The **extended real number line** is obtained from the real number line **R** by adding two elements: +∞ and −∞ (which are not considered to be real numbers). It is useful in mathematical analysis, especially in integration theory. The extended real number line is denoted by **R** or [−∞,+∞].

The extended real number line turns into a totally ordered set by defining −∞ ≤ *a* ≤ +∞ for all *a*. This order has the nice property that every subset has a supremum and an infimum: it is a complete lattice. The total order induces a topology on **R**. In this topology, a set *U* is a neighborhood of +∞ if and only if it contains a set {*x* : *x* ≥ *a*} for some real number *a*, and analogously for the neighborhoods of −∞. **R** is a compact Hausdorff space homeomorphic to the unit interval [0,1].

The arithmetical operations of **R** can be partly extended to **R** as follows:

*a* + ∞ = ∞ + *a* = ∞ if *a* ≠ −∞
*a* − ∞ = −∞ + *a* = −∞ if *a* ≠ +∞
*a* × +∞ = +∞ × *a* = +∞ if *a* > 0
*a* × +∞ = +∞ × *a* = −∞ if *a* < 0
*a* × −∞ = −∞ × *a* = −∞ if *a* > 0
*a* × −∞ = −∞ × *a* = +∞ if *a* < 0
*a* / ±∞ = 0 if −∞ < *a* < +∞
- ±∞ /
*a* = ±∞ if 0 < *a* < +∞
- +∞ /
*a* = −∞ if −∞ < *a* < 0
- −∞ /
*a* = +∞ if −∞ < *a* < 0

The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ are usually left undefined. Also, 1 / 0 is **not** defined as +∞ (because −∞ would be just as good a candidate). These rules are modeled on the laws for infinite limits.

Note that with these definitions, **R** is **not** a field and not even a ring. However, it still has several convenient properties:

*a* + (*b* + *c*) and (*a* + *b*) + *c* are either equal or both undefined.
*a* + *b* and *b* + *a* are either equal or both undefined.
*a* × (*b* × *c*) and (*a* × *b*) × *c* are either equal or both undefined.
*a* × *b* and *b* × *a* are either equal or both undefined
*a* × (*b* + *c*) and (*a* × *b*) + (*a* × *c*) are equal if both are defined.
- if
*a* ≤ *b* and if both *a* + *c* and *b* + *c* are defined, then *a* + *c* ≤ *b* + *c*.
- if
*a* ≤ *b* and *c* > 0 and both *a* × *c* and *b* × *c* are defined, then *a* × *c* ≤ *b* × *c*.

In general, all laws of arithmetic are valid in **R** as long as all occurring expressions are defined.

By using the intuition of limits, several functions can be naturally extended to **R**. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = ∞ etc.