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In mathematics, the affinely extended real number system is obtained from the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, especially in the theory of measure and integration. The affinely extended real number system is denoted \overline or \infty, +\infty/math> or It is the
Dedekind–MacNeille completion In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed ...
of the real numbers. When the meaning is clear from context, the symbol +\infty is often written simply as


Motivation


Limits

It is often useful to describe the behavior of a function f, as either the argument x or the function value f gets "infinitely large" in some sense. For example, consider the function f defined by :f(x) = \frac. The graph of this function has a horizontal
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
at y = 0. Geometrically, when moving increasingly farther to the right along the x-axis, the value of / approaches . This limiting behavior is similar to the limit of a function \lim_ f(x) in which the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
x approaches x_0, except that there is no real number to which x approaches. By adjoining the elements +\infty and -\infty to \R, it enables a formulation of a "limit at infinity", with
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
properties similar to those for \R. To make things completely formal, the Cauchy sequences definition of \R allows defining +\infty as the set of all sequences \left\ of rational numbers, such that every M \in \R is associated with a corresponding N \in \N for which a_n > M for all n > N. The definition of -\infty can be constructed similarly.


Measure and integration

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 \R that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering
improper integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
s, such as :\int_1^\frac the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as :f_n(x) = \begin 2n\left(1-nx\right), & \mbox 0 \leq x \leq \frac \\ 0, & \mbox \frac < x \leq 1 \end Without allowing functions to take on infinite values, such essential results as the
monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Infor ...
and the
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
would not make sense.


Order and topological properties

The affinely extended real number system can be turned into a
totally ordered set In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
, by defining -\infty \leq a \leq +\infty for all a. With this
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
, \overline has the desirable property of compactness: Every subset of \overline\R has a supremum and an
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
(the infimum of the empty set is +\infty, and its supremum is -\infty). Moreover, with this topology, \overline\R is homeomorphic to the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Thus the topology is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
, 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 \R. In this topology, a set U is a neighborhood of +\infty, if and only if it contains a set \ for some real number a. The notion of the neighborhood of -\infty can be defined similarly. Using this characterization of extended-real neighborhoods,
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
with x tending to +\infty or -\infty, and limits "equal" to +\infty and -\infty, reduce to the general topological definition of limits—instead of having a special definition in the real number system.


Arithmetic operations

The arithmetic operations of \R can be partially extended to \overline\R as follows: : \begin a + \infty = +\infty + a & = +\infty, & a & \neq -\infty \\ a - \infty = -\infty + a & = -\infty, & a & \neq +\infty \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\ \frac & = 0, & a & \in \mathbb \\ \frac & = \pm\infty, & a & \in (0, +\infty) \\ \frac & = \mp\infty, & a & \in (-\infty, 0) \end For exponentiation, see . Here, a + \infty means both a + (+\infty) and a - (-\infty), while a - \infty means both a - (+\infty) and a + (-\infty). The expressions \infty - \infty, 0 \times (\pm\infty) and \pm\infty/\pm\infty (called indeterminate forms) are usually left Defined and undefined, undefined. These rules are modeled on the laws for Limit_of_a_function#Limits_involving_infinity, infinite limits. However, in the context of probability or measure theory, 0 \times \pm\infty is often defined as When dealing with both positive and negative extended real numbers, the expression 1/0 is usually left undefined, because, although it is true that for every real nonzero sequence f that converges to 0, the reciprocal sequence 1/f is eventually contained in every neighborhood of \, it is ''not'' true that the sequence 1/f must itself converge to either -\infty or \infty. Said another way, if a continuous function f achieves a zero at a certain value x_0, then it need not be the case that 1/f tends to either -\infty or \infty in the limit as x tends to x_0. This is the case for the limits of the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
f(x) = x when x tends to 0, and of f(x) = x^2 \sin \left( 1/x \right) (for the latter function, neither -\infty nor \infty is a limit of 1/f(x), 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 1/0 = +\infty. For example, when working with power series, the
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
with coefficients a_n is often defined as the reciprocal of the limit-supremum of the sequence \left\. Thus, if one allows 1/0 to take the value +\infty, then one can use this formula regardless of whether the limit-supremum is 0 or not.


Algebraic properties

With these definitions, \overline\R is not even a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
, let alone a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
or a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
as in the case of \R. However, it has several convenient properties: * a + (b + c) and (a + b) + c are either equal or both undefined. * a + b and b + a are either equal or both undefined. * a \cdot (b \cdot c) and (a \cdot b) \cdot c are either equal or both undefined. * a \cdot b and b \cdot a are either equal or both undefined * a \cdot (b + c) and (a \cdot b) + (a \cdot c) are equal if both are defined. * If a \leq b and if both a + c and b + c are defined, then a + c \leq b + c. * If a \leq b and c > 0 and if both a \cdot c and b \cdot c are defined, then a \cdot c \leq b \cdot c. In general, all laws of arithmetic are valid in \overline\R—as long as all occurring expressions are defined.


Miscellaneous

Several functions can be continuously extended to \overline\R by taking limits. For instance, one may define the extremal points of the following functions as: :\exp(-\infty) = 0, :\ln(0) = -\infty, :\tanh(\pm\infty) = \pm 1, :\arctan(\pm\infty) = \pm\frac. Some singularities may additionally be removed. For example, the function 1/x^2 can be continuously extended to \overline\R (under ''some'' definitions of continuity), by setting the value to +\infty for x = 0, and 0 for x = +\infty and x = -\infty. On the other hand, the function 1/x can''not'' be continuously extended, because the function approaches -\infty as x approaches 0 from below, and +\infty as x approaches 0 from above. A similar but different real-line system, the
projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standar ...
, does not distinguish between +\infty and -\infty (i.e. infinity is unsigned). As a result, a function may have limit \infty 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 1/x at x = 0. On the other hand, \lim_ and \lim_ 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 e^x and \arctan(x) cannot be made continuous at x = \infty on the projectively extended real line.


See also

*
Division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
* Extended complex plane * Extended natural numbers *
Improper integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
* Infinity *
Log semiring In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are d ...
* Series (mathematics) *
Projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standar ...
* Computer representations of extended real numbers, see and
IEEE floating point The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in ...


Notes


References


Further reading

* * {{Large numbers Infinity Real numbers