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{{about|"bracketing" punctuation marks}}
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{{Punctuation marks|[&nbsp;]|Brackets<!--
-->|variant1=(&nbsp;)|caption1=Parentheses<!--
-->|variant2={&nbsp;}|caption2=Braces or curly brackets<!--
-->|variant3={{angbr|&nbsp;}}|caption3=Chevrons or angle brackets}}
'''Brackets''' are tall [[punctuation]] marks used in matched pairs within text, to set apart or interject other text. Used unqualified, brackets refer to different types of brackets in different parts of the world and in different  contexts.
 
==List of types==
 
* (&nbsp;) &mdash; parentheses, brackets (UK, New Zealand, and Australia), parens, round brackets, soft brackets, or circle brackets{{ctn|date=June 2013}}
* [&nbsp;] &mdash; square brackets, closed brackets, hard brackets, or brackets (US){{ctn|date=June 2013}}
* {&nbsp;} &mdash; braces (UK and US), flower brackets (India), French brackets, curly brackets, definite brackets, swirly brackets, curly braces, birdie brackets, Scottish brackets, squirrelly brackets, gullwings, seagulls, squiggly brackets, Tuborg brackets (DK), accolades (NL), pointy brackets, or fancy brackets{{ctn|date=June 2013}}
* {{angbr|&nbsp;}} &mdash; pointy brackets, angle brackets, triangular brackets, diamond brackets, tuples, or chevrons{{ctn|date=June 2013}}
* &lt;&nbsp;&gt; &mdash; inequality signs, pointy brackets, or brackets. Sometimes referred to as ''angle brackets'', in such cases as [[HTML]] markup. Occasionally known as ''broken brackets'' or ''brokets''.<ref>{{cite web|url=http://catb.org/jargon/html/B/broket.html |title=broket |publisher=Catb.org |date= |accessdate=2013-02-13}}</ref>
* ⸤ ⸥; ｢ ｣ &mdash; corner brackets{{ctn|date=September 2013}}
Characters ‹&nbsp;› and «&nbsp;», known as [[guillemet]]s or ''angular quote brackets'', are actually [[quotation mark glyphs]] used in several European languages.{{ctn|date=June 2013}}
 
==History==
The chevron was the earliest type of bracket to appear in [[written English]]. [[Desiderius Erasmus]] coined the term '''''lunula''''' to refer to the rounded parentheses (), recalling the shape of the crescent [[moon]].<ref>Truss, Lynne. ''Eats, Shoots & Leaves'', 2003. p. 161. ISBN 1-59240-087-6.</ref>
 
==Usage==
 
In addition to referring to the class of all types of brackets, the unqualified word ''bracket'' is most commonly used to refer to a specific type of bracket. In modern American usage this is usually the square bracket and in modern British usage this is usually the parenthesis.
 
In American usage, parentheses are usually considered separate from other brackets, and calling them "brackets" at all is unusual even though they serve a similar function.
 
In more formal usage, "[[Parenthesis (rhetoric)|parenthesis]]" may refer to the entire bracketed text, not just to the punctuation marks used (so all the text in this set of round brackets may be said to be "[[Parenthesis (rhetoric)|a parenthesis]]", "a parenthetical", or "a parenthetical phrase").<ref>{{cite web|url=http://www.thefreedictionary.com/parenthetical |title=The Free Online Dictionary |publisher=Thefreedictionary.com |date= |accessdate=2013-02-13}}</ref>
 
According to early typographic practice, brackets are never set in [[italics]], even when the surrounding characters are italic.<ref>Robert Bringhurst, ''The Elements of Typographic Style'', §5.3.2.</ref>
{{further|Parenthesis (rhetoric)}}
===Types===
 
==== Parentheses ( ) {{Anchor|Parentheses|Parenthesis}} ====
{{dablink|Various terms redirect here; for other uses, see [[Parenthesis (disambiguation)]], [[( ) (disambiguation)]], [[Parenthetical referencing]], and [[Parenthetical Girls]]}}
{{dablink|Due to [[Wikipedia:Naming conventions (technical restrictions)|technical restrictions]], titles like "<tt>:)</tt>" redirect here. For typographical portrayals of faces, see [[Emoticon]].}}
'''Parentheses''' {{IPAc-en|p|ə|ˈ|r|ɛ|n|θ|ɨ|s|iː|z}} (singular, '''parenthesis''' {{IPAc-en|p|ə|ˈ|r|ɛ|n|θ|ɨ|s|ɨ|s}}) (also called simply '''brackets''', or '''round brackets''', '''curved brackets''', '''oval brackets''', or, colloquially, '''parens''' {{IPAc-en|p|ə|ˈ|r|ɛ|n|z}}) contain material that could be omitted without destroying or altering the meaning of a sentence (in most writing, overuse of parentheses is usually a sign of a badly structured text){{According to whom|date=April 2013}}. A milder effect may be obtained by using a pair of commas as the [[delimiter]], though if the sentence contains commas for other purposes, visual confusion may result.
 
Parentheses may be used in formal writing to add supplementary information, such as "Sen. John McCain (R., Arizona) spoke at length." They can also indicate shorthand for "[[uncertain plural|either singular or plural]]" for nouns&nbsp;– e.g., "the claim(s)"&nbsp;– or for "either masculine or feminine" in some languages with [[grammatical gender]].<ref>[[Slash (punctuation)#Gender-neutrality in Spanish and Portuguese]]</ref>
 
Parenthetical phrases have been used extensively in informal writing and stream of consciousness literature. Examples include the southern American author [[William Faulkner]] (see ''[[Absalom, Absalom!]]'' and [[The Sound and the Fury#Part 2: June 2, 1910|the Quentin section of ''The Sound and the Fury'']]) as well as poet [[E. E. Cummings]].
Parentheses have historically been used where the [[Em dash|dash]] is currently used—that is, in order to depict alternatives, such as "parenthesis)(parentheses". Examples of this usage can be seen in editions of ''[[A Dictionary of Modern English Usage|Fowler's]]''.
 
Parentheses may be nested (generally with one set (such as this) inside another set). This is not commonly used in formal writing (though sometimes other brackets [especially square brackets] will be used for one or more inner set of parentheses, [in other words, secondary {or even tertiary} phrases can be found within the main parenthetical sentence]).<ref>{{cite web|last=Fogarty|first=Mignon|title=Parentheses, Brackets, and Braces|url=http://grammar.quickanddirtytips.com/parentheses-brackets-and-braces.aspx|publisher=Quick and Dirty Tips|accessdate=27 March 2011}}</ref>
 
Any punctuation inside parentheses or other brackets is independent of the rest of the text: "Mrs. Pennyfarthing (What? Yes, that was her name!) was my landlady." In this usage, the explanatory text in the parentheses is a [[Parenthesis (rhetoric)|parenthesis]]. (Parenthesized text is usually short and within a single sentence. Where several sentences of supplemental material are used in parentheses the final [[full stop]] would be within the parentheses. Again, the parenthesis implies that the meaning and flow of the text is supplemental to the rest of the text and the whole would be unchanged were the parenthesized sentences removed.)
 
Parentheses are included in the syntaxes of many computer programming languages. Typically needed to denote an argument; to tell the compiler what data type the Method/Function needs to look for first in order to initialise.
 
Parentheses in [[mathematics]] signify a different [[Order of operations|precedence]] of operators. Normally, {{math|2 + 3 × 4}} would be 14, since the multiplication is done before the addition. On the other hand {{math|(2 + 3) × 4}} is 20, because the parentheses override normal precedence, causing the addition to be done first. Some authors follow the convention in mathematical equations that, when parentheses have one level of nesting, the inner pair are parentheses and the outer pair are square brackets. Example:
 
:<math>[(2+3)\times4]^2=400</math>
 
A related convention is that when parentheses have two levels of nesting, curly brackets (braces) are the outermost pair. Following this convention, when more than three levels of nesting are needed, often a cycle of parentheses, square brackets, and curly brackets will continue. This helps to distinguish between one such level and the next.{{ctn|date=September 2013}}
 
Parentheses are also used to set apart the [[Parameter|argument]]s in mathematical [[Function (mathematics)|function]]s. For example, {{math|''f''(''x'')}} is the function ''{{math|f}}'' applied to the [[Variable (mathematics)|variable]] ''{{math|x}}''. In [[coordinate system]]s parentheses are used to denote a set of coordinates; so in the [[Cartesian coordinate system]] {{math|(4, 7)}} may represent the point located at 4 on the ''x''-axis and 7 on the ''y''-axis. Parentheses may also represent [[interval (mathematics)|interval]]s; {{open-open|0,5}}, for example, is the interval between 0 and 5, not including 0 or 5.
 
Parentheses may also be used to represent a [[binomial coefficient]], and in chemistry to denote a [[polyatomic ion]].
 
In [[Chinese language|Chinese]] and [[Japanese language|Japanese]], 【 】, a combination of brackets and parentheses called 方頭括號 and ''sumitsuki'', are used for inference in Chinese and used in titles and headings in Japanese.
 
;Unpaired parenthesis
Lowercase [[latin letters]] used as indexes, rather than [[Bullet (typography)|bullets]] or numbers, followed by unpaired parenthesis, are used in ordered [[wiktionary:list|list]]s especially in:<br>
:a) [[education]]al testing,
:b) technical writing and diagrams,
:c) [[market research]], and
:d) [[elections]]
 
====Square brackets [ ] {{Anchor|square bracket|Square brackets}}====
Square brackets&nbsp;– also called crotchets or simply '''brackets''' (US)&nbsp; – are mainly used to insert explanatory material or to mark where a passage was omitted from an original material by someone other than the original author, or to mark modifications in quotations.<ref>''The Chicago Manual of Style, 15th ed.'', The University of Chicago Press, 2003, §6.104</ref>
 
A bracketed [[ellipsis]] '''['''…''']''' is often used to indicate omitted material: "I'd like to thank '''['''several unimportant people''']''' for their tolerance..."<ref>[http://www.bartleby.com/68/60/960.html The Columbia Guide to Standard American English]</ref>
Bracketed comments inserted into a quote indicate when the original has been modified for clarity: "I appreciate it '''['''the honor''']''', but I must refuse", and "the future of psionics '''['''see definition''']''' is in doubt". Or one can quote the original statement "I hate to do laundry" with a modification inserted in the middle of it: He "hate'''['''s''']''' to do laundry".
 
Additionally, a small letter can be replaced by a capital one, when the beginning of the original text is omitted for [[succinctness]], for example, when referring to a [[verbose]] original: "To the extent that policymakers and elite opinion in general have made use of economic analysis at all, they have, as the saying goes, done so the way a drunkard uses a lamppost: for support, not illumination", it can be quoted succinctly as: "'''['''P''']'''olicymakers (&hellip;) made use of economic analysis (&hellip;) the way a drunkard uses a lamppost: for support, not illumination." When nested parentheses are needed, brackets are used as a substitute for the inner pair of parentheses within the outer pair.<ref>''The Chicago Manual of Style, 15th ed.'', The University of Chicago Press, 2003, §6.102 and §6.106</ref> When deeper levels of nesting are needed, convention is to alternate between parentheses and brackets at each level.
 
Alternatively, empty square brackets can also indicate omitted material, usually single letter only. The original "Reading is also a process and it also changes you." can be rewritten in a quote as: It has been suggested that reading can "also change'''[]''' you".
 
The bracketed expression “[[sic|[''sic'']]]” is used after a quote or reprinted text to indicate the passage appears exactly as in the original source, where it may otherwise appear that a mistake has been made in reproduction.
 
In translated works, brackets are used to signify the same word or phrase in the original language to avoid ambiguity.<ref>''The Chicago Manual of Style, 15th ed.'', The University of Chicago Press, 2003, §6.105</ref>
For example: ''He is trained in the way of the open hand [karate].''
 
In linguistics, [[phonetic transcription]]s are generally enclosed within brackets,<ref>''The Chicago Manual of Style, 15th ed.'', The University of Chicago Press, 2003, §6.107</ref> often using the [[International Phonetic Alphabet]], whereas [[phoneme|phonemic]] transcriptions typically use paired [[Slash (punctuation)|slash]]es. Pipes (| |) are often used to indicate a [[Morphophoneme|morphophonemic]] rather than phonemic representation. Other conventions are double slashes (// //), double pipes (|| ||) and curly brackets ({ }).
 
Brackets (called ''move-left symbols'' or ''move right symbols'') are added to the sides of text in [[proofreading]] to indicate changes in indentation:
 
<center>
{| class="wikitable"
|-
!  style="width:15%; text-align:left;"| Move left
|  style="width:85%; text-align:left;"| [To Fate I sue, of other means bereft, the only refuge for the wretched left.
|-
!  style="width:15%; text-align:left;"| Center
|  style="width:85%; text-align:left;"| ]Paradise Lost[
|-
!  style="width:15%; text-align:left;"| Move up
|  style="width:85%; text-align:left;"| [[File:Quote to be Moved Up.svg|330px]]
|}
</center>
 
Brackets are used to denote parts of the text that need to be checked when preparing drafts prior to finalizing a document. They often denote points that have not yet been agreed to in legal drafts and the year in which a report was made for certain [[case law]] decisions.
 
Brackets are used in [[mathematics]] in a variety of notations, including standard notations for [[interval (mathematics)|interval]]s, [[commutator]]s, the [[Floor and ceiling functions|floor function]], the [[Lie bracket of vector fields|Lie bracket]], the [[Iverson bracket]], and [[Matrix (mathematics)|matrices]].
 
Brackets can also be used in [[chemistry]] to represent the [[concentration]] of a [[chemical substance]] or to denote distributed charge in a [[Complex (chemistry)|complex ion]].
 
Brackets are used in many computer [[programming language]]s, especially those [[list of C-based programming languages|derived or inspired by]] the [[C (programming language)|C language]], to indicate [[array data structure|array]] indexing operators. In this context, the opening bracket is often pronounced as "sub", indicating a ''sub''script.
 
[[File:Brackets.svg|thumb|145px|From the top: brackets, braces, parentheses, angle brackets/chevrons, and inequality signs]]
 
===={{Anchor|Braces|Curly bracket|Curly brackets}} Curly brackets or braces <nowiki>{ }</nowiki>====
These are used immediately before or after, and span, a list of items where there precedes, or follows, respectively, one or more other items that are common to that list. This usage, however, is precluded in text-editing software that has no provision for such column- or row-spanning characters.{{Citation needed|date=October 2013}}
 
Curly brackets – also called '''braces''' or '''squiggly brackets''' in the US – are used in specialized ways in poetry and music (to mark repeats or joined lines). The musical terms for this mark joining staves are [[accolade (notation)|accolade]] and "[[brace (music)|brace]]", and connect two or more lines of music that are played simultaneously.<ref>[http://www.decodeunicode.org/u+007B Decodeunicode.org > U+007B LEFT CURLY BRACKET] Retrieved on May 3, 2009</ref> In [[mathematics]] they delimit [[Set (mathematics)|set]]s, and in writing, they may be used similarly, "Select your animal {goat, sheep, cow, horse} and follow me". In many programming languages, they enclose groups of [[Statement (programming)|statement]]s. Such languages ([[C (programming language)|C]] being one of the best-known examples) are therefore called [[curly bracket language]]s. Some people use a brace to signify movement in a particular direction.{{Citation needed|date=July 2011}}{{clarify|date=December 2013}}
 
Presumably due to the similarity of the words ''brace'' and ''bracket'' (although they do not share an [[etymology]]), many people mistakenly treat ''brace'' as a [[synonym]] for ''bracket''. Therefore, when it is necessary to avoid any possibility of confusion, such as in computer programming, it may be best to use the term ''curly bracket'' rather than ''brace''. However, general usage in North American English favours the latter form.{{Citation needed|date=May 2008}} Indian programmers often use the name "flower bracket".<ref>K R Venugopa, Rajkumar Buyya, T Ravishankar. ''Mastering C++'', 1999. p. 34. ISBN 0-07-463454-2.</ref>
 
In [[classical mechanics]], curly brackets are often also used to denote the [[Poisson bracket]] between two quantities.
 
===={{anchor|Angle brackets}} Angle brackets or chevrons {{angbr| }}====
{{refimprove section|date=November 2012}}
 
<!--Please do not replace &#x27E8; and &#x27E9; in this section with < and >, even if your computer displays &#27E8; and &#x27E9; as question marks or boxes. Also, do not convert the entities to characters. Doing so breaks the display.-->
 
Chevrons {{Unicode|&#x27E8;&nbsp;&#x27E9;}};<ref>Some fonts don't display these characters correctly. Please refer to the image on the right instead.</ref> are often used to enclose highlighted material.
 
In physical sciences, chevrons are used to denote an average over time or over another continuous parameter. For example,
 
:<math>\left\langle V(t)^2 \right\rangle = \lim_{T\to\infty} \frac{1}{T}\int_{-T/2}^{T/2} V(t)^2\,{\rm{d}}t. </math>
 
The [[inner product]] of two vectors is commonly written as <math>\langle a, b\rangle</math>, but the notation (a, b) is also used.
 
In mathematical physics, especially [[quantum mechanics]], it is common to write the inner product between elements as <math>\langle a | b\rangle</math>, as a short version of <math>\langle a |\cdot| b\rangle</math>, or <math>\langle a | \hat{O} | b\rangle</math>, where <math>\hat{O}</math> is an [[Operator (physics)|operator]]. This is known as [[Dirac notation]] or bra-ket notation.
 
In [[set theory]], chevrons or parentheses are used to denote [[ordered pair]]s and other [[tuple]]s, whereas curly brackets are used for unordered sets.
 
In [[linguistics]], chevrons indicate [[grapheme]]s (i.e., written letters) or [[orthography]], as in “The English word {{IPA|/kæt/}} is spelled {{Unicode|&#x27E8;cat&#x27E9;}}.” In [[epigraphy]], they may be used for mechanical transliterations of a text into the Latin script.
 
In [[textual criticism]], and hence in many editions of pre-modern works, chevrons denote sections of the text which are illegible or otherwise lost; the editor will often insert his own reconstruction where possible within them.
 
Chevrons are infrequently used to denote words that are thought instead of spoken, such as:
:{{Unicode|&#x27E8; What an unusual flower! &#x27E9;}}
 
The mathematical or logical symbols for ''greater-than'' ([[Greater-than sign|&gt;]]) and ''less-than'' ([[Less-than sign|&lt;]]) are [[inequality (mathematics)|inequality]] symbols, and are not punctuation marks when so used. Nevertheless, true chevrons are not available on a typical [[keyboard (computing)|computer keyboard]], but the less-than and greater-than symbols are, so they are often substituted. They are loosely referred to as ''angled brackets'' or ''chevrons'' in this case.{{citation needed|date=November 2012}}
 
Single and double pairs of comparison operators (&lt;&lt;,&nbsp;&gt;&gt;) (meaning ''much smaller than'' and ''much greater than'') are sometimes used instead of [[guillemet]]s ({{unicode|«}}, {{unicode|»}}) (used as [[non-English usage of quotation marks|quotation marks in many languages]]) when the proper characters are not available.
 
In [[comic book]]s, chevrons are often used to mark dialogue that has been translated notionally from another language; in other words, if a character is speaking another language, instead of writing in the other language and providing a translation, one writes the translated text within chevrons. Of course, since no foreign language is actually written, this is only ''notionally'' translated.{{Citation needed|date=December 2008}}
 
Chevron-like symbols are part of standard [[Chinese language|Chinese]], and [[Korean language|Korean]] punctuation, where they generally enclose the titles of books: {{Unicode|︿}} and {{Unicode|﹀}} or {{Unicode|︽}} and {{Unicode|︾}} for traditional [[tategaki|vertical printing]], and {{Unicode|〈}} and {{Unicode|〉}} or {{Unicode|《}} and {{Unicode|》}} for [[yokogaki|horizontal]] printing. See also [[non-English usage of quotation marks]].
 
In [[continuum mechanics]], chevrons may be used as [[Macauley Brackets]].
 
====Angles ｢｣ ====
In [[non-English usage of quotation marks#Chinese, Japanese and Korean quotation marks|East Asian punctuation]], angle brackets are used as quotation marks.
 
==== Floor and ceiling corners ⌊ ⌋, ⌈ ⌉ ====
The floor corner brackets ⌊ and ⌋, the ceiling corner brackets ⌈ and ⌉ are  used to denote the integer [[floor and ceiling functions]].
 
==== Quine corners ⌞⌟, ⌜⌝, and half brackets ⸤⸥, ⸢⸣ ====
The Quine corners ⌜ and ⌝ have at least two uses in mathematical logic: either as [[quasi-quotation]], a generalization of quotation marks, or to denote the [[Gödel numbering|Gödel number]] of the enclosed expression.
 
Half brackets are used in English to mark added text, such as in translations: "Bill saw ⸤her⸥".
 
In editions of [[papyrology|papyrological]] texts, half brackets, ⸤ and ⸥ or ⸢ and ⸣, enclose text which is lacking in the papyrus due to damage, but can be restored by virtue of another source, such as an ancient quotation of the text transmitted by the papyrus.<ref>M.L. West (1973) ''Textual Criticism and Editorial Technique'' (Stuttgart) 81.</ref> For example, [[Callimachus]] ''Iambus'' 1.2 reads: ἐκ τῶν ὅκου βοῦν κολλύ⸤βου π⸥ιπρήσκουσιν. A hole in the papyrus has obliterated βου π, but these letters are supplied by an ancient commentary on the poem. Second intermittent sources can be between ⸢ and ⸣.  Quine corners are sometimes used instead of half brackets.<ref name="Unicode Miscellaneous Technical"/>
 
====Double brackets ⟦&nbsp;⟧ {{Anchor|Double brackets}}====
 
In [[Semantics of programming languages|formal semantics]], double brackets, ⟦&nbsp;⟧, also called Strachey brackets, are used to indicate the ''semantic evaluation function.''
 
===Computing===
 
====Encoding====
Representations of various kinds of brackets in [[ASCII]], [[Unicode]], and [[HTML]] are given below.
 
{|class="wikitable"
|-
!scope="col"| Usage !!colspan="2" scope="col"| Unicode !!scope="col"| SGML/[[List of HTML and XML character entities|HTML/XML entities]] !!scope="col"| Sample
|-
|rowspan="4"| General purpose
| U+0028 || Left parenthesis || &amp;#40; &amp;lparen; ||rowspan="2"| (parentheses)
|-
| U+0029 || Right parenthesis || &amp;#41; &amp;rparen;
|-
| U+005B || Left square bracket || &amp;#91; ||rowspan="2"| [''[[sic]]'']
|-
| U+005D || Right square bracket || &amp;#93;
|-
|rowspan="4"| Technical/mathematical<br>(common)
| U+003C || Less-than sign || &amp;#60; &amp;lt; ||rowspan="2"| &lt;HTML&gt;
|-
| U+003E || Greater-than sign || &amp;#62; &amp;gt;
|-
| U+007B || Left curly bracket || &amp;#123; ||rowspan="2"| {round, square, curly}
|-
| U+007D || Right curly bracket || &amp;#125;
|-
<!-- not brackets, see ==List of types==-->
|rowspan="4"| Quotation<br>(Western texts)
| U+00AB || Left double guillemet || &amp;#171; ||rowspan="2"| «&thinsp;quote&thinsp;»
|-
| U+00BB || Right double guillemet || &amp;#187;
|-
| U+2039|| Left single guillemet || &amp;#8249; ||rowspan="2"| ‹&thinsp;''{{math|x}}''&thinsp;›
|-
| U+203A || Right single guillemet || &amp;#8250;
|-
|rowspan="4"| [[Floor and ceiling functions]]<ref name="Unicode Miscellaneous Technical">{{citation
| url=http://www.unicode.org/charts/PDF/U2300.pdf
| title=Miscellaneous Technical | work=The Unicode Standard, Version 6.3
| year=2013 | accessdate=2012-11-02
}}</ref>
| U+2308 || Left ceiling || &amp;#8968; ||rowspan="2"| ⌈''ceiling''⌉
|-
| U+2309 || Right ceiling || &amp;#8969;
|-
| U+230A || Left floor || &amp;#8970; ||rowspan="2"| ⌊''floor''⌋
|-
| U+230B || Right floor || &amp;#8971;
|-
|rowspan="4"| [[quasi-quotation|Quine corners]]<ref name="Unicode Miscellaneous Technical"/>
| U+231C || Top right corner || &amp;#8988; ||rowspan="2"| ⌜''quasi-quotation''⌝<br>⌜''editorial notation''⌝
|-
| U+231D || Top left corner || &amp;#8989;
|-
| U+231E || Bottom right corner || &amp;#8990; ||rowspan="2"| ⌞''editorial notation''⌟
|-
| U+231F || Bottom left corner || &amp;#8991;
|-
|rowspan="32"| Technical/mathematical<br>(specialized)<ref name="Unicode Miscellaneous Technical" />
| U+239B || Left parenthesis upper hook || &amp;#9115; ||rowspan="6"|
{|
| ⎛<br/>⎜<br/>⎝
| large parentheses
|⎞<br/>⎟<br/>⎠
|}
|-
| U+239C || Left parenthesis extension || &amp;#9116;
|-
| U+239D || Left parenthesis lower hook || &amp;#9117;
|-
| U+239E || Right parenthesis upper hook || &amp;#9118;
|-
| U+239F || Right parenthesis extension || &amp;#9119;
|-
| U+23A0 || Right parenthesis lower hook || &amp;#9120;
|-
| U+23A1 || Left square bracket upper corner || &amp;#9121; ||rowspan="6"|
{|
| ⎡<br/>⎢<br/>⎣
| large square brackets
| ⎤<br/>⎥<br/>⎦
|}
|-
| U+23A2 || Left square bracket extension || &amp;#9122;
|-
| U+23A3 || Left square bracket lower corner || &amp;#9123;
|-
| U+23A4 || Right square bracket upper corner || &amp;#9124;
|-
| U+23A5 || Right square bracket extension || &amp;#9125;
|-
| U+23A6 || Right square bracket lower corner || &amp;#9126;
|-
| U+23A7 || Left curly bracket upper hook || &amp;#9127; ||rowspan="6"|
{|
| ⎧<br/>⎨<br/>⎩
| large curly brackets
| ⎫<br/>⎬<br/>⎭
|}
|-
| U+23A8 || Left curly bracket middle piece || &amp;#9128;
|-
| U+23A9 || Left curly bracket lower hook || &amp;#9129;
|-
| U+23AB || Right curly bracket upper hook || &amp;#9131;
|-
| U+23AC || Right curly bracket middle piece || &amp;#9132;
|-
| U+23AD || Right curly bracket lower hook || &amp;#9133;
|-
| U+23AA || Curly bracket extension || &amp;#9130; || ⎪
|-
| U+23B0 || Upper left or lower right curly bracket section || &amp;#9136; ||rowspan="2"| ⎰&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⎱<br>⎱&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⎰
|-
| U+23B1 || Upper right or lower left curly bracket section || &amp;#9137;
|-
| U+23B4 || Top square bracket || &amp;#9140; ||rowspan="2"|
{| style="text-align: center;"
|-
| ⎴
|-
| horizontal square brackets
|-
| ⎵
|}
|-
| U+23B5 || Bottom square bracket || &amp;#9141;
|-
| U+23B6 || Bottom square bracket over top square bracket || &amp;#9142; ||
{|style="text-align: center;"
|-
| ⎴ || ⎴ || ⎴ || ⎴ || ⎴ || ⎴ || ⎴ || ⎴ || ⎴
|-
| t || e || r || m || i || n || a || l ||
|-
| ⎶ || ⎶ || ⎶ || ⎶ || ⎶ || ⎶ || ⎶ || ⎶ || ⎶
|-
| e || m || u || l || a || t || i || o || n
|-
| ⎵ || ⎵  || ⎵  || ⎵  || ⎵  || ⎵  || ⎵  || ⎵  || ⎵
|}
|-
| U+23B8 || Left vertical box line || &amp;#9144; ||rowspan="2"|⎸boxed text⎹
|-
| U+23B9 || Right vertical box line || &amp;#9145;
|-
| U+23DC || Top parenthesis || &amp;#9180; ||rowspan="2"|
{| style="text-align: center;"
|-
| &#9180;
|-
| horizontal parentheses
|-
| &#9181;
|}
|-
| U+23DD || Bottom parenthesis || &amp;#9181;
|-
| U+23DE || Top curly bracket || &amp;#9182; ||rowspan="2"|
{| style="text-align: center;"
|-
| &#9182;
|-
| horizontal curly brackets
|-
| &#9183;
|}
|-
| U+23DF || Bottom curly bracket || &amp;#9183;
|-
| U+23E0 || Top tortoise shell bracket || &amp;#9184; ||rowspan="2"|
{| style="text-align: center;"
|-
| &#9184;
|-
| tortoise shell brackets
|-
| &#9185;
|}
|-
| U+23E1 || Bottom tortoise shell bracket || &amp;#9185;
|-
|rowspan="32"| Technical mathematical<br>symbols<ref name="Unicode Miscellaneous Mathematical Symbols-A">{{citation
| url=http://www.unicode.org/charts/PDF/U27C0.pdf
| title=Miscellaneous Mathematical Symbols-A | work=The Unicode Standard, Version 6.3
| year=2013 | accessdate=2013-11-02
}}</ref><ref name="Unicode Miscellaneous Mathematical Symbols-B">{{citation
| url=http://www.unicode.org/charts/PDF/U2980.pdf
| title=Miscellaneous Mathematical Symbols-B | work=The Unicode Standard, Version 6.3
| year=2013 | accessdate=2013-11-02
}}</ref>
| U+27E6 || Mathematical left white square bracket || &amp;#10214; ||rowspan="2"| ⟦white square brackets⟧
|-
| U+27E7 || Mathematical right white square bracket || &amp;#10215;
|-
| U+27E8 || Mathematical left angle bracket || &amp;#10216; &amp;lang;<ref name="lang-rang" group="e"/> ||rowspan="2"| {{angbr|''{{math|a}}'', ''{{math|b}}''}}
|-
| U+27E9 || Mathematical right angle bracket || &amp;#10217; &amp;rang;<ref name="lang-rang" group="e"/>
|-
| U+27EA || Mathematical left double angle bracket || &amp;#10218; ||rowspan="2"| ⟪''{{math|A}}'', ''{{math|B}}''⟫
|-
| U+27EB || Mathematical right double angle bracket || &amp;#10219;
|-
| U+27EC || Mathematical left white tortoise shell bracket || &amp;#10220; ||rowspan="2"| ⟬white tortoise shell brackets⟭
|-
| U+27ED || Mathematical right white tortoise shell bracket || &amp;#10221;
|-
| U+27EE || Mathematical left flattened parenthesis || &amp;#10222; ||rowspan="2"| ⟮flattened parentheses⟯
|-
| U+27EF || Mathematical right flattened parenthesis || &amp;#10223;
|-
| U+2983 || Left white curly bracket || &amp;#10627; ||rowspan="2"| ⦃white curly brackets⦄
|-
| U+2984 || Right white curly bracket || &amp;#10628;
|-
| U+2985 || Left white parenthesis || &amp;#10629; ||rowspan="2"| ⦅white/double parentheses⦆
|-
| U+2986 || Right white parenthesis || &amp;#10630;
|-
| U+2987 || [[Z notation]] left image bracket || &amp;#10631; ||rowspan="2"| ''{{math|R}}''⦇''{{math|S}}''⦈
|-
| U+2988 || Z notation right image bracket || &amp;#10632;
|-
| U+2989 || Z notation left binding bracket || &amp;#10633; ||rowspan="2"| ''{{math|A}}''⦉''{{math|B}}''⦊
|-
| U+298A || Z notation right binding bracket || &amp;#10634;
|-
| U+298B || Left square bracket with underbar || &amp;#10635; ||rowspan="2"| ⦋underlined square brackets⦌
|-
| U+298C || Right square bracket with underbar || &amp;#10636;
|-
| U+298D || Left square bracket with tick in top corner || &amp;#10637; ||rowspan="2"| ⦍ticked square brackets⦎
|-
| U+298E || Right square bracket with tick in bottom corner || &amp;#10638;
|-
| U+298F || Left square bracket with tick in bottom corner || &amp;#10639; ||rowspan="2"| ⦏ticked square brackets⦐
|-
| U+2990 || Right square bracket with tick in top corner || &amp;#10640;
|-
| U+2991 || Left angle bracket with dot || &amp;#10641; ||rowspan="2"| ⦑dotted angle brackets⦒
|-
| U+2992 || Right angle bracket with dot || &amp;#10642;
|-
| U+2993 || Left arc less-than bracket || &amp;#10643; ||rowspan="2"| ⦓[[Inequality (mathematics)|inequality]] sign brackets⦔
|-
| U+2994 || Right arc greater-than bracket || &amp;#10644;
|-
| U+2995 || Double left arc greater-than bracket || &amp;#10645; ||rowspan="2"| ⦕inequality sign brackets⦖
|-
| U+2996 || Double right arc less-than bracket || &amp;#10646;
|-
| U+2997 || Left black tortoise shell bracket || &amp;#10647; ||rowspan="2"| ⦗black tortoise shell brackets⦘
|-
| U+2998 || Right black tortoise shell bracket || &amp;#10648;
|-
|rowspan="4"| Half brackets<ref name="Unicode Miscellaneous Technical">{{citation
| url=http://www.unicode.org/charts/PDF/U2E00.pdf
| title=Supplemental Punctuation | work=The Unicode Standard, Version 6.3
| year=2013 | accessdate=2013-11-02
}}</ref>
| U+2E22 || Top left half bracket || &amp;#11810; ||rowspan="2"| ⸢''editorial notation''⸣
|-
| U+2E23 || Top right half bracket || &amp;#11811;
|-
| U+2E24 || Bottom left half bracket || &amp;#11812; ||rowspan="2"| ⸤''editorial notation''⸥
|-
| U+2E25 || Bottom right half bracket || &amp;#11813;
|-
|rowspan="4"| Quotation<br>(halfwidth East-Asian texts)
| U+2329 || Left pointing angle bracket || &amp;#9001; &amp;lang;<ref name="lang-rang" group="e"/> ||rowspan="2"| &#x2329;deprecated&#x232A;
|-
| U+232A || Right pointing angle bracket || &amp;#9002; &amp;rang;<ref name="lang-rang" group="e"/>
|-
| U+FF62 || Halfwidth left corner bracket || &amp;#65378; ||rowspan="2"| ｢ｶﾀｶﾅ｣
|-
| U+FF63 || Halfwidth right corner angle bracket || &amp;#65379;
|-
|rowspan="10"| Quotation<br>(fullwidth East-Asian texts)
| U+3008 || Left angle bracket || &amp;#12296; ||rowspan="2"| 〈한〉
|-
| U+3009 || Right angle bracket || &amp;#12297;
|-
| U+300A || Left double angle bracket || &amp;#12298; ||rowspan="2"| 《한한》
|-
| U+300B || Right double angle bracket || &amp;#12299;
|-
| U+300C || Left corner bracket || &amp;#12300; ||rowspan="2"| 「白八櫨」
|-
| U+300D || Right corner bracket || &amp;#12301;
|-
| U+300E || Left corner bracket || &amp;#12302; ||rowspan="2"| 『カタカナ』
|-
| U+300F || Right corner bracket || &amp;#12303;
|-
| U+3010 || Left thick square bracket || &amp;#12304; ||rowspan="2"| 【ひらがな】
|-
| U+3011 || Right thick square bracket || &amp;#12305;
|-
|rowspan="4"| General purpose<br> (fullwidth East-Asian)
| U+FF08 || Fullwidth left parenthesis || &amp;#65288; ||rowspan="2"| （Ｗｉｋｉ）
|-
| U+FF09 || Fullwidth right parenthesis || &amp;#65289;
|-
| U+FF3B || Fullwidth left square bracket || &amp;#65339; ||rowspan="2"| ［''ｓｉｃ''］
|-
| U+FF3D || Fullwidth right square bracket || &amp;#65341;
|-
|rowspan="4"| Technical/mathematical<br>(fullwidth East-Asian)
| U+FF1C || Fullwidth less-than sign || &amp;#65308; ||rowspan="2"| ＜ＨＴＭＬ＞
|-
| U+FF1E || Fullwidth greater-than sign || &amp;#65310;
|-
| U+FF5B || Fullwidth left curly bracket || &amp;#65371; ||rowspan="2"| ｛１、２｝
|-
| U+FF5D || Fullwidth right curly bracket || &amp;#65373;
|}
 
<references group="e">
<ref name="lang-rang">&amp;lang; and &amp;rang; were tied to the deprecated symbols U+2329 and U+232A in HTML4 and MathML2, but are being migrated to U+27E8 and U+27E9 for HTML5 and MathML3, as defined in [http://www.w3.org/TR/xml-entity-names/#diff-xhtml1 XML Entity Definitions for Characters].
</ref>
</references>
 
Braces (curly brackets) first became part of a character set with the 8-bit code of the [[IBM 7030 Stretch]].<ref>{{cite web |last=Bob |first=Bemer |title=The Great Curly Brace Trace Chase |url=http://www.bobbemer.com/BRACES.HTM |accessdate =2009-09-05 |postscript=<!--None-->}}</ref>
 
The angle brackets or chevrons at U+27E8 and U+27E9 are for mathematical use and Western languages, whereas U+3008 and U+3009 are for East Asian languages. The chevrons at U+2329 and U+232A are deprecated in favour of the U+3008 and U+3009 East Asian angle brackets. Unicode discourages their use for mathematics and in Western texts,<ref name="Unicode Miscellaneous Technical" /> because they are canonically equivalent to the CJK code points U+300x and thus likely to render as double-width symbols. The ''less-than'' and ''greater-than'' symbols are often used as replacements for chevrons.
 
These various bracket characters are frequently used in many [[computer language]]s as operators or for other syntax markup. The more common uses follow.
 
====Uses of "(" and ")"====
* Parentheses are often used to define the syntactic structure of expressions, overriding [[operator precedence]]: <code>a*(b+c)</code> has subexpressions <code>a</code> and <code>b+c</code>, whereas <code>a*b+c</code> has subexpressions <code>a*b</code> and <code>c</code>.
* They are used for passing [[Parameter (computer science)|parameter]]s or arguments to [[Function (computer science)|function]]s, especially in [[C (programming language)|C]] and similar languages, and invoking a function or function-like construct: <code>substring($val,10,1)</code>.
* In [[Lisp programming language|Lisp]], they open and close [[s-expression]]s and therefore function applications: <code>(cons a b)</code>.
* In many [[regular expression]] syntaxes, parentheses create a ''capturing group'', allowing the matched portion inside to be retrieved by the user.
* In [[Forth (programming language)|Forth]], they open and close [[Comment (computer programming)|comment]]s in the code.
* In [[Fortran]]-family and [[COBOL]] languages, they are also used for array references.
* In the [[Perl]] programming language through Perl 5, they are used to define ''lists'', static array-like structures; this idiom is extended to their use as containers of subroutine (function) arguments.
* In the [[Perl 6]] programming language, they define ''captures'', a structure that defers contextual interpretation. This usage extends to ordinary parentheses as well. They are also used to indicate arguments to function calls and to declare signatures of formal parameters or other variables.
* In [[Python (programming language)|Python]] they are used to disambiguate [[tuple]] literals ([[Immutable object|immutable]] ordered lists), which are usually formed by commas, in places where parentheses and commas would otherwise be a part of a function call.
* In [[Tcl]], they are used to enclose the name of an element of an [[associative array]] variable.
* In joined brackets in a table going vertically downwards, a ")" refers to repetition of a term for the number of items towards the left of this joined list of brackets.
 
====Uses of "[" and "]"====
[[File:MediaWiki logo without tagline.png|thumb|The [[MediaWiki]] logo shows two pairs of brackets, one of the best known features of the software's markup.]]
* Square brackets are often used to refer to elements of an [[array (data type)|array]] or [[associative array]], and sometimes to define the number of elements in an array, especially in [[C (programming language)|C]]-like languages: <code>queue[3]</code>.
* In many languages, they may be used to define a literal anonymous array or list: <code class="nowrap">[5, 10, 15]</code>.
* In most [[regular expression]] syntaxes, brackets denote a character class: a set of possible characters to choose from.
* In [[Forth (programming language)|Forth]], "[" causes the system to enter interpretation state and "]" causes the system to enter [[compilation state]]. For example, within a definition, <code class="nowrap">[ 2 + 3 ] literal</code> causes the compiler to switch to the interpreter mode, calculate the expression 2 + 3, leave the result on stack and resume compilation. As a result, a literal constant "5" will be compiled into the definition, instead of the whole expression.
* In [[Tcl]], they enclose a sub-script to be evaluated and the result substituted.
* In some of the [[List of CLI languages|CLI languages]], most notably [[C sharp (programming language)|C#]] and [[C++]], they are used to denote [[Metadata (CLI)|metadata attributes]].
* In [[C++11]] they introduce [[Anonymous function#C++|Lambda expression]]s and hold an optional capture clause.
* In x86 assembly implementations such as [[FASM]], they are used to distinguish pointers from their data.
* In [[Smalltalk]], brackets are used to delineate "blocks" or "block closures", grouping of code that can be executed immediately or later via messages send such as "value" sent to the block. Blocks are full first class objects in Smalltalk.
* In [[Objective-C]], brackets are used to send a message to (i.e. call a [[Method (computer science)|method]] on) an object.
* On [[Unix]], "[" is a shorthand for the [[test (Unix)|test]] command.
* In [[JSON]] they are used to define an array (ordered sequence of comma-separated values).
* In programming documentation and metalanguages (e.g. in descriptions of operator or command syntax), optional elements are enclosed in square brackets. For example, "echo [-n] [-e] <text>" means that the -n and -e parameters are optional.
* Delimiting [[IPv6 address]]es in [[URL]]s, for example: <code>http://[2001:db8:3c4d:15::abcd:ef12]:8080</code>.
* In [[MediaWiki]], pairs of brackets create links from one page to another within a given wiki (e.g. <code>[&#91;Bracket]]</code>), and single brackets create links to a specified URL (e.g. <code>&#91;http:&#47;/en.wikipedia.org/wiki/Bracket]</code>).
 
====Uses of "{" and "}"====
* Curly brackets are used in some programming languages to define the beginning and ending of [[block (programming)|block]]s of code or data. Languages which use this convention are said to belong to the [[curly bracket programming language|curly bracket family]] of programming languages.
* They are used to represent certain type definitions or literal data values, such as a [[Composite type|composite structure]] or associative array.
* In [[mathematics]], they enclose [[Element (mathematics)|element]]s of sets and denote [[Set (mathematics)|set]]s.
* In [[Curl (programming language)|Curl]] they are used to delimit [[Expression (programming)|expression]]s and [[Statement (programming)|statement]]s (similar to [[Lisp (programming language)|Lisp]]'s use of parenthesis).
* In [[Pascal programming language|Pascal]] they define the beginning and ending of [[Comment (computer programming)|comment]]s.
* In most [[regular expression]] syntaxes, they are used as quantifiers, matching ''n'' repetitions of the previous group.
* In [[Perl]] they are also used to refer to elements of an associative array.
* In [[PHP]] they are used to determine structures.
* In [[Tcl]] they enclose a string to be substituted without any internal substitutions being performed.
* In [[Python programming language|Python]] and [[Ruby programming language|Ruby]] they are used for [[Associative array|dictionaries]] (a [[mutable object|mutable]] set of key: value pairs, separated by commas) and for [[set (computer science)|set]]s.
* In [[TeX]]/[[LaTeX]] they can be used for grouping parts sharing the same local format, wrap parameters, or definitions, depending on the local catcode value.
* In [[JSON]] they are used to define an object (an unordered collection of key:value pairs).
* In metalanguages (e.g. in descriptions of operator or command syntax), possible alternatives are enclosed in braces, if at least one is mandatory.
* In [[Verilog]] they are used to specify a list of bit and bit vectors being concatenated.
* These are also used in music at the start of a [[Staff (music)|stave]].
 
====Uses of "&lt;" and "&gt;"====
{{main|less-than sign|greater-than sign}}
These symbols are used in pairs as if they are brackets.
 
* Greater-than and less-than signs are used to set apart [[Uniform resource locator|URL]]s and [[e-mail address]]es in text, such as "I found it on Example.com <nowiki><http://www.example.com/></nowiki>" and "This photo is copyrighted by John Smith <johnsmith@example.com>". This is also the computer-readable form for addresses in [[e-mail header]]s, specified by RFC 2822.
* In documentation, they are often used to specify parameters or other user-specified information (e.g. "The command 'echo <text>' can be used to display <text>").
* To enclose code tags in [[Standard Generalized Markup Language|SGML]], [[HTML]], and [[XML]] (e.g. <code>&lt;div&gt;</code>).
* To target children of a parent element in [[CSS]] (e.g. <code>ul&#46;main&gt;li</code> whereas all ''direct'' child selectors of the <code>ul&#46;main</code> tag are targeted).
* In the [[C++]], [[C Sharp (programming language)|C#]], and [[Java (programming language)|Java]] programming languages, (among others) they delimit [[generic programming|generic argument]]s.
* In Perl through Perl 5 they are used to read a line from an input source.
* In Perl 6 they combine quoting and associative array lookup.
* In [[Backus–Naur Form|BNF]], they are used to denote nonterminals (e.g. <code><name> ::= <first-name> <last-name></code>).
* In [[ABAP]] they denote field symbols&nbsp;– placeholders or symbolic names for other fields, which can point to any data object.
* To indicate an action or status (e.g. <Waves> or <Offline>), particularly in online, real-time text-based discussions (instant messaging, bulletin boards, etc.). (Here, [[asterisk]]s can also be used to signify an action.)
 
When the signs are not used in pairs to delimit text (not acting as brackets):
 
* They are used as less-than and greater-than [[relational operator]]s, possibly in combination with other marks. In some languages the pair together as <code>&lt;&gt;</code> denotes an [[inequation]] ("not equal to").
* When doubled as <code>&lt;&lt;</code> or <code>&gt;&gt;</code> they may represent [[bit shift]] operators, or in C++, also stream input/output operators.
* They indicate the [[redirection (computing)|redirection]] of [[input/output]] in various [[shell (computing)|command shell]]s.<ref>[[Randal Bryant|Bryant, Randal E.]]; O'Hallaron, David. ''Computer Systems: A Programmer's Perspective'', 2003. p. 794. ISBN 0-13-034074-X.</ref>
* Right-angle brackets are used in nested [[Usenet quoting]] and various e-mail formats, and as such are standard [[quotation mark glyphs]].
* A pair of right-angle brackets followed by the character's name and a colon are used in some production scripts and translated closed captioning to denote when there is a change of speaker. This is so a performer can easily scan for their lines when rehearsing a script.
* In translating [[manga]], it is common to use right-angle brackets to indicate that the text was originally in the language it is shown in, and was thus not translated.
 
====Layout styles====
{{main|Indent style}}
In normal writing (prose) an opening bracket is rarely left hanging at the end of a line of text nor is a closing bracket permitted to start one. However, in computer code this is often done intentionally to aid readability. For example, a bracketed list of items separated by semicolons may be written with the brackets on separate lines, and the items, followed by the semicolon, each on one line.
 
A common error in programming is mismatching braces; accordingly, many [[Integrated development environment|IDE]]s have [[braces matching]] to highlight matching pairs.
 
===Mathematics===
{{main|Bracket (mathematics)}}
In addition to the use of [[#Parentheses ( )|parentheses]] to specify the [[order of operations]], both parentheses and brackets are used to denote an [[interval (mathematics)|interval]], also referred to as a half-open range. The notation {{closed-open|{{mvar|a}},{{mvar|c}}}} is used to indicate an interval from {{mvar|a}} to {{mvar|c}} that is inclusive of {{mvar|a}} but exclusive of {{mvar|c}}. That is, {{closed-open|5, 12}} would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any [[finite set|finite]] number of 9s), but 12.0 is not included. In Europe, the notation {{math|[5, 12[}} is also used for this. The endpoint adjoining the bracket is known as ''closed'', whereas the endpoint adjoining the parenthesis is known as ''open''. If both types of brackets are the same, the entire interval may be referred to as ''closed'' or ''open'' as appropriate. Whenever [[Extended real number line|+∞]] or −∞ is used as an endpoint, it is normally considered ''open'' and adjoined to a parenthesis. See [[Interval (mathematics)]] for a more complete treatment.
 
In [[quantum mechanics]], chevrons are also used as part of [[Paul Dirac|Dirac's]] formalism, [[bra-ket notation]], to note vectors from the [[dual space]]s of the Bra&nbsp;{{angbr|{{math|''A''&#124;}} and the Ket&nbsp;{{math|&#124;''B''}}}}. Mathematicians will also commonly write {{angbr|{{math|''a'', ''b''}}}} for the [[inner product]] of two vectors. In statistical mechanics, chevrons denote ensemble or time average. Chevrons are used in [[group theory]] to write [[group presentation]]s, and to denote the [[group generators|subgroup generated]] by a collection of elements. Note that [[obtuse angle]]d chevrons are not always (and even not by all users) distinguished from a pair of less-than and greater-than signs &lt;&gt;, which are sometimes used as a [[typographic approximation]] of chevrons.
 
In [[group theory]] and [[ring theory]], brackets denote the [[commutator]]. In group theory, the commutator {{math|[{{mvar|g}}, {{mvar|h}}]}} is commonly defined as {{math|{{mvar|g}}<sup> −1</sup> {{mvar|h}}<sup> −1</sup> {{mvar|g}} {{mvar|h}} }}. In ring theory, the commutator {{math|[{{mvar|a}}, {{mvar|b}}]}} is defined as {{math|{{mvar|a}} {{mvar|b}} − {{mvar|b}} {{mvar|a}} }}. Furthermore, in ring theory, braces denote the [[anticommutator]] where {{math|{{{mvar|a}}, {{mvar|b}}}}} is defined as {{math|{{mvar|a}} {{mvar|b}} + {{mvar|b}} {{mvar|a}} }}. The bracket is also used to denote the [[Lie derivative]], or more generally the Lie bracket in any [[Lie algebra]].
 
Various notations, like the [[vinculum (symbol)|vinculum]] have a similar effect to brackets in specifying order of operations, or otherwise grouping several characters together for a common purpose.
 
In the [[Z notation|Z]] [[formal specification]] language, braces define a set and chevrons define a sequence.
 
===Accounting===
Traditionally in [[Accountancy|accounting]], negative amounts are placed in parentheses.
 
===Law===
Brackets are used in some countries in the citation of [[law report]]s to identify parallel citations to non-official reporters. For example: Chronicle Pub. Co. v. [[Superior Court]], (1998) 54 Cal.2d 548, [7 Cal.Rptr. 109]. In some other countries (such as [[England and Wales]]), square brackets are used to indicate that the year is part of the citation and parentheses are used to indicate the year the judgment was given. For example, National Coal Board v England [1954] AC 403, is in the 1954 volume of the Appeal Cases reports although the decision may have been given in 1953 or earlier, whereas (1954) 98 Sol Jo 176 reports a decision from 1954, in volume 98 of the Solicitor's Journal which may be published in 1955 or later.
 
When quoted material is in any way altered, the alterations are enclosed in brackets within the quotation. For example: Plaintiff asserts his cause is just, stating, "[m]y causes is [[sic|{{bracket|''sic''}}]] just." Although in the original quoted sentence the word "my" was capitalized, it has been modified in the quotation and the change signalled with brackets. Similarly, where the quotation contained a grammatical error, the quoting author signalled that the error was in the original with "[''sic'']" (Latin for 'thus'). (''California Style Manual'', section 4:59 (4th ed.))
 
===Sports===
[[Bracket (tournament)|Tournament brackets]], the diagrammatic representation of the series of games played during a tournament usually leading to a single winner, are so named for their resemblance to brackets or braces.
 
==See also==
* [[Emoticon]]
* [[Japanese typographic symbols]]
 
==References==
{{reflist|2}}
 
==Bibliography==
* {{cite book |authorlink=John Lennard |last=Lennard |first=John |title=But I Digress: The Exploitation of Parentheses in English Printed Verse |year=1991 |isbn=0-19-811247-5 |publisher=Clarendon Press |location=Oxford}}
* {{cite book |last=Turnbull |author2=et al. |title=The Graphics of Communication |publisher=Holt |location=New York |year=1964}} States that what are depicted as brackets above are called braces and braces are called brackets. This was the terminology in US printing prior to computers.
</div>
 
==External links==
*{{commons category inline|Brackets}}
*{{wiktionary-inline|bracket}}
 
[[Category:Punctuation]]
[[Category:Mathematical notation]]

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