Editing Extended real number line (section)

==Arithmetic operations==

The arithmetic operations of <math>\R</math> can be partially extended to <math>\overline\R</math> as follows:<ref name=":0">{{Cite web|url=http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html|title=Affinely Extended Real Numbers|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>

:<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 {{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</math>, <math>0\times(\pm\infty)</math>, and <math>\pm\infty/\pm\infty</math> (called [[indeterminate form]]s) are usually left [[Defined and undefined|undefined]]. These rules are modeled on the laws for [[Limit_of_a_function#Limits_involving_infinity|infinite limits]]. However, in the context of [[probability theory|probability]] or measure theory, <math>0\times\pm\infty</math> is often defined as 0.<ref name=":2" />

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

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