Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system $ℝ$ by adding two elements: $+ \infty$ and $- \infty$ (read as positive infinity and negative infinity respectively), where the infinities are treated as actual numbers.[1] It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration.[2] The affinely extended real number system is denoted ${\overline {\mathbb {R} }}$ or $[-\infty, +\infty]$ or $ℝ \cup {-\infty, +\infty}$ .[3]

When the meaning is clear from context, the symbol $+\infty$ is often written simply as $\infty$ .[3]

Motivation

Limits

It is often useful to describe the behavior of a function $f(x)$ , as either the argument $x$ or the function value $f(x)$ gets "infinitely large" in some sense. For example, consider the function

f(x)=x^{-2}.

The graph of this function has a horizontal asymptote at y = 0. Geometrically, when moving increasingly farther to the right along the $x$ -axis, the value of $\textstyle {\frac {1}{x^{2}}}$ 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 $x$ approaches.

By adjoining the elements $+\infty$ and $-\infty$ to $\mathbb {R}$ , it enables a formulation of a "limit at infinity", with topological properties similar to those for $\mathbb {R}$ .

To make things completely formal, the Cauchy sequences definition of $\mathbb {R}$ allows defining $+\infty$ as the set of all sequences $\{a_{n}\}$ of rational numbers, such that every $M\in \mathbb {R}$ is associated with a corresponding $N\in \mathbb {N}$ 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 $\mathbb {R}$ 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

\int _{1}^{\infty }{\frac {dx}{x}}

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

f_{n}(x)={\begin{cases}2n(1-nx),&{\mbox{if }}0\leq x\leq {\frac {1}{n}}\\0,&{\mbox{if }}{\frac {1}{n}}<x\leq 1\end{cases}}

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 affinely extended real number system can be turned into a totally ordered set, by defining $-\infty \leq a\leq +\infty$ for all $a$ . With this order topology, ${\overline {\mathbb {R} }}$ has the desirable property of compactness: every subset of ${\overline {\mathbb {R} }}$ has a supremum and an infimum[4] (the infimum of the empty set is $+\infty ,$ and its supremum is $-\infty$ ). Moreover, with this topology, ${\overline {\mathbb {R} }}$ is homeomorphic to the unit interval $[0,1]$ . 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 $\mathbb {R}$ .

In this topology, a set $U$ is a neighborhood of $+\infty$ , if and only if it contains a set $\{x:x>a\}$ for some real number $a$ . The notion of the neighborhood of $-\infty$ can be defined similarly. Using this characterization of extended-real neighborhoods, the specially defined limits for $x$ tending to $+\infty$ and $-\infty$ , and the specially defined concepts of limits equal to $+\infty$ and $-\infty$ , reduce to the general topological definition of limits.

Arithmetic operations

The arithmetic operations of $\mathbb {R}$ can be partially extended to ${\overline {\mathbb {R} }}$ as follows:[3]

{\begin{aligned}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 {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{aligned}}

For exponentiation, see Exponentiation#Limits of powers. 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 undefined. These rules are modeled on the laws for infinite limits. However, in the context of probability or measure theory, $0\times \pm \infty$ is often defined as $0$ .[5]

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 $\{\infty ,-\infty \}$ , 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 $f(x)=x$ when $x$ tends to 0, and of $f(x)=x^{2}\sin(1/x)$ (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 of a power series with coefficients $a_{n}$ is often defined as the reciprocal of the limit-supremum of the sequence $\{|a_{n}|^{1/n}\}$ . 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, ${\overline {\mathbb {R} }}$ is not even a semigroup, let alone a group, a ring or a field as in the case of $\mathbb {R}$ . 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 ${\overline {\mathbb {R} }}$ —as long as all occurring expressions are defined.

Miscellaneous

Several functions can be continuously extended to ${\overline {\mathbb {R} }}$ by taking limits. For instance, one may define the extremal points of the following functions as follow: $\exp(-\infty )=0,\ \ln(0)=-\infty ,\ \tanh(\pm \infty )=\pm 1,\ \arctan(\pm \infty )=\pm {\frac {\pi }{2}}$

Some singularities may additionally be removed. For example, the function $1/x^{2}$ can be continuously extended to ${\overline {\mathbb {R} }}$ (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, does not distinguish between $+\infty$ and $-\infty$ (i.e., infinity is unsigned).[6] 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 _{x\to -\infty }{f(x)}

and

\lim _{x\to +\infty }{f(x)}

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.

References

"The Definitive Glossary of Higher Mathematical Jargon — Infinite". Math Vault. 2019-08-01. Retrieved 2019-12-03.
Wilkins, David (2007). "Section 6: The Extended Real Number System" (PDF). maths.tcd.ie. Retrieved 2019-12-03.
Weisstein, Eric W. "Affinely Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03.
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.
"extended real number in nLab". ncatlab.org. Retrieved 2019-12-03.
Weisstein, Eric W. "Projectively Extended Real Numbers". mathworld.wolfram.com. Retrieved 2019-12-03.