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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
Contents
- Dividing by zero
- Extensions of the real line
- Order
- Geometry
- Motivation for arithmetic operations
- Arithmetic operations that are defined
- Arithmetic operations that are left undefined
- Algebraic properties
- Intervals and topology
- Interval arithmetic
- Calculus
- Neighbourhoods
- Basic definitions of limits
- Comparison with limits in R displaystyle mathbb R
- Extended definition of limits
- Continuity
- As a projective range
- References
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 zero
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 line
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
Order
The order relation cannot be extended to
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
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
In addition to the standard operations on the subset
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. Consequently, they are left undefined:
Algebraic properties
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
Intervals and topology
The concept of an interval can be extended to
With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints.
The open intervals as base define a topology on
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 arithmetic
Interval arithmetic extends to
irrespective of whether either interval includes
Calculus
The tools of calculus can be used to analyze functions of
Neighbourhoods
Let
Basic definitions of limits
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
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
It can be shown that
Comparison with limits in R {displaystyle mathbb {R} }
The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements,
Extended definition of limits
Let
Let
This corresponds to the regular topological definition of continuity, applied to the subspace topology on
Continuity
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
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
then tan is continuous in
Thus 1/x is continuous on
As a 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.