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.
A half-open interval includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. For example, (0,1] means greater than 0 and less than or equal to 1, while [0,1) means greater than or equal to 0 and less than 1.
A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a,a]). 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 (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 0 (or left undefined).
The centre (midpoint) of bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length |a − b|/2. These concepts are undefined for empty or unbounded intervals.
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 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 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.
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. In countries where numbers are written with a decimal comma, a semicolon may be used as a separator to avoid ambiguity.
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. Thus, in set builder notation,
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. That is why Bourbaki introduced the notation ]a, b[ to denote the open interval. 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 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 [−∞, b] , (−∞, b] , [a, +∞] , and [a, +∞) are all meaningful and distinct. In particular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals.
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 [a .. b) or [a .. b[ are rarely used for integer intervals.
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 (for p = 2).
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 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.
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.