Extended real number line

thumb|right|Extended real numbers (top) vs [[projectively extended real numbers (bottom)]]

In mathematics, the extended real number system is obtained from the real number system <math>\R</math> by adding two elements denoted <math>+\infty</math> and <math>-\infty</math> 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 <math>(1,2,\ldots)</math> 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 <math>+\infty</math> as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of <math>+\infty</math> and <math>-\infty</math> as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers.

The extended real number system is denoted <math>\overline{\R}</math>, <math>[-\infty,+\infty]</math>, or <math>\R\cup\left\{-\infty,+\infty\right\}</math>. (the infimum of the empty set is <math>+\infty</math>, and its supremum is <math>-\infty</math>). Moreover, with this topology, <math>\overline\R</math> is homeomorphic to the unit interval <math>[0,1]</math>. 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 <math>\R</math>.

In this topology, a set <math>U</math> is a neighborhood of <math>+\infty</math> if and only if it contains a set <math>\{x:x>a\}</math> for some real number <math>a</math>. The notion of the neighborhood of <math>-\infty</math> can be defined similarly. Using this characterization of extended-real neighborhoods, limits with <math>x</math> tending to <math>+\infty</math> or <math>-\infty</math>, and limits "equal" to <math>+\infty</math> and <math>-\infty</math>, 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 <math>\R</math> can be partially extended to <math>\overline\R</math> as follows:

:<math display="block">\begin{align}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{align}</math>

For exponentiation, see . 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</math>, <math>0\times(\pm\infty)</math>, and <math>\pm\infty/\pm\infty</math> (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, <math>0\times\pm\infty</math> is often defined as 0. As a result, a function may have limit <math>\infty</math> 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 <math>1/x</math> at <math>x=0</math>. On the other hand, on the projectively extended real line, <math>\lim_{x\to-\infty}{f(x)}</math> and <math>\lim_{x\to+\infty}{f(x)}</math> 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 <math>e^x</math> and <math>\arctan(x)</math> cannot be made continuous at <math>x=\infty</math> on the projectively extended real line.

Extended real number line

Arithmetic operations

See also

Notes

References