Editing Interval (mathematics) (section)

==Notations for intervals==
The interval of numbers between {{mvar|a}} and {{mvar|b}}, including {{mvar|a}} and {{mvar|b}}, is often denoted {{closed-closed|''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.

===Including or excluding endpoints===
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]],

:<math>\begin{align}
(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R \mid a<x<b\},    \\[5mu]
[a,b) = \mathopen{[}a,b\mathclose{[} &= \{x\in\R \mid a\le x<b\}, \\[5mu]
(a,b] = \mathopen{]}a,b\mathclose{]} &= \{x\in\R \mid a<x\le b\}, \\[5mu]
[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R \mid a\le x\le b\}.
\end{align}</math>

Each interval {{open-open|''a'', ''a''}}, {{closed-open|''a'', ''a''}}, and {{open-closed|''a'', ''a''}} represents the [[empty set]], whereas {{closed-closed|''a'', ''a''}} denotes the singleton set&nbsp;{{math|{''a''}{{null}}}}.  When {{math|''a'' > ''b''}}, all four notations are usually taken to represent the empty set.

Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation {{math|(''a'', ''b'')}} is often used to denote an [[tuple|ordered pair]] in set theory, the [[coordinates]] of a [[point (geometry)|point]] or [[vector (mathematics)|vector]] in [[analytic geometry]] and [[linear algebra]], or (sometimes) a [[complex number]] in [[algebra]]. That is why [[Nicolas Bourbaki|Bourbaki]] introduced the notation {{math|]''a'', ''b''[}} to denote the open interval.<ref>{{cite web|url=http://hsm.stackexchange.com/a/193|title=Why is American and French notation different for open intervals (''x'', ''y'') vs. ]''x'', ''y''[?|website=hsm.stackexchange.com|access-date=28 April 2018}}</ref> The notation {{math|[''a'', ''b'']}} too is occasionally used for ordered pairs, especially in [[computer science]].

Some authors such as Yves Tillé use {{math|]''a'', ''b''[}} to denote the complement of the interval&nbsp;{{open-open|''a'', ''b''}}; namely, the set of all real numbers that are either less than or equal to {{mvar|a}}, or greater than or equal to {{mvar|b}}.

===Infinite endpoints===
In some contexts, an interval may be defined as a subset of the [[extended real number line|extended real numbers]], the set of all real numbers augmented with {{math|−∞}} and {{math|+∞}}.

In this interpretation, the notations {{closed-closed|−∞, ''b''}} , {{open-closed|−∞, ''b''}} , {{closed-closed|''a'', +∞}} , and {{closed-open|''a'', +∞}} are all meaningful and distinct. In particular, {{open-open|−∞, +∞}} denotes the set of all ordinary real numbers, while {{closed-closed|−∞, +∞}} denotes the extended reals. 

Even in the context of the ordinary reals, one may use an [[infinity (mathematics)|infinite]] endpoint to indicate that there is no bound in that direction. For example, {{open-open|0, +∞}} is the set of [[positive real numbers]], also written as <math>\mathbb{R}_+.</math> The context affects some of the above definitions and terminology. For instance, the interval {{open-open|−∞, +∞}}&nbsp;=&nbsp;<math>\R</math> is closed in the realm of ordinary reals, but not in the realm of the extended reals.

===Integer intervals===
When {{mvar|a}} and {{mvar|b}} are [[integer]]s, the notation ⟦''a, b''⟧, or {{closed-closed|''a'' .. ''b''}} or {{math|{''a'' .. ''b''}{{null}}}} or just {{math|''a'' .. ''b''}}, is sometimes used to indicate the interval of all ''integers'' between {{mvar|a}} and {{mvar|b}} included. The notation {{closed-closed|''a'' .. ''b''}} is used in some [[programming language]]s; in [[Pascal programming language|Pascal]], for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid [[Indexed family|indices]] of an [[Array data type|array]].

Another way to interpret integer intervals are as [[Set-builder_notation#Sets_defined_by_enumeration|sets defined by enumeration]], using [[ellipsis]] notation.

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 {{math|''a'' .. ''b'' − 1}} ,  {{math|''a'' + 1 .. ''b''}} , or  {{math|''a'' + 1 .. ''b'' − 1}}.  Alternate-bracket notations like {{closed-open|''a'' .. ''b''}} or {{math|[''a'' .. ''b''[}} are rarely used for integer intervals.{{citation needed|date=February 2014}}