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Projectively extended real line

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 number line by a point denoted . It is thus the set (where is the set of the real numbers) with the standard arithmetic operations extended where possible, sometimes denoted by 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 points have been assigned specific values (e.g. 0, 1 and ). The projectively extended real line must not be confused with the extended real number line, in which +∞ and −∞ are distinct.

Dividing by zeroEdit

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

 

for nonzero a. In particular 1/0 = ∞, and moreover 1/∞ = 0, making reciprocal, 1/x, a total function in this structure. The structure, however, is not a field, and none of the binary arithmetic operations are total, as witnessed for example by 0⋅∞ being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, a vertical line has infinite slope.

Extensions of the real lineEdit

The projectively extended real 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  .

In contrast, the extended real number line (also called the two-point compactification of the real line) distinguishes between   and  .

OrderEdit

The order relation cannot be extended to   in a meaningful way. Given a number  , there is no convincing argument to define either   or that  . Since   can't be compared with any of the other elements, there's no point in retaining this relation on  . However, order on   is used in definitions in  .

GeometryEdit

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  .

Arithmetic operationsEdit

Motivation for arithmetic operationsEdit

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 definedEdit

In addition to the standard operations on the subset   of  , the following operations are defined for  , with exceptions as indicated:

 

Arithmetic operations that are left undefinedEdit

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.[a] Consequently, they are left undefined:

 

Algebraic propertiesEdit

The following equalities mean: Either both sides are undefined, or both sides are defined and equal. This is true for any  .

 

The following is true whenever the right-hand side is defined, for any  .

 

In general, all laws of arithmetic that are valid for   are also valid for   whenever all the occurring expressions are defined.

Intervals and topologyEdit

The concept of an interval can be extended to  . 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  ):

 

With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints.

  and the empty set are each also an interval, as is   excluding any single point.[b]

The open intervals as base define a topology on  . Sufficient for a base are the finite open intervals in   and the intervals   for all   such that  .

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  .

Interval arithmeticEdit

Interval arithmetic extends to   from  . 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.[c] In particular, we have, for every  :

 

irrespective of whether either interval includes   and  .

CalculusEdit

The tools of calculus can be used to analyze functions of  . The definitions are motivated by the topology of this space.

NeighbourhoodsEdit

Let  .

  • A is a neighbourhood of x, if and only if A contains an open interval B and  .
  • A is a right-sided neighbourhood of x, if and only if there is   such that A contains  .
  • A is a left-sided neighbourhood of x, if and only if there is   such that A contains  .
  • A is a (right-sided, left-sided) punctured neighbourhood of x, if and only if there is   such that B is a (right-sided, left-sided) neighbourhood of x, and  .

LimitsEdit

Basic definitions of limitsEdit

Let  .

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

 

if and only if for every neighbourhood A of L, there is a punctured neighbourhood B of p, such that   implies  .

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

      

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

It can be shown that   if and only if both   and  .

Comparison with limits in  Edit

The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements,  , the first limit is as defined above, and the second limit is in the usual sense:

  •   is equivalent to  .
  •   is equivalent to  .
  •   is equivalent to  .
  •   is equivalent to  .
  •   is equivalent to  .
  •   is equivalent to  .

Extended definition of limitsEdit

Let  . Then p is a limit point of A if and only if every neighbourhood of p includes a point   such that  .

Let  , 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   implies  .

This corresponds to the regular topological definition of continuity, applied to the subspace topology on  , and the restriction of f to  .

ContinuityEdit

Let

 

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

 

Let

 

f is continuous in A if and only if for every  , 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  . Also, If tan is extended so that

 

then tan is continuous in  . However, many elementary functions, such as trigonometric and exponential functions, are discontinuous at  . For example, sin is continuous in   but discontinuous at  .

Thus 1/x is continuous on   but not on the affinely extended real number system  . Conversely, the function arctan can be extended continuously on  , but not on  .

As a projective rangeEdit

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.

NotesEdit

  1. ^ An extension does however exist in which all the algebraic properties, when restricted to defined operations in  , resolve to the standard rules: see Wheel theory.
  2. ^ If consistency of complementation is required, such that   and   for all   (where the interval on either side is defined), all intervals excluding   and   may be naturally represented using this notation, with   being interpreted as  , and half-open intervals with equal endpoints, e.g.  , remaining undefined.
  3. ^ For example, the ratio of intervals   contains   in both intervals, and since   is undefined, the result of division of these intervals is undefined.

See alsoEdit

External linksEdit