In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, the affinely extended real number system is obtained from the real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

system $\backslash R$ by adding two infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything t ...

elements: $+\backslash infty$ and $-\backslash infty,$ where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various s in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

and mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

, especially in the theory of measure and . The affinely extended real number system is denoted $\backslash overline$ or $;\; href="/html/ALL/s/\backslash infty,\_+\backslash infty.html"\; ;"title="\backslash infty,\; +\backslash infty">\backslash infty,\; +\backslash infty$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 :$f(x)\; =\; \backslash frac.$ The graph of this function has a horizontalasymptote
250px, A curve intersecting an asymptote infinitely many times.
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc ...

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
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches ze ...

$\backslash lim\_\; f(x)$ in which the real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$x$ approaches $x\_0,$ except that there is no real number to which $x$ approaches.
By adjoining the elements $+\backslash infty$ and $-\backslash infty$ to $\backslash R,$ it enables a formulation of a "limit at infinity", with topological
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

properties similar to those for $\backslash R.$
To make things completely formal, the Cauchy sequences definition of $\backslash R$ allows defining $+\backslash infty$ as the set of all sequences $\backslash left\backslash $ of rational numbers, such that every $M\; \backslash in\; \backslash R$ is associated with a corresponding $N\; \backslash in\; \backslash N$ for which $a\_n\; >\; M$ for all $n\; >\; N.$ The definition of $-\backslash infty$ can be constructed similarly.
Measure and integration

Inmeasure theory
Measure is a fundamental concept of . Measures provide a mathematical abstraction for common notions like , /, , , of events, and — after — . These seemingly distinct concepts are innately very similar and may, in many cases, be treated a ...

, 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 $\backslash R$ 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

s, such as
:$\backslash int\_1^\backslash frac$
the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as
:$f\_n(x)\; =\; \backslash begin\; 2n\backslash left(1-nx\backslash right),\; \&\; \backslash mbox\; 0\; \backslash leq\; x\; \backslash leq\; \backslash frac\; \backslash \backslash \; 0,\; \&\; \backslash mbox\; \backslash frac\; <\; x\; \backslash leq\; 1\; \backslash end$
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 $-\backslash infty\; \backslash leq\; a\; \backslash leq\; +\backslash infty$ for all $a.$ With this order topology, $\backslash overline$ has the desirable property of Compact space, compactness: Every subset of $\backslash overline\backslash R$ has a supremum and an infimum (the infimum of the empty set is $+\backslash infty$, and its supremum is $-\backslash infty$). Moreover, with this topology, $\backslash overline\backslash R$ is Homeomorphism, 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, however, that is an extension of the ordinary metric on $\backslash R.$ In this topology, a set $U$ is a Neighborhood (topology), neighborhood of $+\backslash infty$, if and only if it contains a set $\backslash $ for some real number $a.$ The notion of the neighborhood of $-\backslash infty$ can be defined similarly. Using this characterization of extended-real neighborhoods, Limit of a function, limits with $x$ tending to $+\backslash infty$ or $-\backslash infty$, and limits "equal" to $+\backslash infty$ and $-\backslash infty$, 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 $\backslash R$ can be partially extended to $\backslash overline\backslash R$ as follows: :$\backslash begin\; a\; +\; \backslash infty\; =\; +\backslash infty\; +\; a\; \&\; =\; +\backslash infty,\; \&\; a\; \&\; \backslash neq\; -\backslash infty\; \backslash \backslash \; a\; -\; \backslash infty\; =\; -\backslash infty\; +\; a\; \&\; =\; -\backslash infty,\; \&\; a\; \&\; \backslash neq\; +\backslash infty\; \backslash \backslash \; a\; \backslash cdot\; (\backslash pm\backslash infty)\; =\; \backslash pm\backslash infty\; \backslash cdot\; a\; \&\; =\; \backslash pm\backslash infty,\; \&\; a\; \&\; \backslash in\; (0,\; +\backslash infty]\; \backslash \backslash \; a\; \backslash cdot\; (\backslash pm\backslash infty)\; =\; \backslash pm\backslash infty\; \backslash cdot\; a\; \&\; =\; \backslash mp\backslash infty,\; \&\; a\; \&\; \backslash in\; [-\backslash infty,\; 0)\; \backslash \backslash \; \backslash frac\; \&\; =\; 0,\; \&\; a\; \&\; \backslash in\; \backslash mathbb\; \backslash \backslash \; \backslash frac\; \&\; =\; \backslash pm\backslash infty,\; \&\; a\; \&\; \backslash in\; (0,\; +\backslash infty)\; \backslash \backslash \; \backslash frac\; \&\; =\; \backslash mp\backslash infty,\; \&\; a\; \&\; \backslash in\; (-\backslash infty,\; 0)\; \backslash end$ For exponentiation, see . Here, $a\; +\; \backslash infty$ means both $a\; +\; (+\backslash infty)$ and $a\; -\; (-\backslash infty),$ while $a\; -\; \backslash infty$ means both $a\; -\; (+\backslash infty)$ and $a\; +\; (-\backslash infty).$ The expressions $\backslash infty\; -\; \backslash infty,\; 0\; \backslash times\; (\backslash pm\backslash infty)$ and $\backslash pm\backslash infty/\backslash pm\backslash infty$ (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, $0\; \backslash times\; \backslash pm\backslash infty$ 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 $\backslash ,$ it is ''not'' true that the sequence $1/f$ must itself converge to either $-\backslash infty$ or $\backslash infty.$ 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 $-\backslash infty$ or $\backslash infty$ 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 $f(x)\; =\; x^2\; \backslash sin\; \backslash left(\; 1/x\; \backslash right)$ (for the latter function, neither $-\backslash infty$ nor $\backslash infty$ 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 $1/0\; =\; +\backslash infty.$ 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 $\backslash left\backslash .$ Thus, if one allows $1/0$ to take the value $+\backslash infty,$ then one can use this formula regardless of whether the limit-supremum is $0$ or not.Algebraic properties

With these definitions, $\backslash overline\backslash R$ 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 $\backslash R.$ 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. * $a\; \backslash cdot\; (b\; \backslash cdot\; c)$ and $(a\; \backslash cdot\; b)\; \backslash cdot\; c$ are either equal or both undefined. * $a\; \backslash cdot\; b$ and $b\; \backslash cdot\; a$ are either equal or both undefined * $a\; \backslash cdot\; (b\; +\; c)$ and $(a\; \backslash cdot\; b)\; +\; (a\; \backslash cdot\; c)$ are equal if both are defined. * If $a\; \backslash leq\; b$ and if both $a\; +\; c$ and $b\; +\; c$ are defined, then $a\; +\; c\; \backslash leq\; b\; +\; c.$ * If $a\; \backslash leq\; b$ and $c\; >\; 0$ and if both $a\; \backslash cdot\; c$ and $b\; \backslash cdot\; c$ are defined, then $a\; \backslash cdot\; c\; \backslash leq\; b\; \backslash cdot\; c.$ In general, all laws of arithmetic are valid in $\backslash overline\backslash R$—as long as all occurring expressions are defined.Miscellaneous

Several function (mathematics), functions can be continuity (topology), continuously extended to $\backslash overline\backslash R$ by taking limits. For instance, one may define the extremal points of the following functions as: :$\backslash exp(-\backslash infty)\; =\; 0,$ :$\backslash ln(0)\; =\; -\backslash infty,$ :$\backslash tanh(\backslash pm\backslash infty)\; =\; \backslash pm\; 1,$ :$\backslash arctan(\backslash pm\backslash infty)\; =\; \backslash pm\backslash frac.$ Some Singularity (mathematics), singularities may additionally be removed. For example, the function $1/x^2$ can be continuously extended to $\backslash overline\backslash R$ (under ''some'' definitions of continuity), by setting the value to $+\backslash infty$ for $x\; =\; 0,$ and $0$ for $x\; =\; +\backslash infty$ and $x\; =\; -\backslash infty.$ On the other hand, the function $1/x$ can ''not'' be continuously extended, because the function approaches $-\backslash infty$ as $x$ approaches $0$ from below, and $+\backslash infty$ as $x$ approaches $0$ from above. A similar but different real-line system, the projectively extended real line, does not distinguish between $+\backslash infty$ and $-\backslash infty$ (i.e. infinity is unsigned). As a result, a function may have limit $\backslash infty$ 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, $\backslash lim\_$ and $\backslash lim\_$ 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 $\backslash arctan(x)$ cannot be made continuous at $x\; =\; \backslash infty$ 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 and IEEE floating pointNotes

References

Further reading

* * {{Large numbers Infinity Real numbers