +\infty$
- $\forall x \in \R: -\infty < x$

### Definition 2

The **extended real number line** $\overline \R$ is the order completion of the set of real numbers $\R$.

The greatest element of $\overline \R$ is often denoted by $+\infty$ and its least element by $-\infty$.

## Also defined as

Some sources define $\overline \R$ as $\R \cup \set \infty$, that is, without the **negative infinity $-\infty$**.

This is the Alexandroff extension of $\R$.

This is isomorphic to the topological group of complex numbers with norm $1$ under multiplication.

This has the benefit that extended real addition is defined on all of $\overline \R$.

A drawback is that not all suprema and infima exist.

Depending on the context one may decide which form is most suitable.

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## Also known as

This structure can be referred to as:

- the
**extended real line** - the
**extended (set of) real numbers**

Also, the notations $\sqbrk {-\infty, +\infty}$ and $\closedint {-\infty} {+\infty}$ can be encountered, extending the notation for real intervals.

## Also see

- Equivalence of Definitions of Extended Real Number Line
- Definition:Neighborhood of Infinity (Real Analysis)
- Definition:Alexandroff Extension of Real Number Line
- Definition:Projective Real Line
- Definition:Extended Natural Number

### Structures on $\overline \R$

$\overline \R$ can be endowed with the following structures:

- An ordering extending the standard ordering on $\R$
- Extended Real Addition, extending real addition $+$
- Extended Real Subtraction, extending real subtraction $-$
- Extended Real Multiplication, extending real multiplication $\cdot$
- A topology extending the usual (Euclidean) topology on $\R$
- A $\sigma$-algebra extending the Borel $\sigma$-algebra $\map \BB \R$ on $\R$

- Results about
**extended real numbers**can be found here.