Extended real number line

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In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: + ∞ and − ∞ (read as positive infinity and negative infinity respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or [−∞, +∞] or ℝ ∪ {−∞, +∞}.

When the meaning is clear from context, the symbol +∞ is often written simply as .

Motivation[edit]

Limits[edit]

We often wish to describe the behavior of a function , as either the argument or the function value gets "infinitely large" in some sense. For example, consider the function

The graph of this function has a horizontal asymptote at y = 0. Geometrically, as we move increasingly farther to the right along the -axis, the value of approaches 0. This limiting behavior is similar to the limit of a function at a real number, except that there is no real number to which approaches.

By adjoining the elements and to , we allow a formulation of a "limit at infinity" with topological properties similar to those for .

To make things completely formal, the Cauchy sequences definition of allows us to define as the set of all sequences of rational numbers such that every is associated with a corresponding for which for all . We define similarly.

Measure and integration[edit]

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 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[edit]

The affinely extended real number system turns into a totally ordered set by defining for all . With this order topology, has the desirable property of compactness: every nonempty subset of has a supremum and an infimum. Moreover, with this topology, 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 is a neighborhood of if and only if it contains a set for some real number , and analogously for the neighborhoods of . Using this characterization of extended-real neighborhoods, the specially defined limits for tending to and , and the specially defined concepts of limits equal to and , reduce to the general topological definition of limits.

Arithmetic operations[edit]

The arithmetic operations of can be partially extended to as follows:

For exponentiation, see Exponentiation#Limits of powers. Here, "" means both "" and "", while "" means both "" and "".

The expressions and (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, is often defined as .[citation needed]

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 that converges to , the reciprocal sequence is eventually contained in every neighborhood of , it is not true that the sequence must itself converge to either or . Said another way, if a continuous function achieves a zero at a certain value , then it need not be the case that tends to either or in the limit . This is the case for the limits when of the identity function and of (for the latter function, neither nor is a limit of even if only positive values of x 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 is often defined as the reciprocal of the limit-supremum of the sequence . Thus, if one allows to take the value , then one can use this formula regardless of whether the limit-supremum is or not.

Algebraic properties[edit]

With these definitions is not even a semigroup, let alone a group, a ring or a field, like is one. However, it has several convenient properties:

  • and are either equal or both undefined.
  • and are either equal or both undefined.
  • and are either equal or both undefined.
  • and are either equal or both undefined
  • and are equal if both are defined.
  • If and if both and are defined, then .
  • If and and if both and are defined, then .

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

Miscellaneous[edit]

Several functions can be continuously extended to by taking limits. For instance, one may define , etc.

Some singularities may additionally be removed. For example, the function can be continuously extended to (under some definitions of continuity) by setting the value to for , and for and . The function can not be continuously extended because the function approaches as approaches from below, and as approaches from above.

Compare the projectively extended real line, which does not distinguish between and . As a result, on one hand a function may have limit 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 at . On the other hand

and

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 and cannot be made continuous at on the projectively extended real line.

See also[edit]

Further reading[edit]

  • Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), Principles of Real Analysis (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29, ISBN 0-12-050257-7, MR 1669668
  • David W. Cantrell. "Affinely Extended Real Numbers". MathWorld.