Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and [a] 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.[1] The affinely extended real number system is denoted or or [2] It is the Dedekind–MacNeille completion of the real numbers.

When the meaning is clear from context, the symbol is often written simply as [2]

MotivationEdit

LimitsEdit

It is often useful 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   defined by

 

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

By adjoining the elements   and   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   allows defining   as the set of all sequences   of rational numbers, such that every   is associated with a corresponding   for which   for all   The definition of   can be constructed similarly.

Measure and integrationEdit

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 propertiesEdit

The affinely extended real number system can be turned into a totally ordered set, by defining   for all   With this order topology,   has the desirable property of compactness: Every subset of   has a supremum and an infimum[3] (the infimum of the empty set is  , and its supremum is  ). 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, however, 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   The notion of the neighborhood of   can be defined similarly. Using this characterization of extended-real neighborhoods, limits with   tending to   or  , and limits "equal" to   and  , reduce to the general topological definition of limits—instead of having a special definition in the real number system.

Arithmetic operationsEdit

The arithmetic operations of   can be partially extended to   as follows:[2]

 

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  [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   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 as   tends to   This is the case for the limits of the identity function   when   tends to   and of   (for the latter function, neither   nor   is a limit of   even if only positive values of   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 propertiesEdit

With these definitions,   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:

  •   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.

MiscellaneousEdit

Several functions can be continuously extended to   by taking limits. For instance, one may define the extremal points of the following functions as:

 
 
 
 

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   On the other hand, the function   cannot be continuously extended, because the function approaches   as   approaches   from below, and   as   approaches   from above.

A similar but different real-line system, the projectively extended real line, does not distinguish between   and   (i.e. infinity is unsigned).[5] As a result, 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, the functions   and   cannot be made continuous at   on the projectively extended real line.

See alsoEdit

NotesEdit

  1. ^ read as positive infinity and negative infinity respectively

ReferencesEdit

  1. ^ Wilkins, David (2007). "Section 6: The Extended Real Number System" (PDF). maths.tcd.ie. Retrieved 2019-12-03.
  2. ^ a b c Weisstein, Eric W. "Affinely Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03.
  3. ^ Oden, J. Tinsley; Demkowicz, Leszek (16 January 2018). Applied Functional Analysis (3 ed.). Chapman and Hall/CRC. p. 74. ISBN 9781498761147. Retrieved 8 December 2019.
  4. ^ "extended real number in nLab". ncatlab.org. Retrieved 2019-12-03.
  5. ^ Weisstein, Eric W. "Projectively Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03.

Further readingEdit