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, , by a point denoted ∞.[1] It is thus the set with the standard arithmetic operations extended where possible,[1] and is sometimes denoted by [2] or 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:
for nonzero a. In particular, 1 / 0 = ∞ and 1 / ∞ = 0, making the
reciprocalfunction1 / 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]
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 ∞.
The
order relation cannot be extended to 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 .[2] However, order on is used in definitions in .
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
invertiblematrices 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 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 of , the following operations are defined for , with exceptions as indicated:[3][2]
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.[a] Consequently, they are left undefined:
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 expressions involved are 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 topology
The concept of an
interval can be extended to . 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
):[2][additional citation(s) needed]
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]
and the
empty set are also intervals, as is excluding any single point.[b]
The open intervals as a
base define a
topology on . Sufficient for a base are the
bounded 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 arithmetic
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 0 and ∞.
Calculus
The tools of
calculus can be used to analyze functions of . The definitions are motivated by the topology of this space.
Neighbourhoods
Let and .
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 and A contains the semi-open interval .
A is a left-sided neighbourhood of x, if there is a real number y such that and A contains the semi-open interval .
A is a
punctured neighbourhood (resp. a right-sided or a left-sided punctured neighbourhood) of x, if and is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of x.
Limits
Basic definitions of limits
Let and .
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
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 limits
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
is
continuous at p if and only if f is defined at p and
If the function
is continuous in A if and only if, for every , f is defined at p and the limit of as x tends to p through A is
Every
rational functionP(x)/Q(x), where P and Q are
polynomials, can be prolongated, in a unique way, to a function from to that is continuous in In particular, this is the case of
polynomial functions, which take the value at if they are not
constant.
then is continuous in but cannot be prolongated further to a function that is continuous in
Many
elementary functions that are continuous in cannot be prolongated to functions that are continuous in This is the case, for example, of the
exponential function and all
trigonometric functions. For example, the
sine function is continuous in but it cannot be made continuous at As seen above, the tangent function can be prolongated to a function that is continuous in but this function cannot be made continuous at
Many discontinuous functions that become continuous when the
codomain is extended to remain discontinuous if the codomain is extended to the
affinely extended real number system This is the case of the function On the other hand, some functions that are continuous in and discontinuous at become continuous if the
domain is extended to This is the case for the
arctangent.
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.
^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.
^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.
^For example, the ratio of intervals contains 0 in both intervals, and since 0 / 0 is undefined, the result of division of these intervals is undefined.
References
^
abcdeNBU, DDE (2019-11-05).
PG MTM 201 B1. Directorate of Distance Education, University of North Bengal.