Editing Projectively extended real line (section)

{{Short description|Real numbers with an added point at infinity}}
{{about|the extension of the reals by a single point at infinity|the extension by {{math|+∞}} and {{math|–∞}}|Extended real number line}}
{{More cn|date=January 2023}}[[Image:Real projective line.svg|right|thumb|
The projectively extended real line can be visualized as the real number line wrapped around a [[circle]] (by some form of [[stereographic projection]]) with an additional [[point at infinity]].]]

In [[real analysis]], the '''projectively extended real line''' (also called the [[one-point compactification]] of the [[real line]]), is the extension of the [[set (mathematics)|set]] of the [[real number]]s, <math>\mathbb{R}</math>, by a point denoted {{math|∞}}.<ref name=":0">{{Cite book |last=NBU |first=DDE |url=https://books.google.com/books?id=4i7eDwAAQBAJ&dq=%22Projectively+extended+real+line%22+-wikipedia&pg=PA62 |title=PG MTM 201 B1 |date=2019-11-05 |publisher=Directorate of Distance Education, University of North Bengal |language=en}}</ref> It is thus the set <math>\mathbb{R}\cup\{\infty\}</math> with the standard arithmetic operations extended where possible,<ref name=":0" /> and is sometimes denoted by <math>\mathbb{R}^*</math><ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |title=Projectively Extended Real Numbers |url=https://mathworld.wolfram.com/ |access-date=2023-01-22 |website=mathworld.wolfram.com |language=en}}</ref> or <math>\widehat{\mathbb{R}}.</math> The added point is called the [[point at infinity]], because it is considered as a neighbour of both [[End (topology)|ends]] of the real line. More precisely, the point at infinity is the [[limit of a sequence|limit]] of every [[sequence]] of real numbers whose [[absolute value]]s are [[Sequence#Increasing and decreasing|increasing]] and [[bounded function|unbounded]].

The projectively extended real line may be identified with a [[real projective line]] in which three points have been assigned the specific values {{math|0}}, {{math|1}} and {{math|∞}}. The projectively extended real number line is distinct from the [[affinely extended real number line]], in which {{math|+∞}} and {{math|−∞}} are distinct.