In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers , the set of all negative real numbers, and the empty set.
In fact, intervals are meaningful in any (totally or partially) ordered set, not just in the reals; so one can have intervals of rational numbers, integers, computerrepresentable floating point numbers, or subsets of a set (ordered by inclusion), for example.
Real intervals play an important role in the theory of integration, 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 and eventually to the Lebesgue measure.
Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.
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The interval of numbers between a and b, including a and b, is often denoted [a,b]. The two numbers are called the endpoints of the interval.
To indicate that one of the endpoints is to be excluded from the set, many writers substitute a parenthesis for the corresponding square bracket. Thus, in set builder notation,
Note that (a,a), [a,a), and (a,a] denote the empty set, whereas [a,a] denotes the set {a}. When a > b, all four notations are usually assumed to represent the empty set.
International standard ISO 3111 also defines another notation for intervals, which seems to be more commonly taught in Europe^{[citation needed]} and South America. It uses an inwards pointing bracket to indicate inclusion of the endpoint, and outwardspointing bracket for exclusion:
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a,b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. The notation [a,b] too is occasionally used for ordered pairs, especially in computer science.
Some authors use ]a,b[ to denote the complement of the interval (a,b); namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.
In countries where numbers are written with a decimal comma, a semicolon may be used as a separator, to avoid ambiguity.
In both styles of notation, one may use an infinite endpoint to indicate that there is no bound in that direction. Specifically, one may use or (or both). For example, is the set of all positive real numbers, and is the set of real numbers.
The notations , , , and are ambiguous. For authors who define intervals as subsets of the real numbers, those notations are either meaningless, or equivalent to the open variants. In the latter case, the interval comprising all real numbers is both open and closed, .
When a and b are integers, the notation , , or just is sometimes used to indicate the interval of all integers between a and b, including both. This notation is used in some programming languages; in Pascal, for example, it is used to define the set of valid indices of a vector.
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 . Alternatebracket notations like or are rarely used for integer intervals.
An open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greater than 0 and less than 1. Conversely, a closed interval includes its endpoints, and is denoted with square brackets. For example [0,1] means greater than or equal to 0 and less than or equal to 1.
A degenerate interval is any set consisting of a single real number. Some authors include the empty set in this definitions. An interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements.
An interval is said to be leftbounded or rightbounded 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 rightbounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be halfbounded. 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 (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, 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 0 or left undefined.
The centre of bounded interval with endpoints a and b is (a + b) / 2, and its radius is the halflength . These concepts are undefined for empty or unbounded intervals.
An interval is said to be leftopen if and only if it has no minimum (an element that is smaller than all other elements); rightopen if it has no maximum; and open if it has both properties. The interval , for example, is leftclosed and rightopen. The empty set and the set of all reals are open intervals, while the set of nonnegative reals, for example, is a rightopen but not leftopen interval. The open intervals coincide with the open sets of the real line in its standard topology.
An interval is said to be leftclosed if it has a minimum element, rightclosed if it has a maximum, and simply closed if it has both. These definitions are usually extended to include the empty set and to the (left or right) unbounded intervals, so that the closed intervals coincide with closed sets in that topology.
The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which are not endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented with its finite endpoints.
For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X and does not properly contain any other interval that also contains X.
The intervals of real rumbers can be classified into eleven different types, listed below; where a and b are real numbers, with a < b:
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.
This choice affects some of the above definitions and terminology. For instance, the interval is closed in the realm of ordinary reals, but not in the realm of the extended reals.
The intervals are precisely the connected subsets of . 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.
The intervals are also the convex subsets of . The interval enclosure of a subset is also the convex hull 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 nonempty intersection or an open endpoint of one interval is a closed endpoint of the other (e.g., ).
If is viewed as a metric space, its open balls are the open bounded sets (a,b), and its closed balls are the closed bounded sets [a,b].
Any element x of an interval I defines a partition of I into three disjoint intervals I_{1},I_{2},I_{3}: respectively, the elements of I that are less than x, the singleton [x,x] = {x}, and the elements that are greater than x. The parts I_{1} and I_{3} are both nonempty (and have nonempty interiors) if and only if x is in the interior of I. This is an interval version of the trichotomy principle.
A dyadic interval is a bounded real interval whose endpoints are and , where j and n are integers. Depending on the context, either endpoint may or may not be included in the interval.
Dyadic intervals have some nice properties, such as the following:
The dyadic intervals thus have a structure very similar to an infinite binary tree.
Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigrid methods, and wavelet analysis.
In many contexts, an ndimensional interval is defined as a subset of that is the Cartesian product of n intervals, , one on each coordinate axis.
For n = 2, this generally defines a rectangle whose sides are parallel to the coordinate axes; for n = 3, it defines an axisaligned rectangular box.
A facet of such an interval I is the result of replacing any nondegenerate 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 .
Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.^{[1]}
In mathematics an interval is a group of numbers that includes all numbers between the beginning and the end. Numbers that are larger than the beginning number and smaller than the end number are inside the interval, and numbers that are smaller than the beginning number or larger than the end number are not in the interval. The beginning number and end number may or may not be inside the interval. An example of an interval might be from 3.3 to 15. Numbers like 4, 8, 9.5, 14, and even 14.999 are inside this interval. Numbers like 4, 2, 3.2, 20, and 15.000001 are not inside this interval.
To write an interval, write either a square bracket ( [ ) or a parenthesis ( ( ), the beginning number, a comma ( , ), the end number, and either a closing square bracket ( ] ) or a closing parenthesis ( ) ). Examples of intervals are (4, 9.6), [100, 100], [30, 4).
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Intervals can be separated by how they act at their ends. Intervals can be closed, open, or mixed.
An interval that is closed also includes the beginning and the end. A closed interval that has 3 as the beginning and 5.4 as the end would include 3, 5.4, and every number between 3 and 5.4. To write a closed interval, use square brackets ( [ and ] ). An example of an closed interval is [136, 450].
An interval that is open does not include the beginning or the end. An open interval that has 3 as the beginning and 5.4 as the end would include every number between 3 and 5.4, but it would not include 3 or 5.4. To write an open interval, use parenthesis ( ( and ) ). An example of an open interval is (2, 5).
A mixed interval is open at one end and closed at the other end. This means that the interval may include the beginning but not the end, or it may include the end but not the beginning. The interval [9, 23) would include 9, but it would not include 23.
