Extended real number line

In mathematics, the extended real number system[a] is obtained from the real number system

This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities.

of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has

as its least upper bound and as its limit (an actual infinity).

as actual limits extends significantly the possible computations.

The extended real number system is denoted

[2] There is also a distinct projectively extended real line where

are not distinguished, i.e., there is a single actual infinity for both infinitely increasing sequences and infinitely decreasing sequences that is denoted as just

The extended number line is often useful to describe the behavior of a function

defined by The graph of this function has a horizontal asymptote at

This allows proving and writing In measure theory, it is often useful to allow sets that have infinite measure and integrals whose value may be infinite.

that agrees with the usual length of intervals, this measure must be larger than any finite real number.

Also, when considering improper integrals, such as the value "infinity" arises.

Finally, it is often useful to consider the limit of a sequence of functions, such as Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem would not make sense.

, can be turned into a totally ordered set by defining

Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval.

(called indeterminate forms) are usually left undefined.

These rules are modeled on the laws for infinite limits.

However, in the context of probability or measure theory,

[4] When dealing with both positive and negative extended real numbers, the expression

is usually left undefined, because, although it is true that for every real nonzero sequence

However, in contexts where only non-negative values are considered, it is often convenient to define

is not even a semigroup, let alone a group, a ring or a field as in the case of

However, it has several convenient properties: In general, all laws of arithmetic are valid in

For instance, one may define the extremal points of the following functions as: Some singularities may additionally be removed.

(under some definitions of continuity), by setting the value to

approaches 0 from above, i.e., the function not converging to the same value as its independent variable approaching to the same domain element from both the positive and negative value sides.

A similar but different real-line system, the projectively extended real line, does not distinguish between

on the projectively extended real line, while in the extended real number system only the absolute value of the function has a limit, e.g. in the case of the function

On the other hand, on the projectively extended real line,