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Wikipedia / Extended real number line

In mathematics, the extended real number system is obtained from the real number 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 as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers.

,..

There is also a distinct projectively extended real line.

Motivation

Limits

The extended number line is often useful to describe the behavior of a function when either the argument defined by

  1. .

The graph of this function has a horizontal asymptote approaches 0. This limiting behavior is similar to the limit of a function in which the real number). This allows proving and writing

Measure and integration

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

  1. .

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

, can be turned into a totally ordered set. With this order topology has the desirable property of compactness: Every subset has a supremum and an infimum (the infimum of the empty set). Moreover, with this topology is homeomorphic to the unit interval. Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric.

is a neighborhood can be defined similarly. Using this characterization of extended-real neighborhoods, limits, reduce to the general topological definition of limits—instead of having a special definition in the real number system.

Arithmetic operations

as follows:

For exponentiation, see .

(called indeterminate forms) are usually left undefined. These rules are modeled on the laws for infinite limits. However, in the context of probability is often defined as 0.

that converges to 0, the reciprocal, it is not Said another way, if a continuous function. This is the case for the limits of the identity function are considered).

. 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, then one can use this formula regardless of whether the limit-supremum is 0 or not.

Algebraic properties

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

  • are either equal or both undefined.

  • are either equal or both undefined.

  • are either equal or both undefined.

  • are either equal or both undefined

  • are equal if both are defined.

  • .

  • .

as long as all occurring expressions are defined.

Miscellaneous

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

  1. ,

  2. ,

  3. ,

  4. .

Some singularities (under some cannot approaches 0 from below 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 (i.e. infinity is unsigned). on the projectively extended real line, while in the extended real number system only the absolute value on the projectively extended real line.

See also

Notes

References

Further reading