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OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a (real) interval is a set 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 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 $\R$, the set of nonnegative real numbers, the set of positive real numbers, the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. 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 usi ...
, 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 set, such as integers or
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
. The notation of integer intervals is considered in the special section below.

# Terminology

An does not include its endpoints, and is indicated with parentheses. For example, means greater than and less than . This means . This interval can also be denoted by , see below. A 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 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 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 are bounded sets, in the sense that their
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid ...
(which is equal to the absolute difference 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 In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dime ...
) of a 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 (an element that is smaller than all other elements); right-open if it contains no
maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
; and open if it contains neither. The interval , for example, is left-closed and right-open. The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are open sets of the real line in its standard topology, and form a base of the open sets. An interval is said to be left-closed if it has a minimum element or is left-unbounded, right-closed if it has a maximum or is right unbounded; it is simply closed if it is both left-closed and right closed. So, the closed intervals coincide with the closed sets 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, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of . An interval is a proper subinterval of if is a
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
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 a
decimal comma A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The cho ...
, a semicolon 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 in International standard
ISO 31-11 ISO 31-11:1992 was the part of international standard ISO 31 that defines ''mathematical signs and symbols for use in physical sciences and technology''. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-8000 ...
. Thus, in set builder notation, : $\begin a,b = \mathopena,b\mathclose &= \, \\ a,b = \mathopen a,b\mathclose &= \, \\ a,b = \mathopena,b\mathclose &= \, \\ a,b = \mathopen a,b\mathclose &= \. \end$ Each interval , , and represents the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
, 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 set theory, the
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
of a point or
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. 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. Some authors such as Yves Tillé 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 the extended real numbers, 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 endpoint to indicate that there is no bound in that direction. For example, is the set of positive real numbers, also written as $\mathbb_+$. The context affects some of the above definitions and terminology. For instance, the interval  = $\R$ is closed in the realm of ordinary reals, but not in the realm of the extended reals.

## Integer intervals

When and are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 languages; in Pascal, 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 An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
. 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: * Degenerate: * Proper and bounded: ** Open: $\left(a,b\right) = \$ ** Closed: ** Left-closed, right-open: ** Left-open, right-closed: $\left(a,b= \$ * Left-bounded and right-unbounded: ** Left-open: $\left(a,+\infty\right) = \$ ** Left-closed: * Left-unbounded and right-bounded: ** Right-open: $\left(-\infty,b\right) = \$ ** Right-closed: $\left(-\infty,b= \$ * Unbounded at both ends (simultaneously open and closed): $\left(-\infty,+\infty\right) = \R$:

# Properties of intervals

The intervals are precisely the
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
subsets of $\R$. It follows that the image of an interval by any continuous function 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 , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two imp ...
. The intervals are also the convex subsets of $\R$. The interval enclosure of a subset $X\subseteq \R$ is also the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ...
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., . If $\R$ is viewed as a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, its
open ball In mathematics, a ball is the solid figure 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 defi ...
s are the open bounded sets , and its closed balls are the closed bounded sets . Any element  of an interval  defines a partition of  into three disjoint intervals 1, 2, 3: respectively, the elements of  that are less than , the singleton , 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.

A ''dyadic interval'' is a bounded real interval whose endpoints are $\frac$ and $\frac$, 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 infinite binary tree. Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigrid methods and wavelet analysis. Another way to represent such a structure is
p-adic analysis In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of ...
(for ).

# Generalizations

## Multi-dimensional intervals

In many contexts, an $n$-dimensional interval is defined as a subset of $\R^n$ that is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
of $n$ intervals, $I = I_1\times I_2 \times \cdots \times I_n$, one on each
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
axis. For $n=2$, this can be thought of as region bounded by a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
or rectangle, 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, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
or a
rectangular cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
. In higher dimensions, the Cartesian product of $n$ intervals is bounded by an
n-dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
or hyperrectangle. 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 $\R^n$.

## Complex intervals

Intervals of complex numbers can be defined as regions of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, either
rectangular In 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 parallelogram containi ...
or
circular Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning Circular may refer to: * The shape ...
.

# Topological algebra

Intervals can be associated with points of the plane, and hence regions of intervals can be associated with regions of the plane. Generally, an interval in mathematics corresponds to an ordered pair (''x,y'') taken from the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
R × R of real numbers with itself, where it is often assumed that ''y'' > ''x''. For purposes of mathematical structure, 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 ring formed by the direct sum of R with itself, where addition and multiplication are defined component-wise. The direct sum algebra $\left( R \oplus R, +, \times\right)$ has two ideals, and . The
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of this algebra is the condensed interval ,1 If interval 'x,y''is not in one of the ideals, then it has multiplicative inverse /''x'', 1/''y'' Endowed with the usual topology, the algebra of intervals forms a topological ring. The
group of units In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for thi ...
of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component of this group is quadrant I. Every interval can be considered a symmetric interval around its
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dime ...
. 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 $R \oplus R$, the ring of intervals has been identified D. H. Lehmer (1956
Review of "Calculus of Approximations"
from Mathematical Reviews
with the
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
plane by M. Warmus and D. H. Lehmer through the identification : ''z'' = (''x'' + ''y'')/2 + j (''x'' − ''y'')/2. This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.

*
Arc (geometry) In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
* Inequality *
Interval graph In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. I ...
* Interval finite element * Interval (statistics) * Line segment * Partition of an interval *
Unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...

# 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.