m Undid revision 1169677726 by 2600:8805:D609:3800:4D6F:5853:F744:F54B (talk) – repeating this bold text just to substitute "system" for "line" here doesn't really seem worth the space, and we shouldn't make bold titles out of wikilinks
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 '''projectively extended real system''' or '''[[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.
