Inmathematics, theextended real number system^[a] is obtained from thereal number system${\displaystyle \mathbb{R}}$ by adding two elements denoted${\displaystyle +\mathrm{\infty }}$ and${\displaystyle -\mathrm{\infty }}$^[b] that are respectively greater and lower than every real number. This allows for treating thepotential infinities of infinitely increasing sequences and infinitely decreasing series asactual infinities. For example, theinfinite sequence${\displaystyle (1,2,\dots )}$ of thenatural numbers increasesinfinitively and has noupper bound in the real number system (a potential infinity); in the extended real number line, the sequence has${\displaystyle +\mathrm{\infty }}$ as itsleast upper bound and as itslimit (an actual infinity). Incalculus andmathematical analysis, the use of${\displaystyle +\mathrm{\infty }}$ and${\displaystyle -\mathrm{\infty }}$ as actual limits extends significantly the possible computations.^[1] It is theDedekind–MacNeille completion of the real numbers.

The extended real number system is denoted${\displaystyle \overline{\mathbb{R}}}$,${\displaystyle [-\mathrm{\infty },+\mathrm{\infty }]}$, or${\displaystyle \mathbb{R}\cup \left\{-\mathrm{\infty },+\mathrm{\infty }\right\}}$.^[2] When the meaning is clear from context, the symbol${\displaystyle +\mathrm{\infty }}$ is often written simply as${\displaystyle \mathrm{\infty }}$.^[2]

There is also a distinctprojectively extended real line where${\displaystyle +\mathrm{\infty }}$ and${\displaystyle -\mathrm{\infty }}$ 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${\displaystyle \mathrm{\infty }}$ or as${\displaystyle \pm \mathrm{\infty }}$.

The extended number line is often useful to describe the behavior of afunction${\displaystyle f}$ when either theargument${\displaystyle x}$ or the function value${\displaystyle f}$ gets "infinitely large" in some sense. For example, consider the function${\displaystyle f}$ defined by

${\displaystyle f(x)=\frac{1}{x^2}}$.

Thegraph of this function has a horizontalasymptote at${\displaystyle y=0}$. Geometrically, when moving increasingly farther to the right along the${\displaystyle x}$-axis, the value of$1/x^2$approaches 0. This limiting behavior is similar to thelimit of a function$\underset{x\to x_0}{lim}f(x)$ in which thereal number${\displaystyle x}$ approaches${\displaystyle x_0,}$ except that there is no real number that${\displaystyle x}$ approaches when${\displaystyle x}$ increases infinitely. Adjoining the elements${\displaystyle +\mathrm{\infty }}$ and${\displaystyle -\mathrm{\infty }}$ to${\displaystyle \mathbb{R}}$ enables a definition of "limits at infinity" which is very similar to the usual definition of limits, except that is replaced by${\displaystyle x>N}$ (for${\displaystyle +\mathrm{\infty }}$) or (for${\displaystyle -\mathrm{\infty }}$). This allows proving and writing

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Inmeasure theory, it is often useful to allow sets that have infinitemeasure and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure to${\displaystyle \mathbb{R}}$ that agrees with the usual length ofintervals, this measure must be larger than any finite real number. Also, when consideringimproper integrals, such as

${\displaystyle {\int }_1^{\mathrm{\infty }}\frac{dx}{x}}$

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 themonotone convergence theorem and thedominated convergence theorem would not make sense.

The extended real number system${\displaystyle \overline{\mathbb{R}}}$, defined as${\displaystyle [-\mathrm{\infty },+\mathrm{\infty }]}$ or${\displaystyle \mathbb{R}\cup \left\{-\mathrm{\infty },+\mathrm{\infty }\right\}}$, can be turned into atotally ordered set by defining${\displaystyle -\mathrm{\infty }\le a\le +\mathrm{\infty }}$ for all${\displaystyle a\in \overline{\mathbb{R}}}$. With thisorder topology,${\displaystyle \overline{\mathbb{R}}}$ has the desirable property ofcompactness: Everysubset of${\displaystyle \overline{\mathbb{R}}}$ has asupremum and aninfimum^[2] (the infimum of theempty set is${\displaystyle +\mathrm{\infty }}$, and its supremum is${\displaystyle -\mathrm{\infty }}$). Moreover, with thistopology,${\displaystyle \overline{\mathbb{R}}}$ ishomeomorphic to theunit interval${\displaystyle [0,1]}$. Thus the topology ismetrizable, corresponding (for a given homeomorphism) to the ordinarymetric on this interval. There is no metric, however, that is an extension of the ordinary metric on${\displaystyle \mathbb{R}}$.

In this topology, a set${\displaystyle U}$ is aneighborhood of${\displaystyle +\mathrm{\infty }}$ if and only if it contains a set${\displaystyle \{x:x>a\}}$ for some real number${\displaystyle a}$. The notion of the neighborhood of${\displaystyle -\mathrm{\infty }}$ can be defined similarly. Using this characterization of extended-real neighborhoods,limits with${\displaystyle x}$ tending to${\displaystyle +\mathrm{\infty }}$ or${\displaystyle -\mathrm{\infty }}$, and limits "equal" to${\displaystyle +\mathrm{\infty }}$ and${\displaystyle -\mathrm{\infty }}$, reduce to the general topological definition of limits—instead of having a special definition in the real number system.

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

For exponentiation, seeExponentiation § Limits of powers. Here,${\displaystyle a+\mathrm{\infty }}$ means both${\displaystyle a+(+\mathrm{\infty })}$ and${\displaystyle a-(-\mathrm{\infty })}$, while${\displaystyle a-\mathrm{\infty }}$ means both${\displaystyle a-(+\mathrm{\infty })}$ and${\displaystyle a+(-\mathrm{\infty })}$.

The expressions${\displaystyle \mathrm{\infty }-\mathrm{\infty }}$,${\displaystyle 0\times (\pm \mathrm{\infty })}$, and${\displaystyle \pm \mathrm{\infty }/\pm \mathrm{\infty }}$ (calledindeterminate forms) are usually leftundefined. These rules are modeled on the laws forinfinite limits. However, in the context ofprobability or measure theory,${\displaystyle 0\times \pm \mathrm{\infty }}$ is often defined as 0.^[4]

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

However, in contexts where only non-negative values are considered, it is often convenient to define${\displaystyle 1/0=+\mathrm{\infty }}$. For example, when working withpower series, theradius of convergence of a power series withcoefficients${\displaystyle a_n}$ is often defined as the reciprocal of thelimit-supremum of the sequence${\displaystyle \left(|a_n{|}^{1/n}\right)}$. Thus, if one allows${\displaystyle 1/0}$ to take the value${\displaystyle +\mathrm{\infty }}$, then one can use this formula regardless of whether the limit-supremum is 0 or not.

With the arithmetic operations defined above,${\displaystyle \overline{\mathbb{R}}}$ is not even asemigroup, let alone agroup, aring or afield as in the case of${\displaystyle \mathbb{R}}$. However, it has several convenient properties:

• ${\displaystyle a+(b+c)}$ and${\displaystyle (a+b)+c}$ are either equal or both undefined.
• ${\displaystyle a+b}$ and${\displaystyle b+a}$ are either equal or both undefined.
• ${\displaystyle a\cdot (b\cdot c)}$ and${\displaystyle (a\cdot b)\cdot c}$ are either equal or both undefined.
• ${\displaystyle a\cdot b}$ and${\displaystyle b\cdot a}$ are either equal or both undefined
• ${\displaystyle a\cdot (b+c)}$ and${\displaystyle (a\cdot b)+(a\cdot c)}$ are equal if both are defined.
• If${\displaystyle a\le b}$ and if both${\displaystyle a+c}$ and${\displaystyle b+c}$ are defined, then${\displaystyle a+c\le b+c}$.
• If${\displaystyle a\le b}$ and${\displaystyle c>0}$ and if both${\displaystyle a\cdot c}$ and${\displaystyle b\cdot c}$ are defined, then${\displaystyle a\cdot c\le b\cdot c}$.

In general, all laws of arithmetic are valid in${\displaystyle \overline{\mathbb{R}}}$ as long as all occurring expressions are defined.

Several functions can becontinuouslyextended to${\displaystyle \overline{\mathbb{R}}}$ by taking limits. For instance, one may define the extremal points of the following functions as:

${\displaystyle \exp (-\mathrm{\infty })=0}$,

${\displaystyle \ln (0)=-\mathrm{\infty }}$,

${\displaystyle \tanh (\pm \mathrm{\infty })=\pm 1}$,

${\displaystyle \arctan (\pm \mathrm{\infty })=\pm \frac{\pi }{2}}$.

Somesingularities may additionally be removed. For example, the function${\displaystyle 1/x^2}$ can be continuously extended to${\displaystyle \overline{\mathbb{R}}}$ (undersome definitions of continuity), by setting the value to${\displaystyle +\mathrm{\infty }}$ for${\displaystyle x=0}$, and 0 for${\displaystyle x=+\mathrm{\infty }}$ and${\displaystyle x=-\mathrm{\infty }}$. On the other hand, the function${\displaystyle 1/x}$ cannot be continuously extended, because the function approaches${\displaystyle -\mathrm{\infty }}$ as${\displaystyle x}$ approaches 0from below, and${\displaystyle +\mathrm{\infty }}$ as${\displaystyle x}$ 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, theprojectively extended real line, does not distinguish between${\displaystyle +\mathrm{\infty }}$ and${\displaystyle -\mathrm{\infty }}$ (i.e. infinity is unsigned).^[4] As a result, a function may have limit${\displaystyle \mathrm{\infty }}$ on the projectively extended real line, while in the extended real number system only theabsolute value of the function has a limit, e.g. in the case of the function${\displaystyle 1/x}$ at${\displaystyle x=0}$. On the other hand, on the projectively extended real line,${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}f(x)}$ and${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}f(x)}$ 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${\displaystyle e^x}$ and${\displaystyle \arctan (x)}$ cannot be made continuous at${\displaystyle x=\mathrm{\infty }}$ on the projectively extended real line.

• 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, seeFloating-point arithmetic § Infinities andIEEE floating point

• 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.

• Infinity
• Real numbers

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