Extended real number line explained

In mathematics, the extended real number system is obtained from the real number system

by adding two elements denoted

+infty

and

-infty

that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence

(1,2,\ldots)

of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has

+infty

as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of

+infty

and

-infty

as actual limits extends significantly the possible computations.^[1] It is the Dedekind–MacNeille completion of the real numbers.

The extended real number system is denoted

\overline{\R}

[-infty,+infty]

or When the meaning is clear from context, the symbol

+infty

is often written simply as

There is also a distinct projectively extended real line where

+infty

and

-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

infty

or as

\pminfty

Motivation

Limits

when either the argument

or the function value

gets "infinitely large" in some sense. For example, consider the function

defined by

f(x)=

	1
	x²

The graph of this function has a horizontal asymptote at

y=0.

Geometrically, when moving increasingly farther to the right along the

-axis, the value of

/

approaches . This limiting behavior is similar to the limit of a function

\lim_ f(x)

in which the real number

approaches

x_0,

except that there is no real number that

approaches when

increases infinitely. Adjoining the elements

+infty

and

-infty

enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, except that

|x-x_{0|<\varepsilon}

is replaced by

x>N

(for

+infty

) or

x<-N

(for

-infty

). This allows proving and writing

\begin{align} \lim_x

	1{x
	^2}

&=0,\\ \lim_x

	1{x
	^2}

&=0,\\ \lim_x

	1{x
	^2}

&=+infty. \end{align}

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

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

	infty
\int
	1

	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),&if0\leqx\leq

	1
	n

\\ 0,&if

	1
	n

<x\leq1 \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 extended real number system

\overline{\R}

, defined as

[-infty,+infty]

or , can be turned into a totally ordered set by defining

-infty\leqa\leq+infty

for all

a\in\overline{\R}.

With this order topology,

\overline{\R}

has the desirable property of compactness: Every subset of

\overline\R

has a supremum and an infimum^[2] (the infimum of the empty set is

+infty

, and its supremum is

-infty

). Moreover, with this topology,

\overline\R

is 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, however, that is an extension of the ordinary metric on

\R.

In this topology, a set

is a neighborhood of

+infty

if and only if it contains a set

\{x:x>a\}

for some real number

The notion of the neighborhood of

-infty

can be defined similarly. Using this characterization of extended-real neighborhoods, limits with

tending to

+infty

-infty

, and limits "equal" to

+infty

and

-infty

, reduce to the general topological definition of limits—instead of having a special definition in the real number system.

Arithmetic operations

The arithmetic operations of

can be partially extended to

\overline\R

as follows:

$\begina \pm \infty = \pm\infty + a & = \pm\infty, & a & \neq \mp\infty \\a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0)
\end{align}</math>
For exponentiation, see {{Section link|Exponentiation|Limits of powers}}. Here, <math>a + \infty</math> means both <math>a + (+\infty)</math> and <math>a - (-\infty),</math> while <math>a - \infty</math> means both <math>a - (+\infty)</math> and <math>a + (-\infty).</math>
The expressions <math>\infty - \infty, 0 \times (\pm\infty)</math> and <math>\pm\infty/\pm\infty</math> (called [[indeterminate form]]s) are usually left undefined . These rules are modeled on the laws for infinite limits. However, in the context of probability or measure theory,$

0 x \pminfty

is often defined as

When dealing with both positive and negative extended real numbers, the expression

1/0

is usually left undefined, because, although it is true that for every real nonzero sequence

that converges to

the reciprocal sequence

1/f

is eventually contained in every neighborhood of

\{infty,-infty\},

it is not true that the sequence

1/f

must itself converge to either

-infty

infty.

Said another way, if a continuous function

achieves a zero at a certain value

x_0,

then it need not be the case that

1/f

tends to either

-infty

infty

in the limit as

tends to

x_0.

This is the case for the limits of the identity function

f(x)=x

when

tends to

and of

f(x)=x²\sin\left(1/x\right)

(for the latter function, neither

-infty

nor

infty

is a limit of

1/f(x),

even if only positive values of

are considered).

However, in contexts where only non-negative values are considered, it is often convenient to define

1/0=+infty.

For example, when working with power series, the radius of convergence of a power series with coefficients

a_n

is often defined as the reciprocal of the limit-supremum of the sequence

	1/n
\left(\|a
	n\|

\right)

. Thus, if one allows

1/0

to take the value

+infty,

then one can use this formula regardless of whether the limit-supremum is

or not.

Algebraic properties

With the arithmetic operations defined above,

\overline\R

is not even a semigroup, let alone a group, a ring or a field as in the case of

\R.

However, it 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 ⋅ (b ⋅ c)

and

(a ⋅ b) ⋅ c

are either equal or both undefined.

a ⋅ b

and

b ⋅ a

are either equal or both undefined

a ⋅ (b+c)

and

(a ⋅ b)+(a ⋅ c)

are equal if both are defined.

a\leqb

and if both

a+c

and

b+c

are defined, then

a+c\leqb+c.

a\leqb

and

c>0

and if both

a ⋅ c

and

b ⋅ c

are defined, then

a ⋅ c\leqb ⋅ c.

In general, all laws of arithmetic are valid in

\overline\R

as long as all occurring expressions are defined.

Miscellaneous

Several functions can be continuously extended to

\overline\R

by taking limits. For instance, one may define the extremal points of the following functions as:

\exp(-infty)=0,

ln(0)=-infty,

\tanh(\pminfty)=\pm1,

\arctan(\pminfty)=\pm

	\pi
	2

Some singularities may additionally be removed. For example, the function

1/x²

can be continuously extended to

\overline\R

(under some definitions of continuity), by setting the value to

+infty

for

x=0,

and

for

x=+infty

and

x=-infty.

On the other hand, the function

1/x

cannot be continuously extended, because the function approaches

-infty

approaches

from below, and

+infty

approaches

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

+infty

and

-infty

(i.e. infinity is unsigned).^[3] As a result, a function may have limit

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

1/x

x=0.

On the other hand, on the projectively extended real line,

\lim_x{f(x)}

and

\lim_x{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

e^x

and

\arctan(x)

cannot be made continuous at

x=infty

on the projectively extended real line.

References

Web site: Section 6: The Extended Real Number System. Wilkins. David. 2007. maths.tcd.ie. 2019-12-03.
Book: Oden . J. Tinsley . Demkowicz. Leszek. Applied Functional Analysis . 16 January 2018 . Chapman and Hall/CRC . 9781498761147 . 74 . 3 . 8 December 2019 .
Web site: Projectively Extended Real Numbers. Weisstein. Eric W.. mathworld.wolfram.com. en. 2019-12-03.

Extended real number line explained

Motivation

Limits

Measure and integration

Order and topological properties

Arithmetic operations

Algebraic properties

Miscellaneous

See also

References