In

_{1}, _{2}, _{3}: respectively, the elements of that are less than , the singleton $;\; href="/html/ALL/s/,x.html"\; ;"title=",x">,x$, and the elements that are greater than . The parts _{1} and _{3} are both non-empty (and have non-empty interiors), if and only if is in the interior of . This is an interval version of the trichotomy principle.

Review of "Calculus of Approximations"

from Mathematical Reviews with the

"Theory of interval algebra and its application to numerical analysis"

In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29–46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp. 126–143.

American Scientist article

provides an introduction.

Interval Notation

by George Beck,

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 ...

, a (real) interval is a set of 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 contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers $\backslash R$, the set of nonnegative real numbers, the set of positive real numbers, the empty set #REDIRECT Empty set#REDIRECT Empty set
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, cha ...

, and any singleton (set of one element).
Real intervals play an important role in the theory of , because they are the simplest sets whose "size" (or "measure" or "length") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the 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 ...

and eventually to the Lebesgue measureIn measure theory
In mathematics, a measure on a set (mathematics), set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization ...

.
Intervals are central to 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 usin ...

, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.
Intervals are likewise defined on an arbitrary 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 ...

set, such as integers
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

or rational numbers
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). ...

. The notation of integer intervals is considered in the special section below.
Terminology

An open interval does not include its endpoints, and is indicated with parentheses. For example, means greater than and less than . This means . A closed interval is an interval which includes all its limit points, and is denoted with square brackets. For example, means greater than or equal to and less than or equal to . A half-open interval includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. For example, means greater than and less than or equal to , while means greater than or equal to and less than . A degenerate interval is any set consisting of a single real number (i.e., an interval of the form ). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements. An interval is said to be left-bounded or right-bounded, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite intervals. Bounded intervals arebounded set
An artist's impression of a bounded set (top) and of an unbounded set (bottom). The set at the bottom continues forever towards the right.
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology)
In topology
...

s, in the sense that their diameter
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

(which is equal to the absolute difference
The absolute difference of two real number
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 ...

between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as , and the size of the empty interval may be defined as (or left undefined).
The centre (midpoint
282px, The midpoint of the segment (1, 1) to (2, 2)
In geometry, the midpoint is the middle point (geometry), point of a line segment. It is Distance, equidistant from both endpoints, and it is the centroid both of the segment and of the endp ...

) of bounded interval with endpoints and is , and its radius is the half-length . These concepts are undefined for empty or unbounded intervals.
An interval is said to be left-open if and only if it contains no minimum
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 ...

(an element that is smaller than all other elements); right-open if it contains 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 ...

; and open if it has both properties. The interval , for example, is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negative reals, is a right-open but not left-open interval. The open intervals are open set
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 ...

s of the real line in its 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 ...

, and form a base
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

of the open sets.
An interval is said to be left-closed if it has a minimum element, right-closed if it has a maximum, and simply closed if it has both. These definitions are usually extended to include the empty set and the (left- or right-) unbounded intervals, so that the closed intervals coincide with closed set
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

s in that topology.
The interior of an interval is the largest open interval that is contained in ; it is also the set of points in which are not endpoints of . The closure of is the smallest closed interval that contains ; which is also the set augmented with its finite endpoints.
For any set of real numbers, the interval enclosure or interval span of is the unique interval that contains , and does not properly contain any other interval that also contains .
An interval is subinterval of interval if is a 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). ...

of . An interval is a proper subinterval of if is a proper 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). ...

of .
Note on conflicting terminology

The terms segment and interval have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The ''Encyclopedia of Mathematics'' defines ''interval'' (without a qualifier) to exclude both endpoints (i.e., open interval) and ''segment'' to include both endpoints (i.e., closed interval), while Rudin's ''Principles of Mathematical Analysis'' calls sets of the form 'a'', ''b''''intervals'' and sets of the form (''a'', ''b'') ''segments'' throughout. These terms tend to appear in older works; modern texts increasingly favor the term ''interval'' (qualified by ''open'', ''closed'', or ''half-open''), regardless of whether endpoints are included.Notations for intervals

The interval of numbers between and , including and , is often denoted . The two numbers are called the ''endpoints'' of the interval. In countries where numbers are written with adecimal comma
A decimal separator is a symbol used to separate the integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional ...

, a semicolon
The semicolon or semi-colon is a symbol commonly used as orthographic punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to ...

may be used as a separator to avoid ambiguity.
Including or excluding endpoints

To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described inInternational standard
An international standard is a technical standard
A technical standard is an established 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 (philoso ...

ISO 31-11
ISO 31-11:1992 was the part of international standard
An international standard is a technical standard developed by one or more international standards organization, standards organizations. International standards are available for considerati ...

. Thus, in set builder notation
Set, The Set, or SET may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics)
In mathematics, a set is a collection of Distinct (mathematics), distinct Element (mathematics), elements. The elements that make up a set ...

,
: $\backslash begin\; a,b\; =\; \backslash mathopena,b\backslash mathclose\; \&=\; \backslash ,\; \backslash \backslash \; a,b\; =\; \backslash mathopen\; a,b\backslash mathclose\; \&=\; \backslash ,\; \backslash \backslash \; a,b\; =\; \backslash mathopena,b\backslash mathclose\; \&=\; \backslash ,\; \backslash \backslash \; a,b\; =\; \backslash mathopen\; a,b\backslash mathclose\; \&=\; \backslash .\; \backslash end$
Each interval , , and represents the empty set #REDIRECT Empty set#REDIRECT Empty set
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, cha ...

, whereas denotes the singleton set . When , all four notations are usually taken to represent the empty set.
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation is often used to denote an ordered pair
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 set theory, the coordinates
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of a point
Point or points may refer to:
Places
* Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point
Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

or vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

in analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

and linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, or (sometimes) a complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

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 ...

. That is why Bourbaki introduced the notation to denote the open interval. The notation too is occasionally used for ordered pairs, especially in computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

.
Some authors use to denote the complement of the interval ; namely, the set of all real numbers that are either less than or equal to , or greater than or equal to .
Infinite endpoints

In some contexts, an interval may be defined as a subset of theextended real numbers
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and where the infinities are treated as actual numbers. It is useful in describing the algebra on infiniti ...

, the set of all real numbers augmented with and .
In this interpretation, the notations , , , and are all meaningful and distinct. In particular, denotes the set of all ordinary real numbers, while denotes the extended reals.
Even in the context of the ordinary reals, one may use an infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (band), a South Korean boy band
*''Infinite'' (EP), debut EP of American musi ...

endpoint to indicate that there is no bound in that direction. For example, is the set of positive real numbersIn 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 ha ...

, also written as $\backslash mathbb\_+$. The context affects some of the above definitions and terminology. For instance, the interval = $\backslash R$ is closed in the realm of ordinary reals, but not in the realm of the extended reals.
Integer intervals

When and areinteger
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, the notation ⟦''a, b''⟧, or or or just , is sometimes used to indicate the interval of all ''integers'' between and included. The notation is used in some programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s; in Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, French ...

, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array
ARRAY, also known as ARRAY Now, is an independent distribution company launched by film maker and former publicist Ava DuVernay
Ava Marie DuVernay (; born August 24, 1972) is an American filmmaker. She won the directing award in the U.S. dram ...

.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing , , or . Alternate-bracket notations like or are rarely used for integer intervals.
Classification of intervals

The intervals of real numbers can be classified into the eleven different types listed below, where and are real numbers, and $a\; <\; b$: : Empty: $;\; href="/html/ALL/s/,a.html"\; ;"title=",a">,a$ : Degenerate: $;\; href="/html/ALL/s/,a.html"\; ;"title=",a">,a$ : Proper and bounded: :: Open: $(a,b)\; =\; \backslash $ :: Closed: $;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$ :: Left-closed, right-open: $;\; href="/html/ALL/s/,b)\_=\_\backslash $ :: Left-open, right-closed: $(a,b=\; \backslash $ : Left-bounded and right-unbounded: :: Left-open: $(a,+\backslash infty)\; =\; \backslash $ :: Left-closed: $;\; href="/html/ALL/s/,+\backslash infty)\_=\_\backslash $ : Left-unbounded and right-bounded: :: Right-open: $(-\backslash infty,b)\; =\; \backslash $ :: Right-closed: $(-\backslash infty,b=\; \backslash $ : Unbounded at both ends (simultaneously open and closed): $(-\backslash infty,+\backslash infty)\; =\; \backslash R$:Properties of intervals

The intervals are precisely the connected subsets of $\backslash R$. It follows that the image of an interval by anycontinuous 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 gen ...

is also an interval. This is one formulation of the intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if ''f'' is a continuous function whose domain contains the interval 'a'', ''b'' then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point ...

.
The intervals are also the s of $\backslash R$. The interval enclosure of a subset $X\backslash subseteq\; \backslash R$ is also the convex hull
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

of $X$.
The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other (e.g., $(a,b)\; \backslash cup;\; href="/html/ALL/s/,c.html"\; ;"title=",c">,c$).
If $\backslash R$ is viewed as a metric space
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 ...

, its open ball
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 ...

s are the open bounded sets , and its closed ball
In mathematics, a ball is the volume space bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not ...

s are the closed bounded sets .
Any element of an interval defines a partition of into three disjoint intervals Dyadic intervals

A ''dyadic interval'' is a bounded real interval whose endpoints are $\backslash frac$ and $\backslash frac$, where $j$ and $n$ are integers. Depending on the context, either endpoint may or may not be included in the interval. Dyadic intervals have the following properties: * The length of a dyadic interval is always an integer power of two. * Each dyadic interval is contained in exactly one dyadic interval of twice the length. * Each dyadic interval is spanned by two dyadic intervals of half the length. * If two open dyadic intervals overlap, then one of them is a subset of the other. The dyadic intervals consequently have a structure that reflects that of an infinitebinary tree
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , ...

.
Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement
In numerical analysis
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numerical analysis is the study of ...

, multigrid methods and wavelet analysisA wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets a ...

. Another way to represent such a structure is p-adic analysis
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 ...

(for ).
Generalizations

Multi-dimensional intervals

In many contexts, an $n$-dimensional interval is defined as a subset of $\backslash R^n$ that is theCartesian 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 ...

of $n$ intervals, $I\; =\; I\_1\backslash times\; I\_2\; \backslash times\; \backslash cdots\; \backslash times\; I\_n$, one on each coordinate
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

axis.
For $n=2$, this can be thought of as region bounded by a square
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

or rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

, whose sides are parallel to the coordinate axes, depending on whether the width of the intervals are the same or not; likewise, for $n=3$, this can be thought of as a region bounded by an axis-aligned cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

or a rectangular cuboid
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to a ...

.
In higher dimensions, the Cartesian product of $n$ intervals is bounded by an n-dimensional hypercube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

or hyperrectangle
In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. It is formally defined as the Cartesian product of orthogonal interval (mathematics), intervals.
Types
A thr ...

.
A facet of such an interval $I$ is the result of replacing any non-degenerate interval factor $I\_k$ by a degenerate interval consisting of a finite endpoint of $I\_k$. The faces of $I$ comprise $I$ itself and all faces of its facets. The corners of $I$ are the faces that consist of a single point of $\backslash R^n$.
Complex intervals

Intervals ofcomplex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s can be defined as regions of the complex plane
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 ...

, either rectangular
In Euclidean plane geometry, a rectangle is a quadrilateral
A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for ...

or circular
Circular may refer to:
* The shape of a circle
* Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fa ...

.
Topological algebra

Intervals can be associated with points of the plane, and hence regions of intervals can be associated withregion
In geography
Geography (from Ancient Greek, Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and Solar System, planets. The ...

s of the plane. Generally, an interval in mathematics corresponds to an ordered pair (''x,y'') taken from the direct productIn 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 ha ...

R × R of real numbers with itself, where it is often assumed that ''y'' > ''x''. For purposes of mathematical structure
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 ...

, this restriction is discarded, and "reversed intervals" where ''y'' − ''x'' < 0 are allowed. Then, the collection of all intervals 'x,y''can be identified with the topological ringIn 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 ha ...

formed by the 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 ...

of R with itself, where addition and multiplication are defined component-wise.
The direct sum algebra $(\; R\; \backslash oplus\; R,\; +,\; \backslash times)$ has two ideal
Ideal may refer to:
Philosophy
* Ideal (ethics)
An ideal is a principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law
Law is a system
A system is a group of Interaction, interacting ...

s, and . The identity element
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 th ...

of this algebra is the condensed interval ,1 If interval 'x,y''is not in one of the ideals, then it has multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

/''x'', 1/''y'' Endowed with the usual 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 ...

, the algebra of intervals forms a topological ringIn 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 ha ...

. The group of units
In the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...

of this ring consists of four quadrant
Quadrant may refer to:
Companies
* Quadrant Cycle Company, 1899 manufacturers in Britain of the Quadrant motorcar
* Quadrant (motorcycles), one of the earliest British motorcycle manufacturers, established in Birmingham in 1901
* Quadrant Private ...

s determined by the axes, or ideals in this case. The identity component
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 ...

of this group is quadrant I.
Every interval can be considered a symmetric interval around its midpoint
282px, The midpoint of the segment (1, 1) to (2, 2)
In geometry, the midpoint is the middle point (geometry), point of a line segment. It is Distance, equidistant from both endpoints, and it is the centroid both of the segment and of the endp ...

. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" 'x'', −''x''is used along with the axis of intervals 'x,x''that reduce to a point. Instead of the direct sum $R\; \backslash oplus\; R$, the ring of intervals has been identifiedD. H. Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas
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François ...

(1956Review of "Calculus of Approximations"

from Mathematical Reviews with the

split-complex number
In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Cu ...

plane by M. Warmus and D. H. Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas
__NOTOC__
François ...

through the identification
: ''z'' = (''x'' + ''y'')/2 + j (''x'' − ''y'')/2.
This linear mapping of the plane, which amounts of a ring isomorphism
In ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...

, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition
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 ...

.
See also

*Arc (geometry)
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 ...

*Inequality
Inequality may refer to:
Economics
* Attention inequality
Attention inequality is a term used to target the inequality of distribution of attention across users on social networks, people in general, and for scientific papers. Yun Family Foundat ...

*Interval graph
upright=1.35, Seven intervals on the real line and the corresponding seven-vertex interval graph.
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to mo ...

* Interval finite element
*Interval (statistics)In statistics, interval estimation is the use of Sample (statistics), sample data to calculate an interval (mathematics), interval of possible values of an unknown population parameter; this is in contrast to point estimation, which gives a single va ...

*Line segment
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

*Partition of an interval
In mathematics, a partition of an interval (mathematics), interval on the real line is a finite sequence of real numbers such that
:.
In other terms, a partition of a compact space, compact interval is a strictly increasing sequence of numbers ...

*Unit interval
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 ...

References

Bibliography

* T. Sunaga"Theory of interval algebra and its application to numerical analysis"

In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29–46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp. 126–143.

External links

* ''A Lucid Interval'' by Brian Hayes: AAmerican Scientist article

provides an introduction.

Interval Notation

by George Beck,

Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hoste ...

.
*
{{DEFAULTSORT:Interval (Mathematics)
Sets of real numbers
Order theory
Topology