Extended real number line

Real numbers with +∞ and −∞ added

Extended Real Numbers:
a) Affinely extended real numbers
b) Projectively extended real numbers

In mathematics, the extended real number system^[a] is obtained from the real number system ${\displaystyle \mathbb {R} }$ by adding two elements denoted ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$^[b] 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 ${\displaystyle (1,2,\ldots )}$ 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 ${\displaystyle +\infty }$ as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ 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 ${\displaystyle {\overline {\mathbb {R} }}}$ or ${\displaystyle [-\infty ,+\infty ]}$ or ${\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}.}$^[2] When the meaning is clear from context, the symbol ${\displaystyle +\infty }$ is often written simply as ${\displaystyle \infty .}$^[2]

There is also a distinct projectively extended real line where ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ 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 ${\displaystyle \infty }$ or as ${\displaystyle \pm \infty }$.

Motivation

Limits

The extended number line is often useful to describe the behavior of a function ${\displaystyle f}$ when either the argument ${\displaystyle x}$ or the function value ${\displaystyle f}$ gets "infinitely large" in some sense. For example, consider the function ${\displaystyle f}$ defined by

${\displaystyle f(x)={\frac {1}{x^{2}}}.}$

The graph of this function has a horizontal asymptote at ${\displaystyle y=0.}$ Geometrically, when moving increasingly farther to the right along the ${\displaystyle x}$-axis, the value of ${\textstyle {1}/{x^{2}}}$ approaches 0. This limiting behavior is similar to the limit of a function ${\textstyle \lim _{x\to x_{0}}f(x)}$ in which the real number ${\displaystyle x}$ approaches ${\displaystyle x_{0},}$ except that there is no real number that ${\displaystyle x}$ approaches when ${\displaystyle x}$ increases infinitely. Adjoining the elements ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ to ${\displaystyle \mathbb {R} }$ enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, except that ${\displaystyle |x-x_{0}|<\varepsilon }$ is replaced by ${\displaystyle x>N}$ (for ${\displaystyle +\infty }$) or ${\displaystyle x<-N}$ (for ${\displaystyle -\infty }$). This allows proving and writing

${\displaystyle {\begin{aligned}\lim _{x\to +\infty }{\frac {1}{x^{2}}}&=0,\\\lim _{x\to -\infty }{\frac {1}{x^{2}}}&=0,\\\lim _{x\to 0}{\frac {1}{x^{2}}}&=+\infty .\end{aligned}}}$