Projectively extended real line

The real projective line can be thought of as a line whose endpoints meet at infinity.

In real analysis, the projectively extended real line (also called the one-point compactification of the real line, or simply real projective line), is the extension of the number line by a point denoted $\infty$ . It is thus the set $\mathbb {R} \cup \{\infty \}$ (where $\mathbb {R}$ is the set of the real numbers), sometimes denoted by ${\widehat {\mathbb {R} }}.$ 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 the projective line over the reals in which three specific points (e.g. $0$ , $1$ and $\infty$ ) have been chosen. The projectively extended real line must not be confused with the extended real number line, in which $+\infty$ and $-\infty$ are distinct.

Dividing by zero

Unlike most mathematical models of the intuitive concept of 'number', this structure allows division by zero:

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

for nonzero a. In particular $1/0 = \infty$ , and moreover $1/\infty = 0$ , making reciprocal, $1/ x$ , a total function in this structure. The structure, however, is not a field, and division does not retain its original algebraic meaning in it, as witnessed for example by $0\cdot\infty$ being undefined despite reciprocation being total. The geometric interpretation is this: a vertical line has infinite slope.

Extensions of the real line

The real projective line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally $\infty$ .

Compare with the extended real number line (also called the two-point compactification of the real line), which distinguishes between $+\infty$ and $-\infty$ .

Order

The order relation cannot be extended to ${\widehat {\mathbb {R} }}$ in a meaningful way. Given a real number a, there is no convincing reason to decide that $a>\infty$ or that $a<\infty$ . Since $\infty$ can't be compared with any of the other elements, there's no point in using this relation at all. However, order is used to make definitions in ${\widehat {\mathbb {R} }}$ that are based on the properties of reals.

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 R².

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 limits of functions of real numbers.

Arithmetic operations that are defined

{\begin{aligned}\\a+\infty =\infty +a&=\infty ,&a\in \mathbb {R} \\a-\infty =\infty -a&=\infty ,&a\in \mathbb {R} \\a\cdot \infty =\infty \cdot a&=\infty ,&a\in \mathbb {R} ,a\neq 0\\\infty \cdot \infty &=\infty \\{\frac {a}{\infty }}&=0,&a\in \mathbb {R} \\{\frac {\infty }{a}}&=\infty ,&a\in \mathbb {R} \\{\frac {a}{0}}&=\infty ,&a\in \mathbb {R} ,a\neq 0\end{aligned}}

Arithmetic operations that are left undefined

The following cannot be motivated by considering limits of real functions, and any definition of them would require us to give up additional algebraic properties. Therefore, they are left undefined:

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

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 the right-hand side is defined, for any $a,b,c\in {\widehat {\mathbb {R} }}$ .

{\begin{aligned}a\cdot (b+c)&=a\cdot b+a\cdot c\\a&=({\frac {a}{b}})\cdot b&=\,\,&{\frac {(a\cdot b)}{b}}\\a&=(a+b)-b&=\,\,&(a-b)+b\end{aligned}}

In general, all laws of arithmetic are valid as long as 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 an unordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that $a,b\in \mathbb {R} ,a<b$ ):

{\begin{aligned}\left[a,a\right]&=\{a\}\\\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[\infty ,\infty \right]&=\lbrace \infty \rbrace \end{aligned}}

The corresponding open and half-open intervals are obtained by removing the respective endpoints.

${\widehat {\mathbb {R} }}$ itself is also an interval, as is ${\widehat {\mathbb {R} }}$ excluding any single point, but these cannot be represented with this bracket notation.

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

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 R.

Interval arithmetic

Interval arithmetic is trickier in ${\widehat {\mathbb {R} }}$ than in $\mathbb {R}$ . However, the result of an arithmetic operation on intervals is always an interval. 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]

which is true even when the intervals involved include 0.

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

A is a neighbourhood of x, if and only if A contains an open interval B and $x\in B$ .
A is a right-sided neighbourhood of x, if and only if there is $y\in {\widehat {\mathbb {R} }},y>x$ such that A contains $[x,y)$ .
A is a left-sided neighbourhood of x, if and only if there is $y\in {\widehat {\mathbb {R} }},y<x$ such that A contains $(y,x]$ .
A is a (right-sided, left-sided) punctured neighbourhood of x, if and only if there is $B\subseteq {\widehat {\mathbb {R} }}$ such that B is a (right-sided, left-sided) neighbourhood of x, and $A=B\setminus \{x\}$ .

Limits

Basic definitions of limits

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

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

\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 $y\neq x$ .

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

Let

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

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

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

Let

f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},\quad A\subseteq {\widehat {\mathbb {R} }}.

f 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 approaches p through A is f(p).

An interesting feature is that every rational function P(x)/Q(x), where P(x) and Q(x) have no common factor, is continuous in ${\widehat {\mathbb {R} }}$ . Also, If 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 $\mathbb {R}$ . However, many elementary functions, such as trigonometric and exponential functions, are discontinuous at $\infty$ . For example, sin is continuous in $\mathbb {R}$ but discontinuous at $\infty$ .

Thus 1/x is continuous on ${\widehat {\mathbb {R} }}$ but not on the affinely extended real number system R. Conversely, the function arctan can be extended continuously on R, but not on ${\widehat {\mathbb {R} }}$ .

As a projective range

Main article: 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 projectivities preserve the harmonic relation, they form the automorphisms of the real projective line. The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction of a projective line over a ring. Collectively they form the group PGL(2,R).

The projectivities which are their own inverses are called involutions. A hyperbolic involution has two fixed points. Two of these correspond to elementary, arithmetic operations on the real projective line: negation and reciprocation. Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.

External links

Projectively Extended Real Numbers -- From Mathworld

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