In
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, the projectively extended real line (also called the
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
), is the extension of the set of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s,
by a point denoted . It is thus the set
with the standard arithmetic operations extended where possible, and is sometimes denoted by
The added point is called the
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
, because it is considered as a neighbour of both
ends
End, END, Ending, or variation, may refer to:
End
*In mathematics:
**End (category theory)
**End (topology)
**End (graph theory)
** End (group theory) (a subcase of the previous)
** End (endomorphism)
*In sports and games
**End (gridiron football ...
of the real line. More precisely, the point at infinity is the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of every
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of real numbers whose
absolute values are increasing and
unbounded.
The projectively extended real line may be identified with a
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...
in which three points have been assigned the specific values , 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 the intuitive concept of 'number', this structure allows
division by zero:
:
for nonzero ''a''. In particular , and moreover , making
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
, , a
total function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
in this structure. The structure, however, is not a
field, and none of the binary arithmetic operations are total, as witnessed for example by being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, a vertical line has ''infinite''
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
.
Extensions of the real line
The projectively extended real line extends the
field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s in the same way that the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
extends the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, by adding a single point called conventionally
.
In contrast, the
affinely extended real number line (also called the two-point
compactification of the real line) distinguishes between
and
.
Order
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
.
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 mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
, in fact
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. For example the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of 2×2 real
invertible matrices has a
transitive action on it. The
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
may be expressed by
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
s, (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
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
is invariant.
The terminology
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
is appropriate, because the points are in 1-to-1 correspondence with one-dimensional
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
s of
.
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
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 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
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 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.
The open intervals as
base define a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on
. Sufficient for a base are the finite open intervals in
and the intervals
for all
such that
.
As said, the topology is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. Thus it is
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
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
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...
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. In particular, we have, for every
:
:
irrespective of whether either interval includes
and
.
Calculus
The tools of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
can be used to analyze functions of
. The definitions are motivated by the topology of this space.
Neighbourhoods
Let
and
.
* is a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of ''x'', if ''A'' contains an open interval ''B'' that contains .
* is a right-sided neighbourhood of , if there is a real number such that
and contains the semi-open interval