Extended real number line

Summary

In mathematics, the extended real number system[a] is obtained from the real number system by adding two elements denoted and [b] that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence 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). In calculus and mathematical analysis, the use of and as actual limits extends significantly the possible computations.[1] It is the Dedekind–MacNeille completion of the real numbers.

Extended real numbers (top) vs projectively extended real numbers (bottom)

The extended real number system is denoted or or [2] When the meaning is clear from context, the symbol is often written simply as [2]

There is also a distinct projectively extended real line where and 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 or as .

Motivation

edit

Limits

edit

The extended number line is often useful to describe the behavior of a function   when 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 that   approaches when   increases infinitely. Adjoining the elements   and   to   enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, except that   is replaced by   (for  ) or   (for  ). This allows proving and writing

 

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 extended real number system  , defined as   or  , 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 operations

edit

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 properties

edit

With the arithmetic operations defined above,   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.

Miscellaneous

edit

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, 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   and   (i.e. infinity is unsigned).[5] As a result, a function may have limit   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   at   On the other hand, on the projectively extended real line,   and   correspond 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 also

edit

Notes

edit
  1. ^ Some authors use Affinely extended real number system and Affinely extended real number line, although the extended real numbers do not form an affine line.
  2. ^ Read as "positive infinity" and "negative infinity" respectively.

References

edit
  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 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.