Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system $ℝ$ by adding two elements: $+ \infty$ and $- \infty$ (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 $\overline{\mathbb{R}}$ or $[-\infty, +\infty]$ or $ℝ \cup {-\infty, +\infty}$ .

When the meaning is clear from context, the symbol $+\infty$ is often written simply as $\infty$ .

Motivation

Limits

We often wish to describe the behavior of a function $f(x)$ , as either the argument $x$ or the function value $f(x)$ gets "very big" in some sense. For example, consider the function

f(x) = x^{-2}.

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

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

To make things completely formal, the Cauchy sequences definition of $\mathbb {R}$ allows us to define $+\infty$ as the set of all sequences of rationals which, for any $K>0$ , from some point on exceed $K$ . We can define $-\infty$ 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 $\mathbb {R}$ that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals, such as

\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

f_n(x) = \begin{cases} 2n(1-nx), & \mbox{if } 0 \le x \le \frac{1}{n} \\ 0, & \mbox{if } \frac{1}{n} < x \le 1\end{cases}

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 $-\infty\leq a \leq +\infty$ for all $a$ . 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 $\overline{\mathbb{R}}$ . In this topology, a set $U$ is a neighborhood of $+\infty$ if and only if it contains a set $\{x : x > a\}$ for some real number $a$ , and analogously for the neighborhoods of $-\infty$ . $\overline\mathbb R$ is a compact Hausdorff space 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 that is an extension of the ordinary metric on $\mathbb {R}$ .

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

Arithmetic operations

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

\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, " $a + \infty$ " means both " $a + (+\infty)$ " and " $a - (-\infty)$ ", while " $a - \infty$ " means both " $a - (+\infty)$ " and " $a + (-\infty)$ ".

The expressions $\infty - \infty, 0 \times (\pm\infty)$ and $\pm\infty/\pm\infty$ (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, $0 \times \pm\infty$ is often defined as $0$ .

The expression $1/0$ is not defined either as $+\infty$ or $-\infty$ , because although it is true that whenever $f(x) \to 0$ for a continuous function $f(x)$ it must be the case that $1/f(x)$ is eventually contained in every neighborhood of the set $\{-\infty, +\infty\}$ , it is not true that $1/f(x)$ must tend to one of these points. An example is $g(x)=1/\sin(1/x)$ which is of the form $1/f(x)$ but does not tend to either $+\infty$ or $-\infty$ when $\sin(1/x)\to 0$ . For instance, $\lim _{x\to 1/\pi }{\sin(1/x)}=0$ but $\lim _{x\to 1/\pi }{{1} \over {\sin(1/x)}}$ does not exist because $\lim _{x\to {1/\pi }^{+}}{{1} \over {\sin(1/x)}}=+\infty$ but $\lim _{x\to {1/\pi }^{-}}{{1} \over {\sin(1/x)}}=-\infty$ . (The modulus $|1/f(x)|$ , nevertheless, does approach $+\infty$ .)

Algebraic properties

With these definitions $\overline\mathbb{R}$ is not even a semigroup, let alone a group, a ring or a field, like $\mathbb {R}$ is one. However, it still 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.
$a \times (b \times c)$ and $(a \times b) \times c$ are either equal or both undefined.
$a\times b$ and $b \times a$ are either equal or both undefined
$a \times (b + c)$ and $(a \times b) + (a \times c)$ are equal if both are defined.
If $a\leq b$ and if both $a + c$ and $b+c$ are defined, then $a + c \leq b + c$ .
If $a\leq b$ and $c>0$ and if both $a \times c$ and $b \times c$ are defined, then $a \times c \leq b \times c$ .

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

Miscellaneous

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

Some singularities may additionally be removed. For example, the function $1/x^{2}$ can be continuously extended to $\overline\mathbb{R}$ (under some definitions of continuity) by setting the value to $+\infty$ for $x=0$ , and $0$ for $x = +\infty$ and $x = -\infty$ . The function $1/x$ can not be continuously extended because the function approaches $-\infty$ as $x$ approaches 0 from below, and $+\infty$ as $x$ approaches $0$ from above.

Compare the projectively extended real line, which does not distinguish between $+\infty$ and $-\infty$ . As a result, on one hand a function may have limit $\infty$ 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

\lim_{x \to -\infty}{f(x)}

and

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

References

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.

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