Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system formula_1 by adding two infinity elements: formula_2 and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted formula_3 or formula_4 or It is the Dedekind–MacNeille completion of the real numbers.
When the meaning is clear from context, the symbol formula_2 is often written simply as 
Motivation.
Limits.
It is often useful to describe the behavior of a function as either the argument formula_6 or the function value formula_7 gets "infinitely large" in some sense. For example, consider the function
The graph of this function has a horizontal asymptote at Geometrically, when moving increasingly farther to the right along the formula_6-axis, the value of formula_10 approaches 0. This limiting behavior is similar to the limit of a function formula_11 in which the real number formula_6 approaches except that there is no real number to which formula_6 approaches.
By adjoining the elements formula_2 and formula_15 to it enables a formulation of a "limit at infinity", with topological properties similar to those for 
To make things completely formal, the Cauchy sequences definition of formula_1 allows defining formula_2 as the set of all sequences formula_18 of rational numbers, such that every formula_19 is associated with a corresponding formula_20 for which formula_21 for all The definition of formula_15 can be constructed similarly. 
Measure and integration.
In measure theory, it is often useful to allow sets that have infinite measure and integrals whose value may be infinite.
Such measures arise naturally out of calculus. For example, in assigning a measure to formula_1 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.
Order and topological properties.
The affinely extended real number system can be turned into a totally ordered set, by defining formula_26 for all With this order topology, formula_3 has the desirable property of compactness: every subset of formula_28 has a supremum and an infimum (the infimum of the empty set is formula_29 and its supremum is formula_15). Moreover, with this topology, formula_28 is homeomorphic to the unit interval Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric that is an extension of the ordinary metric on 
In this topology, a set formula_32 is a neighborhood of if and only if it contains a set formula_33 for some real number The notion of the neighborhood of formula_15 can be defined similarly. Using this characterization of extended-real neighborhoods, the specially defined limits for formula_6 tending to formula_2 and and the specially defined concepts of limits equal to formula_2 and reduce to the general topological definition of limits.
Arithmetic operations.
The arithmetic operations of formula_1 can be partially extended to formula_28 as follows:
For exponentiation, see . Here, means both and while means both and 
The expressions formula_41 and formula_42 (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, formula_43 is often defined as 
When dealing with both positive and negative extended real numbers, the expression formula_44 is usually left undefined, because, although it is true that for every real nonzero sequence formula_45 that converges to the reciprocal sequence formula_46 is eventually contained in every neighborhood of it is "not" true that the sequence formula_46 must itself converge to either formula_15 or formula_49 Said another way, if a continuous function formula_45 achieves a zero at a certain value then it need not be the case that formula_46 tends to either formula_15 or formula_53 in the limit as formula_6 tends to This is the case for the limits of the identity function formula_55 when formula_6 tends to 0, and of formula_57 (for the latter function, neither formula_15 nor formula_53 is a limit of even if only positive values of formula_6 are considered).
However, in contexts where only non-negative values are considered, it is often convenient to define For example, when working with power series, the radius of convergence of a power series with coefficients formula_61 is often defined as the reciprocal of the limit-supremum of the sequence Thus, if one allows formula_44 to take the value then one can use this formula regardless of whether the limit-supremum is formula_63 or not.
Algebraic properties.
With these definitions, formula_28 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 formula_28—as long as all occurring expressions are defined.
Miscellaneous.
Several functions can be continuously extended to formula_28 by taking limits. For instance, one may define the extremal points of the following functions as follow:formula_84
Some singularities may additionally be removed. For example, the function formula_85 can be continuously extended to formula_28 (under "some" definitions of continuity), by setting the value to formula_2 for and formula_63 for formula_89 and On the other hand, the function formula_90 can "not" be continuously extended, because the function approaches formula_15 as formula_6 approaches formula_63 from below, and formula_2 as formula_6 approaches formula_63 from above.
A similar but different real-line system, the projectively extended real line, does not distinguish between formula_2 and formula_15 (i.e. infinity is unsigned). As a result, a function may have limit formula_53 on the projectively extended real line, while in the affinely extended real number system, only the absolute value of the function has a limit, e.g. in the case of the function formula_90 at On the other hand
correspond on the projectively extended real line to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus, the functions formula_103 and formula_104 cannot be made continuous at formula_105 on the projectively extended real line.