Editing Extended real number line (section)

{{Short description|Real numbers with + and - infinity added}}
{{about|the extension of the reals with {{math|+∞}} and {{math|−∞}}|the extension by a single point at infinity|Projectively extended real line}}
[[File:Extended_Real_Numbers.svg|thumb|right|Extended real numbers (top) vs [[projectively extended real numbers]] (bottom)]]
In [[mathematics]], the '''extended real number system'''{{efn|Some authors use ''Affinely extended real number system'' and ''Affinely extended real number line'', although the extended real numbers do not form an [[affine line]].}} is obtained from the [[real number]] system <math>\R</math> by adding two elements denoted <math>+\infty</math> and <math>-\infty</math>{{efn|Read as "positive infinity" and "negative infinity" respectively.}} that are respectively greater and lower than every real number. This allows for treating the [[potential infinity|potential infinities]] of infinitely increasing sequences and infinitely decreasing series as [[actual infinity|actual infinities]]. For example, the [[infinite sequence]] <math>(1,2,\ldots)</math> of the [[natural number]]s 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 (mathematics)|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.<ref>{{Cite web|url=https://www.maths.tcd.ie/~dwilkins/Courses/221/Extended.pdf|title=Section 6: The Extended Real Number System|last=Wilkins|first=David|date=2007|website=maths.tcd.ie|access-date=2019-12-03}}</ref> It is the [[Dedekind–MacNeille completion]] of the real numbers.

The extended real number system is denoted <math>\overline{\R}</math>,<!--{{math|{{overset|—|ℝ}}}}--> <math>[-\infty,+\infty]</math>, or <math>\R\cup\left\{-\infty,+\infty\right\}</math>.<ref name=":1" /> When the meaning is clear from context, the symbol <math>+\infty</math> is often written simply as <math>\infty</math>.<ref name=":1" />

There is also a distinct [[projectively extended real line]] where <math>+\infty</math> and <math>-\infty</math> 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 <math>\infty</math> or as <math>\pm\infty</math>.