Editing Extended real number line

{{Short description|Real numbers with + and - infinity added}}
{{about|the extension of the reals with {{math|+∞}} and {{math|−∞}}|the extension by a single point at infinity|Projectively extended real line}}
[[File:Extended_Real_Numbers.svg|thumb|right|Extended real numbers (top) vs [[projectively extended real numbers]] (bottom)]]
In [[mathematics]], the '''extended real number system'''{{efn|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]].}} is obtained from the [[real number]] system <math>\R</math> by adding two elements denoted <math>+\infty</math> and <math>-\infty</math>{{efn|Read as "positive infinity" and "negative infinity" respectively.}} that are respectively greater and lower than every real number. This allows for treating the [[potential infinity|potential infinities]] of infinitely increasing sequences and infinitely decreasing series as [[actual infinity|actual infinities]]. For example, the [[infinite sequence]] <math>(1,2,\ldots)</math> of the [[natural number]]s increases ''infinitively'' and has no [[upper bound]] in the real number system (a potential infinity); in the extended real number line, the sequence has <math>+\infty</math> as its [[least upper bound]] and as its [[limit (mathematics)|limit]] (an actual infinity). In [[calculus]] and [[mathematical analysis]], the use of <math>+\infty</math> and <math>-\infty</math> as actual limits extends significantly the possible computations.<ref>{{Cite web|url=https://www.maths.tcd.ie/~dwilkins/Courses/221/Extended.pdf|title=Section 6: The Extended Real Number System|last=Wilkins|first=David|date=2007|website=maths.tcd.ie|access-date=2019-12-03}}</ref> It is the [[Dedekind–MacNeille completion]] of the real numbers.

The extended real number system is denoted <math>\overline{\R}</math>,<!--{{math|{{overset|—|ℝ}}}}--> <math>[-\infty,+\infty]</math>, or <math>\R\cup\left\{-\infty,+\infty\right\}</math>.<ref name=":1" /> When the meaning is clear from context, the symbol <math>+\infty</math> is often written simply as <math>\infty</math>.<ref name=":1" />

There is also a distinct [[projectively extended real line]] where <math>+\infty</math> and <math>-\infty</math> 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 <math>\infty</math> or as <math>\pm\infty</math>.

==Motivation==
===Limits===
The extended number line is often useful to describe the behavior of a [[function (mathematics)|function]] <math>f</math> when either the [[argument of a function|argument]] <math>x</math> or the function value <math>f</math> gets "infinitely large" in some sense. For example, consider the function <math>f</math> defined by

:<math>f(x)=\frac{1}{x^{2}}</math>.

The [[graph of a function|graph]] of this function has a horizontal [[asymptote]] at <math>y=0</math>. Geometrically, when moving increasingly farther to the right along the <math>x</math>-axis, the value of <math display="inline">{1}/{x^2}</math> [[limit of a function|approaches]] 0. This limiting behavior is similar to the [[limit of a function]] <math display="inline">\lim_{x\to x_0}f(x)</math> in which the [[real number]] <math>x</math> approaches <math>x_0,</math> except that there is no real number that <math>x</math> approaches when <math>x</math> increases infinitely. Adjoining the elements <math>+\infty</math> and <math>-\infty</math> to <math>\R</math> enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, except that <math>|x-x_0|<\varepsilon</math> is replaced by <math>x>N</math> (for <math>+\infty</math>) or <math>x<-N</math> (for <math>-\infty</math>). This allows proving and writing
:<math>\begin{align}\lim_{x\to+\infty}\frac1{x^2}&=0,\\\lim_{x\to-\infty}\frac1{x^2}&=0,\\\lim_{x\to0}\frac1{x^2}&=+\infty.\end{align}</math>

===Measure and integration===
{{confusing section|reason=since this is a subsection of section "Motivation", it must be understandable by readers who know nothing more than the basic definition of an integral|date=September 2024}}
In [[measure theory]], it is often useful to allow sets that have infinite [[measure (mathematics)|measure]] and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure to <math>\R</math> that agrees with the usual length of [[interval (mathematics)|intervals]], this measure must be larger than any finite real number. Also, when considering [[improper integral]]s, such as

:<math>\int_1^{\infty}\frac{dx}{x}</math>

the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as      

:<math>f_n(x)=\begin{cases}2n(1-nx),&\mbox{if }0\leq x\leq\frac{1}{n}\\0,&\mbox{if }\frac{1}{n}<x\leq1\end{cases} </math>.

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==

The extended real number system <math>\overline{\R}</math>, defined as <math>[-\infty,+\infty]</math> or <math>\R\cup\left\{-\infty,+\infty\right\}</math>, can be turned into a [[totally ordered set]] by defining <math>-\infty\leq a\leq+\infty</math> for all <math>a\in\overline{\R}</math>. With this [[order topology]], <math>\overline{\R}</math> has the desirable property of [[compact space|compactness]]: Every [[subset]] of <math>\overline\R</math> has a [[supremum]] and an [[infimum]]<ref name=":1">{{cite book |last1=Oden |first1=J. Tinsley |last2= Demkowicz|first2= Leszek|title=Applied Functional Analysis |date=16 January 2018 |publisher=Chapman and Hall/CRC |isbn=9781498761147 |page=74 |edition=3 |access-date=8 December 2019 |url=https://www.crcpress.com/Applied-Functional-Analysis/Oden-Demkowicz/p/book/9781498761147}}</ref> (the infimum of the [[empty set]] is <math>+\infty</math>, and its supremum is <math>-\infty</math>). Moreover, with this [[topological space|topology]], <math>\overline\R</math> is [[homeomorphic]] to the [[unit interval]] <math>[0,1]</math>. Thus the topology is [[metrizable]], corresponding (for a given homeomorphism) to the ordinary [[metric (mathematics)|metric]] on this interval. There is no metric, however, that is an extension of the ordinary metric on <math>\R</math>.

In this topology, a set <math>U</math> is a [[neighborhood (mathematics)|neighborhood]] of <math>+\infty</math> if and only if it contains a set <math>\{x:x>a\}</math> for some real number <math>a</math>. The notion of the neighborhood of <math>-\infty</math> can be defined similarly. Using this characterization of extended-real neighborhoods, [[limit of a function|limits]] with <math>x</math> tending to <math>+\infty</math> or <math>-\infty</math>, and limits "equal" to <math>+\infty</math> and <math>-\infty</math>, 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 <math>\R</math> can be partially extended to <math>\overline\R</math> as follows:<ref name=":0">{{Cite web|url=http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html|title=Affinely Extended Real Numbers|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>

:<math display="block">\begin{align}a\pm\infty=\pm\infty+a&=\pm\infty,&a&\neq\mp\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{a}{\pm\infty}&=0,&a&\in\mathbb{R}\\\frac{\pm\infty}{a}&=\pm\infty,&a&\in(0,+\infty)\\\frac{\pm\infty}{a}&=\mp\infty,&a&\in(-\infty,0)\end{align}</math>

For exponentiation, see {{Section link|Exponentiation|Limits of powers}}. Here, <math>a+\infty</math> means both <math>a+(+\infty)</math> and <math>a-(-\infty)</math>, while <math>a-\infty</math> means both <math>a-(+\infty)</math> and <math>a+(-\infty)</math>.

The expressions <math>\infty-\infty</math>, <math>0\times(\pm\infty)</math>, and <math>\pm\infty/\pm\infty</math> (called [[indeterminate form]]s) 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 theory|probability]] or measure theory, <math>0\times\pm\infty</math> is often defined as 0.<ref name=":2" />

When dealing with both positive and negative extended real numbers, the expression <math>1/0</math> is usually left undefined, because, although it is true that for every real nonzero sequence <math>f</math> that [[limit of a sequence|converges]] to 0, the [[multiplicative inverse|reciprocal]] sequence <math>1/f</math> is eventually contained in every neighborhood of <math>\{\infty,-\infty\}</math>, it is ''not'' true that the sequence <math>1/f</math> must itself converge to either <math>-\infty</math> or <math>\infty.</math> Said another way, if a [[continuous function]] <math>f</math> achieves a zero at a certain value <math>x_0,</math> then it need not be the case that <math>1/f</math> tends to either <math>-\infty</math> or <math>\infty</math> in the limit as <math>x</math> tends to <math>x_0</math>. This is the case for the limits of the [[identity function]] <math>f(x)=x</math> when <math>x</math> tends to 0, and of <math>f(x)=x^2\sin\left(1/x\right)</math> (for the latter function, neither <math>-\infty</math> nor <math>\infty</math> is a limit of <math>1/f(x)</math>, even if only positive values of <math>x</math> are considered).

However, in contexts where only non-negative values are considered, it is often convenient to define <math>1/0=+\infty</math>. For example, when working with [[power series]], the [[radius of convergence]] of a power series with [[coefficient]]s <math>a_n</math> is often defined as the reciprocal of the [[limit inferior and limit superior|limit-supremum]] of the sequence <math>\left(|a_n|^{1/n}\right)</math>. Thus, if one allows <math>1/0</math> to take the value <math>+\infty</math>, then one can use this formula regardless of whether the limit-supremum is 0 or not.

==Algebraic properties==

With the arithmetic operations defined above, <math>\overline\R</math> is not even a [[semigroup]], let alone a [[group (mathematics)|group]], a [[ring (mathematics)|ring]] or a [[field (mathematics)|field]] as in the case of <math>\R</math>. However, it has several convenient properties:
* <math>a+(b+c)</math> and <math>(a+b)+c</math> are either equal or both undefined.
* <math>a+b</math> and <math>b+a</math> are either equal or both undefined.
* <math>a\cdot(b\cdot c)</math> and <math>(a\cdot b)\cdot c</math> are either equal or both undefined.
* <math>a\cdot b</math> and <math>b\cdot a</math> are either equal or both undefined
* <math>a\cdot(b+c)</math> and <math>(a\cdot b)+(a\cdot c)</math> are equal if both are defined.
* If <math>a\leq b</math> and if both <math>a+c</math> and <math>b+c</math> are defined, then <math>a+c\leq b+c</math>.
* If <math>a\leq b</math> and <math>c>0</math> and if both <math>a\cdot c</math> and <math>b\cdot c</math> are defined, then <math>a\cdot c\leq b\cdot c</math>.

In general, all laws of arithmetic are valid in <math>\overline\R</math> as long as all occurring expressions are defined.

==Miscellaneous==

Several functions can be [[continuity (topology)|continuously]] [[restriction (mathematics)|extended]] to <math>\overline\R</math> by taking limits. For instance, one may define the extremal points of the following functions as:
:<math>\exp(-\infty)=0</math>,
:<math>\ln(0)=-\infty</math>,
:<math>\tanh(\pm\infty)=\pm1</math>,
:<math>\arctan(\pm\infty)= \pm\frac{\pi}{2}</math>.

Some [[singularity (mathematics)|singularities]] may additionally be removed. For example, the function <math>1/x^2</math> can be continuously extended to <math>\overline\R</math> (under ''some'' definitions of continuity), by setting the value to <math>+\infty</math> for <math>x=0</math>, and 0 for <math>x=+\infty</math> and <math>x=-\infty</math>. On the other hand, the function <math>1/x</math> can''not'' be continuously extended, because the function approaches <math>-\infty</math> as <math>x</math> approaches 0 [[one-sided limit|from below]], and <math>+\infty</math> as <math>x</math> 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]], does not distinguish between <math>+\infty</math> and <math>-\infty</math> (i.e. infinity is unsigned).<ref name=":2">{{Cite web|url=http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html|title=Projectively Extended Real Numbers|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref> As a result, a function may have limit <math>\infty</math> 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 <math>1/x</math> at <math>x=0</math>. On the other hand, on the projectively extended real line, <math>\lim_{x\to-\infty}{f(x)}</math> and <math>\lim_{x\to+\infty}{f(x)}</math> 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 <math>e^x</math> and <math>\arctan(x)</math> cannot be made continuous at <math>x=\infty</math> on the projectively extended real line.

==See also==
* [[Division by zero]]
* [[Extended complex plane]]
* [[Extended natural numbers]]
* [[Improper integral]]
* [[Infinity]]
* [[Log semiring]]
* [[Series (mathematics)]]
* [[Projectively extended real line]]
* Computer representations of extended real numbers, see {{section link|Floating-point arithmetic|Infinities}} and [[IEEE floating point]]

==Notes==
{{notelist}}

== References ==
<references />

==Further reading==
* {{citation
 | last1 = Aliprantis | first1 = Charalambos D.
 | last2 = Burkinshaw | first2 = Owen
 | edition = 3rd
 | isbn = 0-12-050257-7
 | location = San Diego, CA
 | mr = 1669668
 | page = 29
 | publisher = Academic Press, Inc.
 | title = Principles of Real Analysis
 | url = https://books.google.com/books?id=m40ivUwAonUC&pg=PA29
 | year = 1998}}
* {{MathWorld|author= David W. Cantrell|title=Affinely Extended Real Numbers|urlname=AffinelyExtendedRealNumbers}}

{{Real numbers|state=expanded}}
{{Large numbers}}

[[Category:Infinity]]
[[Category:Real numbers]]