Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system {\displaystyle \mathbb {R} }\mathbb {R} by adding two infinity elements: {\displaystyle +\infty }+\infty and {\displaystyle -\infty ,}{\displaystyle -\infty ,}[a] where the infinities are treated as actual numbers. 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.[1] The affinely extended real number system is denoted {\displaystyle {\overline {\mathbb {R} }}}{\displaystyle {\overline {\mathbb {R} }}} or {\displaystyle [-\infty ,+\infty ]}[-\infty, +\infty] or {\displaystyle \mathbb {R} \cup \left{-\infty ,+\infty \right}.}{\displaystyle \mathbb {R} \cup \left{-\infty ,+\infty \right}.}[2] It is the Dedekind–MacNeille completion of the real numbers.

When the meaning is clear from context, the symbol {\displaystyle +\infty }+\infty is often written simply as {\displaystyle \infty .}\infty .[2]

1 Motivation

1.1 Limits

It is often useful to describe the behavior of a function {\displaystyle f}f, as either the argument {\displaystyle x}x or the function value {\displaystyle f}f gets “infinitely large” in some sense. For example, consider the function {\displaystyle f}f defined by

{\displaystyle f(x)={\frac {1}{x^{2}}}.}{\displaystyle f(x)={\frac {1}{x^{2}}}.}
The graph of this function has a horizontal asymptote at {\displaystyle y=0.}{\displaystyle y=0.} Geometrically, when moving increasingly farther to the right along the {\displaystyle x}x-axis, the value of {\textstyle {1}/{x^{2}}}{\textstyle {1}/{x^{2}}} approaches 0. This limiting behavior is similar to the limit of a function {\textstyle \lim {x\to x{0}}f(x)}{\textstyle \lim {x\to x{0}}f(x)} in which the real number {\displaystyle x}x approaches {\displaystyle x_{0},}x_0, except that there is no real number to which {\displaystyle x}x approaches.

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

To make things completely formal, the Cauchy sequences definition of {\displaystyle \mathbb {R} }\mathbb {R} allows defining {\displaystyle +\infty }+\infty as the set of all sequences {\displaystyle \left{a_{n}\right}}{\displaystyle \left{a_{n}\right}} of rational numbers, such that every {\displaystyle M\in \mathbb {R} }{\displaystyle M\in \mathbb {R} } is associated with a corresponding {\displaystyle N\in \mathbb {N} }N \in \N for which {\displaystyle a_{n}>M}{\displaystyle a_{n}>M} for all {\displaystyle n>N.}{\displaystyle n>N.} The definition of {\displaystyle -\infty }-\infty can be constructed similarly.

1.2 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 {\displaystyle \mathbb {R} }\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

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

{\displaystyle f_{n}(x)={\begin{cases}2n\left(1-nx\right),&{\mbox{if }}0\leq x\leq {\frac {1}{n}}\0,&{\mbox{if }}{\frac {1}{n}} 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.

2 Order and topological properties

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

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

3 Arithmetic operations

The arithmetic operations of {\displaystyle \mathbb {R} }\mathbb {R} can be partially extended to {\displaystyle {\overline {\mathbb {R} }}}{\displaystyle {\overline {\mathbb {R} }}} as follows:[2]

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

The expressions {\displaystyle \infty -\infty ,0\times (\pm \infty )}\infty - \infty, 0 \times (\pm\infty) and {\displaystyle \pm \infty /\pm \infty }\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 }0 \times \pm\infty is often defined as {\displaystyle 0.}{\displaystyle 0.}[4]

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

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

4 Algebraic properties

With these definitions, {\displaystyle {\overline {\mathbb {R} }}}{\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} .}{\displaystyle \mathbb {R} .} However, it has several convenient properties:

{\displaystyle a+(b+c)}a + (b + c) and {\displaystyle (a+b)+c}(a + b) + c are either equal or both undefined.
{\displaystyle a+b}a+b and {\displaystyle b+a}b + a are either equal or both undefined.
{\displaystyle a\cdot (b\cdot c)}a\cdot (b\cdot c) and {\displaystyle (a\cdot b)\cdot c}(a\cdot b)\cdot c are either equal or both undefined.
{\displaystyle a\cdot b}a\cdot b and {\displaystyle b\cdot a}b\cdot a are either equal or both undefined
{\displaystyle a\cdot (b+c)}{\displaystyle a\cdot (b+c)} and {\displaystyle (a\cdot b)+(a\cdot c)}{\displaystyle (a\cdot b)+(a\cdot c)} are equal if both are defined.
If {\displaystyle a\leq b}a\leq b and if both {\displaystyle a+c}a + c and {\displaystyle b+c}b+c are defined, then {\displaystyle a+c\leq b+c.}{\displaystyle a+c\leq b+c.}
If {\displaystyle a\leq b}a\leq b and {\displaystyle c>0}c>0 and if both {\displaystyle a\cdot c}{\displaystyle a\cdot c} and {\displaystyle b\cdot c}{\displaystyle b\cdot c} are defined, then {\displaystyle a\cdot c\leq b\cdot c.}{\displaystyle a\cdot c\leq b\cdot c.}
In general, all laws of arithmetic are valid in {\displaystyle {\overline {\mathbb {R} }}}{\displaystyle {\overline {\mathbb {R} }}}—as long as all occurring expressions are defined.

5 Miscellaneous

Several functions can be continuously extended to {\displaystyle {\overline {\mathbb {R} }}}{\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 \exp(-\infty )=0,}
{\displaystyle \ln(0)=-\infty ,}{\displaystyle \ln(0)=-\infty ,}
{\displaystyle \tanh(\pm \infty )=\pm 1,}{\displaystyle \tanh(\pm \infty )=\pm 1,}
{\displaystyle \arctan(\pm \infty )=\pm {\frac {\pi }{2}}.}{\displaystyle \arctan(\pm \infty )=\pm {\frac {\pi }{2}}.}
Some singularities may additionally be removed. For example, the function {\displaystyle 1/x^{2}}1/x{2} can be continuously extended to {\displaystyle {\overline {\mathbb {R} }}}{\displaystyle {\overline {\mathbb {R} }}} (under some definitions of continuity), by setting the value to {\displaystyle +\infty }+\infty for {\displaystyle x=0,}{\displaystyle x=0,} and {\displaystyle 0}0 for {\displaystyle x=+\infty }x = +\infty and {\displaystyle x=-\infty .}{\displaystyle x=-\infty .} On the other hand, the function {\displaystyle 1/x}1/x cannot be continuously extended, because the function approaches {\displaystyle -\infty }-\infty as {\displaystyle x}x approaches {\displaystyle 0}0 from below, and {\displaystyle +\infty }+\infty as {\displaystyle x}x approaches {\displaystyle 0}0 from above.

A similar but different real-line system, the projectively extended real line, does not distinguish between {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty (i.e. infinity is unsigned).[5] As a result, a function may have limit {\displaystyle \infty }\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 {\displaystyle 1/x}1/x at {\displaystyle x=0.}x=0. On the other hand, {\displaystyle \lim {x\to -\infty }{f(x)}}\lim{x \to -\infty}{f(x)} and {\displaystyle \lim {x\to +\infty }{f(x)}}\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 {\displaystyle e^{x}}e{x} and {\displaystyle \arctan(x)}\arctan(x) cannot be made continuous at {\displaystyle x=\infty }x = \infty on the projectively extended real line.

6 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 Floating-point arithmetic § Infinities and IEEE floating point