Editing Projectively extended real line

{{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.

== Dividing by zero ==

Unlike most mathematical models of numbers, this structure allows [[division by zero]]:
: <math>\frac{a}{0} = \infty</math>
for nonzero ''a''. In particular, {{math|1 / 0 {{=}} ∞}} and {{math|1 / ∞ {{=}} 0}}, making the [[multiplicative inverse|reciprocal]] [[function (mathematics)|function]] {{math|1 / ''x''}} a [[total function]] in this structure.<ref name=":0" /> The structure, however, is not a [[field (mathematics)|field]], and none of the [[binary operation|binary]] arithmetic operations are total – for example, {{math|0 ⋅ ∞}} is undefined, even though the reciprocal is total.<ref name=":0" /> It has usable interpretations, however – for example, in geometry, the [[slope]] of a vertical line is {{math|∞}}.<ref name=":0" />

== Extensions of the real line ==

The projectively extended real line extends the [[field (mathematics)|field]] of [[real number]]s in the same way that the [[Riemann sphere]] extends the field of [[complex number]]s, by adding a single point called conventionally {{math|∞}}.

In contrast, the [[affinely extended real number line]] (also called the two-point [[compactification (mathematics)|compactification]] of the real line) distinguishes between {{math|+∞}} and {{math|−∞}}.

== Order ==

The [[order theory|order]] relation cannot be extended to <math>\widehat{\mathbb{R}}</math> in a meaningful way. Given a number {{math|''a'' ≠ ∞}}, there is no convincing argument to define either {{math|''a'' > ∞}} or that {{math|''a'' < ∞}}. Since {{math|∞}} can't be compared with any of the other elements, there's no point in retaining this relation on <math>\widehat{\mathbb{R}}</math>.<ref name=":1" /> However, order on <math>\mathbb{R}</math> is used in definitions in <math>\widehat{\mathbb{R}}</math>.

== Geometry ==

Fundamental to the idea that {{math|∞}} is a point ''no different from any other'' is the way the real projective line is a [[homogeneous space]], in fact [[homeomorphic]] to a circle. For example the [[general linear group]] of 2&thinsp;×&thinsp;2 real [[invertible matrix|invertible]] [[matrix (mathematics)|matrices]] has a [[transitive action]] on it. The [[group action]] may be expressed by [[Möbius transformation]]s (also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is {{math|0}}, the image is {{math|∞}}.

The detailed analysis of the action shows that for any three distinct points ''P'', ''Q'' and ''R'', there is a linear fractional transformation taking ''P'' to 0, ''Q'' to 1, and ''R'' to {{math|∞}} that is, the [[group (mathematics)|group]] of linear fractional transformations is [[transitive action|triply transitive]] on the real projective line. This cannot be extended to 4-tuples of points, because the [[cross-ratio]] is invariant.

The terminology [[projective line]] is appropriate, because the points are in 1-to-1 correspondence with one-[[dimension (vector space)|dimensional]] [[linear subspace]]s of <math>\mathbb{R}^2</math>.

== Arithmetic operations ==

=== Motivation for arithmetic operations ===
The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the [[limit of a function|limits]] of functions of real numbers.

=== Arithmetic operations that are defined ===
In addition to the standard operations on the [[subset]] <math>\mathbb{R}</math> of <math>\widehat{\mathbb{R}}</math>, the following operations are defined for <math>a \in \widehat{\mathbb{R}}</math>, with exceptions as indicated:<ref>{{Cite book |last=Lee |first=Nam-Hoon |url=https://books.google.com/books?id=l3HgDwAAQBAJ&dq=%22Projectively+extended+real+line%22+-wikipedia&pg=PA255 |title=Geometry: from Isometries to Special Relativity |date=2020-04-28 |publisher=Springer Nature |isbn=978-3-030-42101-4 |language=en}}</ref><ref name=":1" />
:<math>\begin{align}
a + \infty = \infty + a & = \infty, & a \neq \infty \\
a - \infty = \infty - a & = \infty, & a \neq \infty \\
a / \infty = a \cdot 0 = 0 \cdot a & = 0, & a \neq \infty \\
\infty / a & = \infty, & a \neq \infty \\
a / 0 = a \cdot \infty = \infty \cdot a & = \infty, & a \neq 0 \\
0 / a & = 0, & a \neq 0
\end{align}</math>

=== Arithmetic operations that are left undefined ===
The following expressions cannot be motivated by considering limits of real functions, and no definition of them allows the statement of the standard algebraic properties to be retained unchanged in form for all defined cases.{{efn|An extension does however exist in which all the algebraic properties, when restricted to defined operations in <math>\widehat{\mathbb{R}}</math>, resolve to the standard rules: see [[Wheel theory]].}} Consequently, they are left undefined:
:<math>\begin{align}
& \infty + \infty \\
& \infty - \infty \\
& \infty \cdot 0 \\
& 0 \cdot \infty \\
& \infty / \infty \\
& 0 / 0
\end{align}</math>
The [[exponential function]] <math>e^x</math> cannot be extended to <math>\widehat{\mathbb{R}}</math>.<ref name=":1" />

== Algebraic properties ==
The following equalities mean: ''Either both sides are undefined, or both sides are defined and equal.'' This is true for any <math>a, b, c \in \widehat{\mathbb{R}}.</math>
:<math>\begin{align}
(a + b) + c & = a + (b + c) \\
a + b & = b + a \\
(a \cdot b) \cdot c & = a \cdot (b \cdot c) \\
a \cdot b & = b \cdot a \\
a \cdot \infty & = \frac{a}{0} \\
\end{align}</math>
The following is true whenever expressions involved are defined, for any <math>a, b, c \in \widehat{\mathbb{R}}.</math>
:<math>
\begin{align}
a \cdot (b + c) & = a \cdot b + a \cdot c \\
a & = \left(\frac{a}{b}\right) \cdot b & = \,\,& \frac{(a \cdot b)}{b} \\
a & = (a + b) - b & = \,\,& (a - b) + b
\end{align}
</math>
In general, all laws of arithmetic that are valid for <math>\mathbb{R}</math> are also valid for <math>\widehat{\mathbb{R}}</math> whenever all the occurring expressions are defined.

== Intervals and topology ==
The concept of an [[interval (mathematics)|interval]] can be extended to <math>\widehat{\mathbb{R}}</math>. However, since it is not an ordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that 
<math>a, b \in \mathbb{R}, a < b</math>):<ref name=":1" />{{Additional citations needed|date=January 2023}}

: <math>\begin{align}
\left[a, b\right] & = \lbrace x \mid x \in \mathbb{R}, a \leq x \leq b \rbrace  \\
\left[a, \infty\right] & = \lbrace x \mid x \in \mathbb{R}, a \leq x \rbrace  \cup \lbrace \infty \rbrace  \\
\left[b, a\right] & = \lbrace x \mid x \in \mathbb{R}, b \leq x \rbrace  \cup \lbrace \infty \rbrace \cup \lbrace x \mid x \in \mathbb{R}, x \leq a \rbrace  \\
\left[\infty, a\right] & = \lbrace \infty \rbrace \cup \lbrace x \mid x \in \mathbb{R}, x \leq a \rbrace  \\
\left[a, a\right] & = \{ a \}  \\
\left[\infty, \infty\right] & = \lbrace \infty \rbrace 
\end{align}</math>

With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints. This redefinition is useful in [[interval arithmetic]] when dividing by an interval containing 0.<ref name=":1" />

<math>\widehat{\mathbb{R}}</math> and the [[empty set]] are also intervals, as is <math>\widehat{\mathbb{R}}</math> excluding any single point.{{efn|If consistency of complementation is required, such that <math>[a,b]^\complement = (b,a)</math> and <math>(a,b]^\complement = (b,a]</math> for all <math>a, b \in \widehat{\mathbb{R}}</math> (where the interval on either side is defined), all intervals excluding <math>\varnothing</math> and <math>\widehat{\mathbb{R}}</math> may be naturally represented using this notation, with <math>(a,a)</math> being interpreted as <math>\widehat{\mathbb{R}}\setminus \{ a \}</math>, and half-open intervals with equal endpoints, e.g. <math>(a,a]</math>, remaining undefined.}}

The open intervals as a [[base (topology)|base]] define a [[topological space|topology]] on <math>\widehat{\mathbb{R}}</math>. Sufficient for a base are the [[bounded interval|bounded]] open intervals in <math>\mathbb{R}</math> and the intervals <math>(b, a) = \{x \mid x \in \mathbb{R}, b < x\} \cup \{\infty\} \cup \{x \mid x \in \mathbb{R}, x < a\}</math> for all <math>a, b \in \mathbb{R}</math> such that <math>a < b.</math>

As said, the topology is [[homeomorphic]] to a circle. Thus it is [[metrizable]] corresponding (for a given homeomorphism) to the ordinary [[metric (mathematics)|metric]] on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on <math>\mathbb{R}.</math>

== Interval arithmetic ==
[[Interval arithmetic]] extends to <math>\widehat{\mathbb{R}}</math> from <math>\mathbb{R}</math>. The result of an arithmetic operation on intervals is always an interval, except when the intervals with a binary operation contain incompatible values leading to an undefined result.{{efn|For example, the ratio of intervals <math>[0,1]/[0,1]</math> contains {{math|0}} in both intervals, and since {{math|0 / 0}} is undefined, the result of division of these intervals is undefined.}} In particular, we have, for every <math>a, b \in \widehat{\mathbb{R}}</math>:
:<math>x \in [a, b] \iff \frac{1}{x} \in \left[ \frac{1}{b}, \frac{1}{a} \right] \!,</math>
irrespective of whether either interval includes {{math|0}} and {{math|∞}}.

== Calculus ==
The tools of [[calculus]] can be used to analyze functions of <math>\widehat{\mathbb{R}}</math>. The definitions are motivated by the topology of this space.

=== Neighbourhoods ===
Let <math>x \in \widehat{\mathbb{R}}</math> and <math>A \subseteq \widehat{\mathbb{R}}</math>.
* {{mvar|A}} is a [[Neighbourhood (mathematics)|neighbourhood]] of {{math|''x''}}, if {{math|''A''}} contains an open interval {{math|''B''}} that contains {{mvar|x}}.
* {{mvar|A}} is a right-sided neighbourhood of {{mvar|x}}, if there is a real number {{mvar|y}} such that <math>y \neq x </math> and {{mvar|A}} contains the semi-open interval <math>[x, y)</math>.
* {{mvar|A}} is a left-sided neighbourhood of {{mvar|x}}, if there is a real number {{mvar|y}} such that <math>y \neq x </math> and {{mvar|A}} contains the semi-open interval <math>(y, x]</math>.
* {{mvar|A}} is a [[punctured neighbourhood]] (resp. a right-sided or a left-sided punctured neighbourhood) of {{mvar|x}}, if <math>x\not\in A,</math> and <math>A\cup\{x\}</math> is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of {{mvar|x}}.

=== Limits ===

==== Basic definitions of limits ====
Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},</math> <math>p \in \widehat{\mathbb{R}},</math> and <math>L \in \widehat{\mathbb{R}}</math>.

The [[limit of a function|limit]] of ''f''{{hairsp}}(''x'') as {{math|''x''}} approaches ''p'' is ''L'', denoted
: <math>\lim_{x \to p}{f(x)} = L</math>
if and only if for every neighbourhood ''A'' of ''L'', there is a punctured neighbourhood ''B'' of ''p'', such that <math>x \in B</math> implies <math>f(x) \in A</math>.

The [[one-sided limit]] of ''f''{{hairsp}}(''x'') as ''x'' approaches ''p'' from the right (left) is ''L'', denoted
: <math>\lim_{x \to p^{+}}{f(x)} = L \qquad \left( \lim_{x \to p^{-}}{f(x)} = L \right),</math>
if and only if for every neighbourhood ''A'' of ''L'', there is a right-sided (left-sided) punctured neighbourhood ''B'' of ''p'', such that <math>x \in B</math> implies <math>f(x) \in A.</math>

It can be shown that <math>\lim_{x \to p}{f(x)} = L</math> if and only if both <math>\lim_{x \to p^+}{f(x)} = L</math> and <math>\lim_{x \to p^-}{f(x)} = L</math>.

==== Comparison with limits in <math>\mathbb{R}</math> ====
The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, <math>p, L \in \mathbb{R},</math> the first limit is as defined above, and the second limit is in the usual sense:
* <math>\lim_{x \to p}{f(x)} = L</math> is equivalent to <math>\lim_{x \to p}{f(x)} = L</math>
* <math>\lim_{x \to \infty^{+}}{f(x)} = L</math> is equivalent to <math>\lim_{x \to -\infty}{f(x)} = L</math>
* <math>\lim_{x \to \infty^{-}}{f(x)} = L</math> is equivalent to <math>\lim_{x \to +\infty}{f(x)} = L</math>
* <math>\lim_{x \to p}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to p}{|f(x)|} = +\infty</math>
* <math>\lim_{x \to \infty^{+}}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to -\infty}{|f(x)|} = +\infty</math>
* <math>\lim_{x \to \infty^{-}}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to +\infty}{|f(x)|} = +\infty</math>

==== Extended definition of limits ====
Let <math>A \subseteq \widehat{\mathbb{R}}</math>. Then ''p'' is a [[limit point]] of ''A'' if and only if every neighbourhood of ''p'' includes a point <math>y \in A</math> such that <math>y \neq p.</math>

Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}</math>, ''p'' a limit point of ''A''. The limit of ''f''{{hairsp}}(''x'') as ''x'' approaches ''p'' through ''A'' is ''L'', if and only if for every neighbourhood ''B'' of ''L'', there is a punctured neighbourhood ''C'' of ''p'', such that <math>x \in A \cap C</math> implies <math>f(x) \in B.</math>

This corresponds to the regular [[continuity (topology)|topological definition of continuity]], applied to the [[subspace topology]] on <math>A\cup \lbrace p \rbrace,</math> and the [[restriction (mathematics)|restriction]] of ''f'' to <math>A \cup \lbrace p \rbrace.</math>

=== Continuity ===

The function
: <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},\quad p \in \widehat{\mathbb{R}}.</math>
is [[Continuous function|continuous]] at {{math|''p''}} if and only if {{math|''f''}} is defined at {{math|''p''}} and
: <math>\lim_{x \to p}{f(x)} = f(p).</math>

If <math>A \subseteq \widehat\mathbb R,</math> the function
: <math>f : A \to \widehat{\mathbb{R}}</math>
is continuous in {{math|''A''}} if and only if, for every <math>p \in A</math>, {{math|''f''}} is defined at {{math|''p''}} and the limit of <math>f(x)</math> as {{math|''x''}} tends to {{math|''p''}} through {{math|''A''}} is <math>f(p).</math>

Every [[rational function]] {{math|''P''(''x'')/''Q''(''x'')}}, where {{math|''P''}} and {{math|''Q''}} are [[polynomial]]s, can be prolongated, in a unique way, to a function from <math>\widehat{\mathbb{R}}</math> to <math>\widehat{\mathbb{R}}</math> that is continuous in <math>\widehat{\mathbb{R}}.</math> In particular, this is the case of [[polynomial function]]s, which take the value <math>\infty</math> at <math>\infty,</math> if they are not [[constant function|constant]].

Also, if the [[tangent function|tangent]] function <math>\tan</math> is extended so that
: <math>\tan\left(\frac{\pi}{2} + n\pi\right) = \infty\text{ for }n \in \mathbb{Z},</math>
then <math>\tan</math> is continuous in <math>\mathbb{R},</math> but cannot be prolongated further to a function that is continuous in <math>\widehat{\mathbb{R}}.</math>

Many [[elementary function]]s that are continuous in <math>\mathbb R</math> cannot be prolongated to functions that are continuous in <math>\widehat\mathbb{R}.</math> This is the case, for example, of the [[exponential function]] and all [[trigonometric functions]]. For example, the [[sine]] function is continuous in <math>\mathbb{R},</math> but it cannot be made continuous at <math>\infty.</math> As seen above, the tangent function can be prolongated to a function that is continuous in <math>\mathbb{R},</math> but this function cannot be made continuous at <math>\infty.</math>

Many discontinuous functions that become continuous when the [[codomain]] is extended to <math>\widehat{\mathbb{R}}</math> remain discontinuous if the codomain is extended to the [[affinely extended real number system]] <math>\overline{\mathbb{R}}.</math> This is the case of the function <math>x\mapsto \frac 1x.</math> On the other hand, some functions that are continuous in <math>\mathbb R</math> and discontinuous at <math>\infty \in \widehat{\mathbb{R}}</math> become continuous if the [[domain of a function|domain]] is extended to <math>\overline{\mathbb{R}}.</math> This is the case for the [[arctangent]].

== As a projective range ==
{{Main|Projective range}}
When the [[real projective line]] is considered in the context of the [[real projective plane]], then the consequences of [[Desargues' theorem]] are implicit. In particular, the construction of the [[projective harmonic conjugate]] relation between points is part of the structure of the real projective line. For instance, given any pair of points, the [[point at infinity]] is the projective harmonic conjugate of their [[midpoint]].

As [[projectivity|projectivities]] preserve the harmonic relation, they form the [[automorphism]]s of the real projective line. The projectivities are described algebraically as [[homography|homographies]], since the real numbers form a [[ring (mathematics)|ring]], according to the general construction of a [[projective line over a ring]]. Collectively they form the group [[PGL(2,R)|PGL(2,&nbsp;'''R''')]].

The projectivities which are their own inverses are called [[involution (mathematics)#Projective geometry|involutions]]. A '''hyperbolic involution''' has two [[fixed point (mathematics)|fixed point]]s. Two of these correspond to elementary, arithmetic operations on the real projective line: [[additive inverse|negation]] and [[multiplicative inverse|reciprocation]]. Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.

== See also ==
* [[Real projective plane]]
* [[Complex projective plane]]
* [[Wheel theory]]

== Notes ==
{{notelist}}

== References ==
{{Reflist}}

[[Category:Real analysis]]
[[Category:Topological spaces]]
[[Category:Projective geometry]]
[[Category:Infinity]]