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 .

MotivationEdit

LimitsEdit

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 rationals which, for any  , from some point on exceed  . We can define   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 turns into a totally ordered set by defining   for all  . This order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice.

This induces the order topology 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  .   is a compact Hausdorff space 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  .

With this topology, the specially defined limits for   tending to   and  , and the specially defined concepts of limits equal to   and  , reduce to the general topological definitions of limits.

Arithmetic operationsEdit

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, it is often convenient to define the radius of convergence of a power series with coefficients   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, since both the radius of convergence and the limit-supremum are non-negative quantities. As another example, in the theory of differentiable curves, the radius of curvature is defined as the reciprocal of the curvature constant of the curve. Since both quantities are non-negative, this definition can be used even when the curvature constant is not zero, if we define  .

Algebraic propertiesEdit

With these definitions   is not even a semigroup, let alone a group, a ring or a field, like   is one. However, it nevertheless does form a complete metric space, and 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  , 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 alsoEdit

Further readingEdit

  • 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.