Extended real number line

The extended real number system ${\displaystyle {\overline {\mathbb {R} }}}$, defined as ${\displaystyle [-\infty ,+\infty ]}$ or ${\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}}$, can be turned into a totally ordered set by defining ${\displaystyle -\infty \leq a\leq +\infty }$ for all ${\displaystyle a\in {\overline {\mathbb {R} }}.}$ With this order topology, ${\displaystyle {\overline {\mathbb {R} }}}$ has the desirable property of compactness: Every subset of ${\displaystyle {\overline {\mathbb {R} }}}$ has a supremum and an infimum[3] (the infimum of the empty set is ${\displaystyle +\infty }$, and its supremum is ${\displaystyle -\infty }$). Moreover, with this topology, ${\displaystyle {\overline {\mathbb {R} }}}$ is homeomorphic to the unit interval ${\displaystyle [0,1].}$ 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 ${\displaystyle \mathbb {R} .}$

In this topology, a set ${\displaystyle U}$ is a neighborhood of ${\displaystyle +\infty }$ if and only if it contains a set ${\displaystyle \{x:x>a\}}$ for some real number ${\displaystyle a.}$ The notion of the neighborhood of ${\displaystyle -\infty }$ can be defined similarly. Using this characterization of extended-real neighborhoods, limits with ${\displaystyle x}$ tending to ${\displaystyle +\infty }$ or ${\displaystyle -\infty }$, and limits "equal" to ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$, 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:[2]

$${\displaystyle {\begin{aligned}a\pm \infty =\pm \infty +a&=\pm \infty ,&a&\neq \mp \infty \\a\cdot (\pm \infty )=\pm \infty \cdot a&=\pm \infty ,&a&\in (0,+\infty ]\\a\cdot (\pm \infty )=\pm \infty \cdot a&=\mp \infty ,&a&\in [-\infty ,0)\\{\frac {a}{\pm \infty }}&=0,&a&\in \mathbb {R} \\{\frac {\pm \infty }{a}}&=\pm \infty ,&a&\in (0,+\infty )\\{\frac {\pm \infty }{a}}&=\mp \infty ,&a&\in (-\infty ,0)\end{aligned}}}$$

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

The expressions ${\displaystyle \infty -\infty ,0\times (\pm \infty )}$ and ${\displaystyle \pm \infty /\pm \infty }$ (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, ${\displaystyle 0\times \pm \infty }$ is often defined as ${\displaystyle 0.}$[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 ${\displaystyle f}$ that converges to ${\displaystyle 0,}$ the reciprocal sequence ${\displaystyle 1/f}$ is eventually contained in every neighborhood of ${\displaystyle \{\infty ,-\infty \},}$ it is not true that the sequence ${\displaystyle 1/f}$ must itself converge to either ${\displaystyle -\infty }$ or ${\displaystyle \infty .}$ Said another way, if a continuous 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 ${\displaystyle -\infty }$ or ${\displaystyle \infty }$ in the limit as ${\displaystyle x}$ tends to ${\displaystyle x_{0}.}$ This is the case for the limits of the identity function ${\displaystyle f(x)=x}$ when ${\displaystyle x}$ tends to ${\displaystyle 0,}$ and of ${\displaystyle f(x)=x^{2}\sin \left(1/x\right)}$ (for the latter function, neither ${\displaystyle -\infty }$ nor ${\displaystyle \infty }$ 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 ${\displaystyle 1/0=+\infty .}$ For example, when working with power series, the radius of convergence of a power series with coefficients ${\displaystyle a_{n}}$ is often defined as the reciprocal of the limit-supremum of the sequence ${\displaystyle \left(|a_{n}|^{1/n}\right)}$. Thus, if one allows ${\displaystyle 1/0}$ to take the value ${\displaystyle +\infty ,}$ then one can use this formula regardless of whether the limit-supremum is ${\displaystyle 0}$ or not.

With these definitions, ${\displaystyle {\overline {\mathbb {R} }}}$ is not even a semigroup, let alone a group, a ring or a field 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 ${\displaystyle a\leq b}$ and if both ${\displaystyle a+c}$ and ${\displaystyle b+c}$ are defined, then ${\displaystyle a+c\leq b+c.}$
• If ${\displaystyle a\leq b}$ and ${\displaystyle c>0}$ and if both ${\displaystyle a\cdot c}$ and ${\displaystyle b\cdot c}$ are defined, then ${\displaystyle a\cdot c\leq b\cdot c.}$

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

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

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

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

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

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

Some singularities may additionally be removed. For example, the function ${\displaystyle 1/x^{2}}$ can be continuously extended to ${\displaystyle {\overline {\mathbb {R} }}}$ (under some definitions of continuity), by setting the value to ${\displaystyle +\infty }$ for ${\displaystyle x=0,}$ and ${\displaystyle 0}$ for ${\displaystyle x=+\infty }$ and ${\displaystyle x=-\infty .}$ On the other hand, the function ${\displaystyle 1/x}$ cannot be continuously extended, because the function approaches ${\displaystyle -\infty }$ as ${\displaystyle x}$ approaches ${\displaystyle 0}$ from below, and ${\displaystyle +\infty }$ as ${\displaystyle x}$ approaches ${\displaystyle 0}$ 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 ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ (i.e. infinity is unsigned).[5] As a result, a function may have limit ${\displaystyle \infty }$ 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 ${\displaystyle 1/x}$ at ${\displaystyle x=0.}$ On the other hand, on the projectively extended real line, ${\displaystyle \lim _{x\to -\infty }{f(x)}}$ and ${\displaystyle \lim _{x\to +\infty }{f(x)}}$ 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 ${\displaystyle x=\infty }$ 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, see Floating-point arithmetic § Infinities and IEEE 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.