Interval (mathematics)
Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound.
Interval arithmetic consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of input data and rounding errors.
Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers.
The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers.
If one allows an endpoint in the closed side to be an infinity (such as (0,+∞]), the result will not be an interval, since it is not even a subset of the real numbers.
Instead, the result can be seen as an interval in the extended real line, which occurs in measure theory, for example.
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.
The diameter may be called the length, width, measure, range, or size of the interval.
The centre (midpoint) of a bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length |a − b|/2.
The closure of I is the smallest closed interval that contains I; which is also the set I augmented with its finite endpoints.
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.
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.
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.
Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction.
The notation [a .. b] 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.
Another way to interpret integer intervals are as sets defined by enumeration, using ellipsis notation.
is viewed as a metric space, its open balls are the open bounded intervals (c + r, c − r), and its closed balls are the closed bounded intervals [c + r, c − r].
-dimensional Euclidean space, a ball is the set of points whose distance from the center is less than the radius.
If a half-space is taken as a kind of degenerate ball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint.
Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any
That is, the convex components of a subset of a totally ordered set form a partition.
83, Theorem 2.3.23 The concepts of convex sets and convex components are used in a proof that every totally ordered set endowed with the order topology is completely normal[15] or moreover, monotonically normal.
Generally, an interval in mathematics corresponds to an ordered pair (x, y) taken from the direct product
For purposes of mathematical structure, this restriction is discarded,[18] 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
The group of units of this ring consists of four quadrants determined by the axes, or ideals in this case.
the ring of intervals has been identified[19] with the hyperbolic numbers by M. Warmus and D. H. Lehmer through the identification where
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.
