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Interval (mathematics)

An interval is a connected subset of the real numbers containing every point between two endpoints. Intervals can be open, closed, half-open, bounded or unbounded and are fundamental in analysis and topology.

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Overview

An interval is a collection of real numbers that contains all numbers lying between any two of its members. In the real number line, intervals capture the intuitive idea of a continuous chunk of numbers rather than isolated points. They are a basic building block in calculus, real analysis, measure theory and topology and frequently appear whenever one speaks of ranges, domains, or solutions described by inequalities. For further formal background see related mathematical references.

Notation and common types

Intervals are usually specified by their endpoints with brackets or parentheses: square brackets [a, b] denote inclusion of an endpoint and parentheses (a, b) denote exclusion. The main types are:

  • Closed interval [a, b]: contains both endpoints a and b.
  • Open interval (a, b): contains none of the endpoints.
  • Half-open (or half-closed) [a, b) or (a, b]: contains exactly one endpoint.
  • Unbounded intervals such as [a, ∞), (−∞, b), or (−∞, ∞) (the whole line).

Properties and characterizations

Intervals are precisely the connected subsets of the real line: any set that is connected in the standard topology on ℝ is an interval (possibly a single point). Bounded intervals have a finite length or measure equal to the difference b−a when endpoints are finite. Intervals are convex: if x and y lie in an interval then every convex combination tx+(1−t)y (with 0<=t<=1) also lies in it. Many operations preserve interval structure — for example the intersection of intervals is again an interval, while the union of two intervals need not be (unless they overlap or touch).

Notation alternatives and examples

Besides bracket notation, intervals can be described by inequalities or set-builder notation. For example, (3.3, 15] can be written {x∈ℝ : 3.3<x≤15}. Examples include simple bounded sets like (4, 9.6), symmetric ranges like [−100, 100], and one-sided bounds like [−30, ∞). These notations are used to specify domains of functions, solution sets of inequalities, and integration limits.

Historical and conceptual context

The formal study of intervals became clearer with the rigorous construction of the real numbers in the 19th century (for example via Dedekind cuts and Cauchy sequences), which provided a solid foundation for talking about order, completeness and limits. Intervals play a central role in statements such as the Intermediate Value Theorem and in characterizing continuous and monotone functions.

Intervals appear across mathematics and applications: they describe feasible ranges in optimization, confidence intervals in statistics, domains of integration, and constraints in numerical methods. Interval arithmetic treats intervals as numbers to propagate bounds through calculations. In other ordered sets and in topology one generalizes the idea of an interval to order intervals or connected subsets. These variations preserve the essential idea: containing all points between any two elements.

Examples

In the set of natural numbers

\{5,6,7,8,9\}

In this case of a discrete set, the elements of the interval are adjacent.

In the set of real numbers

[0,1]=\{x\in \mathbb{R} \mid 0\leq x\leq 1\},

the set of all numbers between 0 and 1, where the endpoints 0 and 1 are included.

Trivial examples of intervals are the empty set and sets that have exactly one element. If one does not want to include these, then one speaks of real intervals.

The set \{5,6,7,8,9\}can also be considered as a subset of the carrier set of real numbers. In this case, it is not an interval, since the set does not contain, for example, the non-natural numbers lying between 6 and 7.

The carrier set of the real numbers plays a special role among the mentioned carrier sets for intervals insofar as it is order-complete (see also Dedekind cut). Intervals are in this case exactly the subsets connected in the sense of the topology.

n-dimensional intervals

Definition

Analogously, for defines n\in \mathbb {N} in the n-dimensional space \mathbb {R} ^{n}any n-dimensional interval (cuboid).

{\displaystyle I:=I_{1}\times \dotsb \times I_{n}}with arbitrary intervals {\displaystyle I_{1},\dotsc ,I_{n}\subseteq \mathbb {R} .}

Constrained n-dimensional intervals

Now let a,b\in \mathbb{R} ^{n}with {\displaystyle a=(a_{1},\dotsc ,a_{n})}and {\displaystyle b=(b_{1},\dotsc ,b_{n})}, then specifically:

Completed interval

{\displaystyle [a,b]:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}\leq b_{1},\dotsc ,a_{n}\leq x_{n}\leq b_{n}\}.}

Open interval

{\displaystyle (a,b)={]{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}<b_{1},\dotsc ,a_{n}<x_{n}<b_{n}\}.}

Half-open (more precisely right-open) interval

{\displaystyle [a,b)={[{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}<b_{1},\dotsc ,a_{n}\leq x_{n}<b_{n}\}.}

Half-open (more precisely left-open) interval

{\displaystyle (a,b]={]{a,b}]}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}\leq b_{1},\dotsc ,a_{n}<x_{n}\leq b_{n}\}.}

Generalization

In topology, real intervals are examples of connected sets; in fact, a subset of the real numbers is connected precisely if it is an interval. Open intervals are open sets and closed intervals are closed sets. Semi-open intervals are neither open nor closed. Closed bounded intervals are compact.

All notations made here for the real numbers \mathbb {R} can be directly applied to any totally ordered set.

See also

Questions and answers

Q: What is an interval in mathematics?

A: An interval in mathematics is a group of numbers that includes all numbers between the beginning and the end.

Q: How do you determine which numbers are inside an interval?

A: 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.

Q: Do both the beginning and end numbers have to be included in an interval?

A: The beginning number and end number may or may not be inside the interval.

Q: How do you write an interval?

A: To write an interval, write either a square bracket ( [ ) or a parenthesis ( ( ), then include the beginning number, followed by a comma ( , ), then include the end number, followed by either a closing square bracket ( ] ) or a closing parenthesis ( ).

Q: Can you give examples of intervals?

A: Examples of intervals are (4, 9.6), [-100, 100], [-30, -4).

Q: Are negative numbers allowed within an interval?

A: Yes, negative numbers can be included within an interval.

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