TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the real line, or real number line is the
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

whose
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
are the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. That is, the real line is the set of all real numbers, viewed as a
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ...

space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...
, namely the
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

one. It can be thought of as a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(or
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
), a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
, a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, a
measure space A measure space is a basic object of measure theory Measure is a fundamental concept of mathematics. Measures provide a mathematical abstraction for common notions like mass, distance/length, area, volume, probability of events, and — after si ...
, or a
linear continuum In the mathematical field of order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
. Just like the set of real numbers, the real line is usually denoted by the symbol (or alternatively, $\mathbb$, the letter “” in
blackboard bold Image:Blackboard bold.svg, 250px, An example of blackboard bold letters Blackboard bold is a typeface style that is often used for certain symbols in mathematics, mathematical texts, in which certain lines of the symbol (usually vertical or near-v ...

). However, it is sometimes denoted in order to emphasize its role as the first Euclidean space. This article focuses on the aspects of as a geometric space in
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, geometry, and
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

. The real numbers also play an important role in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

as a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, but in this context is rarely referred to as a line. For more information on in all of its guises, see
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
.

As a linear continuum

The real line is a
linear continuum In the mathematical field of order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
under the standard ordering. Specifically, the real line is
linearly ordered Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...
by , and this ordering is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
and has the
least-upper-bound property In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. In addition to the above properties, the real line has no
maximum In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...
or minimum element. It also has a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, namely the set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line. The real line also satisfies the countable chain condition: every collection of mutually ,
nonempty In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by includ ...
open intervals in is countable. In
order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to . This statement has been shown to be
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independent ...
of the standard axiomatic system of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
known as .

As a metric space

The real line forms a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
, with the
distance function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

given by absolute difference: : $d\left(x, y\right) = , x - y, .$ The
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
is clearly the 1-dimensional
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occas ...
. Since the -dimensional Euclidean metric can be represented in matrix form as the -by- identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1. If and , then the -
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
in centered at is simply the open interval . This real line has several important properties as a metric space: * The real line is a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
, in the sense that any
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of points converges. * The real line is
path-connected In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
and is one of the simplest examples of a geodesic metric space. * The
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), ...
of the real line is equal to one.

As a topological space

The real line carries a standard
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, which can be introduced in two different, equivalent ways. First, since the real numbers are
totally ordered In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, they carry an
order topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Second, the real numbers inherit a
metric topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
from the metric defined above. The order topology and metric topology on are the same. As a topological space, the real line is
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to the open interval . The real line is trivially a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological sp ...
of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. Up to homeomorphism, it is one of only two different connected 1-manifolds without
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
, the other being the
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

. It also has a standard differentiable structure on it, making it a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
. (Up to
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, there is only one differentiable structure that the topological space supports.) The real line is a
locally compact spaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
and a
paracompact space In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
, as well as
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base (topology), base. More explicitly, a topological space T is second-countable if there exists some countable ...
and . It is also
path-connected In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
, and is therefore connected as well, though it can be disconnected by removing any one point. The real line is also
contractible In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and as such all of its
homotopy group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s and
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero. This change is required to make statements without some number of exceptional cases ( ...
groups are zero. As a locally compact space, the real line can be compactified in several different ways. The
one-point compactificationIn the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematic ...
of is a circle (namely, the
real projective lineImage:Real projective line.svg, The real projective line can be modeled by the projectively extended real line, which consists of the real line together with a point at infinity; i.e., the one-point compactification of R. In geometry, a real projecti ...

), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two
ends ''Ends'' is a collection of science fiction Science fiction (sometimes shortened to sci-fi or SF) is a of which typically deals with and futuristic concepts such as advanced and , , , , and . It has been called the " of ", and it often ...
, and the resulting end compactification is the extended real line . There is also the
Stone–Čech compactificationIn the mathematical discipline of general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic defin ...
of the real line, which involves adding an infinite number of additional points. In some contexts, it is helpful to place other topologies on the set of real numbers, such as the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topological space, topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and h ...
or the
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
. For the real numbers, the latter is the same as the finite complement topology.

As a vector space

The real line is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
of real numbers (that is, over itself) of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. It has the usual multiplication as an
inner product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, making it a
Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ...
. The
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
defined by this inner product is simply the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.

As a measure space

The real line carries a canonical measure, namely the
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
. This measure can be defined as the completion of a
Borel measure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
defined on , where the measure of any interval is the length of the interval. Lebesgue measure on the real line is one of the simplest examples of a
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfré ...
on a
locally compact group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
.

In real algebras

The real line is a one-dimensional subspace of a real algebra ''A'' where R ⊂ ''A''. For example, in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''z'' = ''x'' + i''y'', the subspace is a real line. Similarly, the algebra of
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s :''q'' = ''w'' + ''x'' i + ''y'' j + ''z'' k has a real line in the subspace . When the real algebra is a
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
$A = R \oplus V,$ then a conjugation on ''A'' is introduced by the mapping $v \mapsto -v$ of subspace ''V''. In this way the real line consists of the fixed points of the conjugation.

*
Cantor–Dedekind axiomIn mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
* Hyperreal number line * Imaginary line (mathematics) *
Line (geometry) In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

*
Projectively extended real line Image:Real projective line.svg, 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 projectiv ...
*
Real projective lineImage:Real projective line.svg, The real projective line can be modeled by the projectively extended real line, which consists of the real line together with a point at infinity; i.e., the one-point compactification of R. In geometry, a real projecti ...

References

* * {{DEFAULTSORT:Real Line Real numbers Topological spaces es:Recta real eu:Zuzen erreal