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{{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]].]]▼
▲[[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}
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
: <math>\frac{a}{0} = \infty</math>▼
for nonzero ''a''.
== Extensions of the real line ==▼
▲:<math>\frac{a}{0} = \infty</math>
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
▲for nonzero ''a''. In particular {{math|1/0 {{=}} ∞}}, and moreover {{math|1/∞ {{=}} 0}}, making [[Multiplicative inverse|reciprocal]], {{math|1/''x''}}, a [[total function]] in this structure. The structure, however, is not a [[field (mathematics)|field]], and none of the binary arithmetic operations are total, as witnessed for example by {{math|0⋅∞}} being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, a vertical line has ''infinite'' [[slope]].
In contrast, the [[affinely extended real number line]] (also called the two-point [[compactification (mathematics)|compactification]] of the real line) distinguishes between
▲==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>\infty</math>.
The [[order theory|order]] relation cannot be extended to <math>\widehat{\mathbb{R}}</math> in a meaningful way. Given a number
▲In contrast, the [[extended real number line]] (also called the two-point [[compactification (mathematics)|compactification]] of the real line) distinguishes between <math>+\infty</math> and <math>-\infty</math>.
==
▲The order relation cannot be extended to <math>\widehat{\mathbb{R}}</math> in a meaningful way. Given a number <math>a \ne \infty</math>, there is no convincing argument to define either <math>a > \infty</math> or that <math>a < \infty</math>. Since <math>\infty</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>. However, order on <math>\mathbb{R}</math> is used in definitions in <math>\widehat{\mathbb{R}}</math>.
Fundamental to the idea that
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''
▲Fundamental to the idea that ∞ is a point ''no different from any other'' is the way the real projective line is a [[homogeneous space]], in fact [[homeomorphism|homeomorphic]] to a [[circle]]. For example the [[general linear group]] of 2×2 real [[invertible]] matrices has a [[transitive action]] on it. The [[Group action (mathematics)|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 0, the image is ∞.
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>.▼
▲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 ∞ that is, the group of linear fractional transformations is triply [[transitive action|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-dimensional [[linear subspace]]s of <math>\mathbb{R}^2</math>.
==
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.▼
===
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" />▼
▲The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the limits of functions of real numbers.
:<math>\begin{align}▼
▲===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:
▲\begin{align}
a + \infty = \infty + a & = \infty, & a \neq \infty \\
a - \infty = \infty - a & = \infty, & a \neq \infty \\
Line 48 ⟶ 46:
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 \\
Line 61 ⟶ 57:
& \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 \\
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a \cdot b & = b \cdot a \\
a \cdot \infty & = \frac{a}{0} \\
\end{align}</math>
The following is true whenever
▲The following is true whenever the right-hand side is 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}
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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
<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 \\
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\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
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
As said, the topology is [[
== 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>x \in [a, b] \iff \frac{1}{x} \in \left[ \frac{1}{b}, \frac{1}{a} \right] \!,</math>
irrespective of whether either interval includes
== 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}}
*
*
*
*
=== 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
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>
* <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
This corresponds to the regular [[continuity (topology)|topological definition of continuity]], applied to the [[subspace topology]] on <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
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>
Also, if the [[tangent function|tangent]]
: <math>\tan\left(\frac{\pi}{2} + n\pi\right) = \infty\text{ for }n \in \mathbb{Z},</math>
then
Many [[
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
== 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
The projectivities which are their own inverses are called [[involution (mathematics)#Projective geometry|involutions]]. A '''hyperbolic involution'''
▲== Notes ==
{{notelist}}▼
== See also ==
Line 191 ⟶ 180:
* [[Wheel theory]]
==
▲{{notelist}}
== References ==
{{Reflist}}
[[Category:Real analysis]]
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