Extended real number line

(Redirected from Negative infinity)

In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: + ∞ and − ∞ (read as positive infinity and negative infinity respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted ${\displaystyle {\overline {\mathbb {R} }}}$ or [−∞, +∞] or ℝ ∪ {−∞, +∞}.

When the meaning is clear from context, the symbol +∞ is often written simply as .

Motivation

Limits

We often wish to describe the behavior of a function ${\displaystyle f(x)}$ , as either the argument ${\displaystyle x}$  or the function value ${\displaystyle f(x)}$  gets "infinitely large" in some sense. For example, consider the function

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

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

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

To make things completely formal, the Cauchy sequences definition of ${\displaystyle \mathbb {R} }$  allows us to define ${\displaystyle +\infty }$  as the set of all sequences of rationals which, for any ${\displaystyle K>0}$ , from some point on exceed ${\displaystyle K}$ . We can define ${\displaystyle -\infty }$  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 ${\displaystyle \mathbb {R} }$  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

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

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(1-nx),&{\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.

Order and topological properties

The affinely extended real number system turns into a totally ordered set by defining ${\displaystyle -\infty \leq a\leq +\infty }$  for all ${\displaystyle a}$ . This order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice.

This induces the order topology on ${\displaystyle {\overline {\mathbb {R} }}}$ . 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 ${\displaystyle a}$ , and analogously for the neighborhoods of ${\displaystyle -\infty }$ . ${\displaystyle {\overline {\mathbb {R} }}}$  is a compact Hausdorff space 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 that is an extension of the ordinary metric on ${\displaystyle \mathbb {R} }$ .

With this topology, the specially defined limits for ${\displaystyle x}$  tending to ${\displaystyle +\infty }$  and ${\displaystyle -\infty }$ , and the specially defined concepts of limits equal to ${\displaystyle +\infty }$  and ${\displaystyle -\infty }$ , reduce to the general topological definitions of limits.

Arithmetic operations

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

{\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}}}

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 ${\displaystyle 0}$ .[citation needed]

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 ${\displaystyle x\to x_{0}}$ . This is the case for the limits when ${\displaystyle x\to 0}$  of the identity function ${\displaystyle f(x)=x,}$  and of ${\displaystyle f(x)=x^{2}\sin(1/x)}$  (for the latter function, neither ${\displaystyle -\infty }$  nor ${\displaystyle \infty }$  is a limit of ${\displaystyle 1/f(x),}$  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 ${\displaystyle 1/0=+\infty }$ . For example, when working with power series, it is often convenient to define the radius of convergence of a power series with coefficients ${\displaystyle a_{n}}$  as the reciprocal of the limit-supremum of the sequence ${\displaystyle \{|(a_{n})^{1/n}|\}}$ . 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, since both the radius of convergence and the limit-supremum are non-negative quantities. As another example, in the theory of differentiable curves, the radius of curvature is defined as the reciprocal of the curvature constant of the curve. Since both quantities are non-negative, this definition can be used even when the curvature constant is not zero, if we define ${\displaystyle 1/0=+\infty }$ .

Algebraic properties

With these definitions ${\displaystyle {\overline {\mathbb {R} }}}$  is not even a semigroup, let alone a group, a ring or a field, like ${\displaystyle \mathbb {R} }$  is one. However, it nevertheless does form a complete metric space, and 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.

Miscellaneous

Several functions can be continuously extended to ${\displaystyle {\overline {\mathbb {R} }}}$  by taking limits. For instance, one may define ${\displaystyle \exp(-\infty )=0,\ \exp(+\infty )=+\infty ,\ \ln(0)=-\infty ,\ \ln(+\infty )=+\infty }$ , etc.

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 ${\displaystyle x=-\infty }$ . The function ${\displaystyle 1/x}$  can not 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.

Compare the projectively extended real line, which does not distinguish between ${\displaystyle +\infty }$  and ${\displaystyle -\infty }$ . As a result, on one hand a function may have limit ${\displaystyle \infty }$  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 ${\displaystyle 1/x}$  at ${\displaystyle x=0}$ . On the other hand

${\displaystyle \lim _{x\to -\infty }{f(x)}}$  and ${\displaystyle \lim _{x\to +\infty }{f(x)}}$

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 ${\displaystyle e^{x}}$  and ${\displaystyle \arctan(x)}$  cannot be made continuous at ${\displaystyle x=\infty }$  on the projectively extended real line.