Projectively_extended_real_line

Projectively extended real line

Real numbers with an added 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 of the real numbers, $\mathbb {R}$ , by a point denoted ∞.^[1] It is thus the set $\mathbb {R} \cup \{\infty \}$ with the standard arithmetic operations extended where possible,^[1] and is sometimes denoted by $\mathbb {R} ^{*}$ ^[2] or ${\widehat {\mathbb {R} }}.$ The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.

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

Dividing by zero

Unlike most mathematical models of numbers, this structure allows division by zero:

{\frac {a}{0}}=\infty

for nonzero a. In particular, 1 / 0 = ∞ and 1 / ∞ = 0, making the reciprocal function 1 / x a total function in this structure.^[1] The structure, however, is not a field, and none of the binary arithmetic operations are total – for example, 0 ⋅ ∞ is undefined, even though the reciprocal is total.^[1] It has usable interpretations, however – for example, in geometry, the slope of a vertical line is ∞.^[1]

Order

The order relation cannot be extended to ${\widehat {\mathbb {R} }}$ in a meaningful way. Given a number a ≠ ∞, there is no convincing argument to define either a > ∞ or that a < ∞. Since ∞ can't be compared with any of the other elements, there's no point in retaining this relation on ${\widehat {\mathbb {R} }}$ .^[2] However, order on $\mathbb {R}$ is used in definitions in ${\widehat {\mathbb {R} }}$ .

Geometry

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 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 may be expressed by Möbius transformations (also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is 0, the image is ∞.

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 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 subspaces of $\mathbb {R} ^{2}$ .

Arithmetic operations

Arithmetic operations that are defined

In addition to the standard operations on the subset $\mathbb {R}$ of ${\widehat {\mathbb {R} }}$ , the following operations are defined for $a\in {\widehat {\mathbb {R} }}$ , with exceptions as indicated:^[3]^[2]

{\begin{aligned}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{aligned}}

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.^{[lower-alpha 1]} Consequently, they are left undefined:

{\begin{aligned}&\infty +\infty \\&\infty -\infty \\&\infty \cdot 0\\&0\cdot \infty \\&\infty /\infty \\&0/0\end{aligned}}

The exponential function $e^{x}$ cannot be extended to ${\widehat {\mathbb {R} }}$ .^[2]

Algebraic properties

The following equalities mean: Either both sides are undefined, or both sides are defined and equal. This is true for any $a,b,c\in {\widehat {\mathbb {R} }}.$

{\begin{aligned}(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{aligned}}

The following is true whenever expressions involved are defined, for any $a,b,c\in {\widehat {\mathbb {R} }}.$

{\begin{aligned}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{aligned}}

In general, all laws of arithmetic that are valid for $\mathbb {R}$ are also valid for ${\widehat {\mathbb {R} }}$ whenever all the occurring expressions are defined.

Intervals and topology

The concept of an interval can be extended to ${\widehat {\mathbb {R} }}$ . 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 $a,b\in \mathbb {R} ,a<b$ ):^[2]^{[additional citation(s) needed]}

{\begin{aligned}\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{aligned}}

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.^[2]

${\widehat {\mathbb {R} }}$ and the empty set are also intervals, as is ${\widehat {\mathbb {R} }}$ excluding any single point.^{[lower-alpha 2]}

The open intervals as a base define a topology on ${\widehat {\mathbb {R} }}$ . Sufficient for a base are the bounded open intervals in $\mathbb {R}$ and the intervals $(b,a)=\{x\mid x\in \mathbb {R} ,b<x\}\cup \{\infty \}\cup \{x\mid x\in \mathbb {R} ,x<a\}$ for all $a,b\in \mathbb {R}$ such that $a<b.$

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

Interval arithmetic

Interval arithmetic extends to ${\widehat {\mathbb {R} }}$ from $\mathbb {R}$ . 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.^{[lower-alpha 3]} In particular, we have, for every $a,b\in {\widehat {\mathbb {R} }}$ :

x\in [a,b]\iff {\frac {1}{x}}\in \left[{\frac {1}{b}},{\frac {1}{a}}\right]\!,

irrespective of whether either interval includes 0 and ∞.

Calculus

The tools of calculus can be used to analyze functions of ${\widehat {\mathbb {R} }}$ . The definitions are motivated by the topology of this space.

Neighbourhoods

Let $x\in {\widehat {\mathbb {R} }}$ and $A\subseteq {\widehat {\mathbb {R} }}$ .

A is a neighbourhood of x, if A contains an open interval B that contains x.
A is a right-sided neighbourhood of x, if there is a real number y such that $y\neq x$ and A contains the semi-open interval $[x,y)$ .
A is a left-sided neighbourhood of x, if there is a real number y such that $y\neq x$ and A contains the semi-open interval $(y,x]$ .
A is a punctured neighbourhood (resp. a right-sided or a left-sided punctured neighbourhood) of x, if $x\not \in A,$ and $A\cup \{x\}$ is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of x.

Limits

Basic definitions of limits

Let $f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},$ $p\in {\widehat {\mathbb {R} }},$ and $L\in {\widehat {\mathbb {R} }}$ .

The limit of f (x) as x approaches p is L, denoted

\lim _{x\to p}{f(x)}=L

if and only if for every neighbourhood A of L, there is a punctured neighbourhood B of p, such that $x\in B$ implies $f(x)\in A$ .

The one-sided limit of f (x) as x approaches p from the right (left) is L, denoted

\lim _{x\to p^{+}}{f(x)}=L\qquad \left(\lim _{x\to p^{-}}{f(x)}=L\right),

if and only if for every neighbourhood A of L, there is a right-sided (left-sided) punctured neighbourhood B of p, such that $x\in B$ implies $f(x)\in A.$

It can be shown that $\lim _{x\to p}{f(x)}=L$ if and only if both $\lim _{x\to p^{+}}{f(x)}=L$ and $\lim _{x\to p^{-}}{f(x)}=L$ .

Comparison with limits in $\mathbb {R}$

The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, $p,L\in \mathbb {R} ,$ the first limit is as defined above, and the second limit is in the usual sense:

$\lim _{x\to p}{f(x)}=L$ is equivalent to $\lim _{x\to p}{f(x)}=L$
$\lim _{x\to \infty ^{+}}{f(x)}=L$ is equivalent to $\lim _{x\to -\infty }{f(x)}=L$
$\lim _{x\to \infty ^{-}}{f(x)}=L$ is equivalent to $\lim _{x\to +\infty }{f(x)}=L$
$\lim _{x\to p}{f(x)}=\infty$ is equivalent to $\lim _{x\to p}{|f(x)|}=+\infty$
$\lim _{x\to \infty ^{+}}{f(x)}=\infty$ is equivalent to $\lim _{x\to -\infty }{|f(x)|}=+\infty$
$\lim _{x\to \infty ^{-}}{f(x)}=\infty$ is equivalent to $\lim _{x\to +\infty }{|f(x)|}=+\infty$

Extended definition of limits

Let $A\subseteq {\widehat {\mathbb {R} }}$ . Then p is a limit point of A if and only if every neighbourhood of p includes a point $y\in A$ such that $y\neq p.$

Let $f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},A\subseteq {\widehat {\mathbb {R} }},L\in {\widehat {\mathbb {R} }},p\in {\widehat {\mathbb {R} }}$ , p a limit point of A. The limit of f (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 $x\in A\cap C$ implies $f(x)\in B.$

This corresponds to the regular topological definition of continuity, applied to the subspace topology on $A\cup \lbrace p\rbrace ,$ and the restriction of f to $A\cup \lbrace p\rbrace .$

Continuity

The function

f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},\quad p\in {\widehat {\mathbb {R} }}.

is continuous at p if and only if f is defined at p and

\lim _{x\to p}{f(x)}=f(p).

If $A\subseteq {\widehat {\mathbb {R} }},$ the function

f:A\to {\widehat {\mathbb {R} }}

is continuous in A if and only if, for every $p\in A$ , f is defined at p and the limit of $f(x)$ as x tends to p through A is $f(p).$

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

Also, if the tangent function $\tan$ is extended so that

\tan \left({\frac {\pi }{2}}+n\pi \right)=\infty {\text{ for }}n\in \mathbb {Z} ,

then $\tan$ is continuous in $\mathbb {R} ,$ but cannot be prolongated further to a function that is continuous in ${\widehat {\mathbb {R} }}.$

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

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

Notes

An extension does however exist in which all the algebraic properties, when restricted to defined operations in ${\widehat {\mathbb {R} }}$ , resolve to the standard rules: see Wheel theory.
If consistency of complementation is required, such that $[a,b]^{\complement }=(b,a)$ and $(a,b]^{\complement }=(b,a]$ for all $a,b\in {\widehat {\mathbb {R} }}$ (where the interval on either side is defined), all intervals excluding $\varnothing$ and ${\widehat {\mathbb {R} }}$ may be naturally represented using this notation, with $(a,a)$ being interpreted as ${\widehat {\mathbb {R} }}\setminus \{a\}$ , and half-open intervals with equal endpoints, e.g. $(a,a]$ , remaining undefined.
For example, the ratio of intervals $[0,1]/[0,1]$ contains 0 in both intervals, and since 0 / 0 is undefined, the result of division of these intervals is undefined.

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[4] An extension does however exist in which all the algebraic properties, when restricted to defined operations in ${\widehat {\mathbb {R} }}$ , resolve to the standard rules: see Wheel theory.

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

[6] For example, the ratio of intervals $[0,1]/[0,1]$ contains 0 in both intervals, and since 0 / 0 is undefined, the result of division of these intervals is undefined.

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Weisstein, Eric W. "Projectively Extended Real Numbers". mathworld.wolfram.com. Retrieved 2023-01-22.

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Lee, Nam-Hoon (2020-04-28). Geometry: from Isometries to Special Relativity. Springer Nature. ISBN 978-3-030-42101-4.

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