Interval (mathematics)

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In mathematics, an (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 Template:Mvar satisfying 0 ≤ x ≤ 1 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.




Notations for intervals

The interval of numbers between Template:Mvar and Template:Mvar, including Template:Mvar and Template:Mvar, is often denoted Template:Closed-closed. The two numbers are called the endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be used as a separator, to avoid ambiguity.

Including or excluding endpoints


Note that Template:Open-open, Template:Closed-open, and Template:Open-closed represent the empty set, whereas Template:Closed-closed denotes the set {a} . When a > b, all four notations are usually assumed 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 of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. The notation too is occasionally used for ordered pairs, especially in computer science.

Some authors use to denote the complement of the interval Template:Open-open; namely, the set of all real numbers that are either less than or equal to Template:Mvar, or greater than or equal to Template:Mvar.

Infinite endpoints

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, Template:Open-open is the set of all positive real numbers, and Template:Open-open is the set of real numbers.

The extended real number line includes −∞ and +∞ as elements. The notations Template:Closed-closed , Template:Closed-open , Template:Closed-closed , and Template:Open-closed may be used in this context. For example Template:Open-closed means the extended real numbers excluding only −∞.

Integer intervals

The notation Template:Closed-closed when Template:Mvar and Template:Mvar are integers, or {a .. b} , or just a .. b is sometimes used to indicate the interval of all integers between Template:Mvar and Template:Mvar, including both. This 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 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 a .. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notations like Template:Closed-open or [a .. b[ are rarely used for integer intervals{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}.


An open interval does not include its endpoints, and is indicated with parentheses. For example Template:Open-open means greater than 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example Template:Closed-closed means greater than or equal to 0 and less than or equal to 1.




The centre (midpoint) of bounded interval with endpoints Template:Mvar and Template:Mvar is (a + b)/2, and its radius is the half-length Template:Mabs/2. These concepts are undefined for empty or unbounded intervals.



The interior of an interval Template:Mvar is the largest open interval that is contained in Template:Mvar; it is also the set of points in Template:Mvar which are not endpoints of Template:Mvar. The closure of Template:Mvar is the smallest closed interval that contains Template:Mvar; which is also the set Template:Mvar augmented with its finite endpoints.

For any set Template:Mvar of real numbers, the interval enclosure or interval span of Template:Mvar is the unique interval that contains Template:Mvar and does not properly contain any other interval that also contains Template:Mvar.

Classification of intervals

The intervals of real numbers can be classified into eleven different types, listed below; where Template:Mvar and Template:Mvar are real numbers, with :

proper and bounded:
left-closed, right-open:
left-open, right-closed:
left-bounded and right-unbounded:
left-unbounded and right-bounded:
unbounded at both ends:

Intervals of the extended real line


In this interpretation, the notations Template:Closed-closed , Template:Closed-open , Template:Closed-closed , and Template:Open-closed are all meaningful and distinct. In particular, Template:Open-open denotes the set of all ordinary real numbers, while Template:Closed-closed denotes the extended reals.

This choice affects some of the above definitions and terminology. For instance, the interval Template:Open-open =  is closed in the realm of ordinary reals, but not in the realm of the extended reals.

Properties of intervals




If is viewed as a metric space, its open balls are the open bounded sets Template:Open-open, and its closed balls are the closed bounded sets Template:Closed-closed.

Any element Template:Mvar of an interval Template:Mvar defines a partition of Template:Mvar into three disjoint intervals Template:Mvar1, Template:Mvar2, Template:Mvar3: respectively, the elements of Template:Mvar that are less than Template:Mvar, the singleton , and the elements that are greater than Template:Mvar. The parts Template:Mvar1 and Template:Mvar3 are both non-empty (and have non-empty interiors) if and only if Template:Mvar is in the interior of Template:Mvar. This is an interval version of the trichotomy principle.

Dyadic intervals


Dyadic intervals have the following properties:





Multi-dimensional intervals


For , this generally defines a rectangle whose sides are parallel to the coordinate axes; for , it defines an axis-aligned rectangular box.


Complex intervals


Topological algebra






See also



External links