Bracket

This article is about the family of punctuation marks. For other uses, seeBracket (disambiguation).

Brackets
( ) [ ] { } ⟨ ⟩ Brackets (BE) or parentheses (AE) or round brackets (BE) Brackets (AE) or square brackets (BE) Braces (BE&AE) or curly brackets (BE) Angle brackets (BE&AE) or chevrons

Abracketis either of two tall fore- or back-facingpunctuationmarks commonly used to isolate a segment of text or data from its surroundings.They come in four main pairs of shapes, as given in the box to the right, which also gives their names, that vary betweenBritishandAmerican English."Brackets", without further qualification, are in British English the(…)marks and in American English the[…]marks.

Other minor bracket shapes exist, such as (for example)slashordiagonalbrackets used by linguists to enclosephonemes.

Brackets are typically deployed in symmetric pairs, and an individual bracket may be identified as a 'left' or 'right' bracket or, alternatively, an "opening bracket" or "closing bracket",respectively, depending on thedirectionalityof the context.

In casual writing and in technical fields such as computing or linguistic analysis ofgrammar,bracketsnest,with segments of bracketed material containing embedded within them other further bracketed sub-segments.The number of opening brackets matches the number of closing brackets in such cases.

Various forms of brackets are used in mathematics,with specific mathematical meanings, often for denoting specificmathematical functionsandsubformulas.

History

Angle brackets orchevrons⟨ ⟩were the earliest type of bracket to appear inwritten English.Erasmuscoined the termlunulato refer to the round brackets orparentheses()recalling the shape of thecrescent moon(Latin:luna).

Most typewriters only had the left and right parentheses. Square brackets appeared with some teleprinters.

Braces (curly brackets) first became part of a character set with the 8-bit code of theIBM 7030 Stretch.

In 1961,ASCIIcontained parentheses, square, and curly brackets, and also less-than and greater-than signs that could be used as angle brackets.

Typography

In English, typographers mostly prefer not to set brackets initalics,even when the enclosed text is italic.However, in other languages like German, if brackets enclose text in italics, they are usually also set in italics.

Parentheses or round brackets

Several terms redirect here. For other uses, seeParenthesis (disambiguation),Paren (disambiguation),Parens (disambiguation),and( ) (disambiguation).

Parenthesis
( ) parentheses (AE) brackets (BE) round brackets (BE)
In Unicode
General purpose(half-width): U+0028(LEFT PARENTHESIS(() U+0029)RIGHT PARENTHESIS()) General purpose(full-width East Asian): U+FF08(FULLWIDTH LEFT PARENTHESIS U+FF09)FULLWIDTH RIGHT PARENTHESIS Arabic script (Quranic quotations) U+FD3E﴾ORNATE LEFT PARENTHESIS U+FD3F﴿ORNATE RIGHT PARENTHESIS Mediaeval studies U+2E28⸨LEFT DOUBLE PARENTHESIS U+2E29⸩RIGHT DOUBLE PARENTHESIS Technical U+207D⁽SUPERSCRIPT LEFT PARENTHESIS U+207E⁾SUPERSCRIPT RIGHT PARENTHESIS U+208D₍SUBSCRIPT LEFT PARENTHESIS U+208E₎SUBSCRIPT RIGHT PARENTHESIS U+239B⎛LEFT PARENTHESIS UPPER HOOK U+239C⎜LEFT PARENTHESIS EXTENSION U+239D⎝LEFT PARENTHESIS LOWER HOOK U+239E⎞RIGHT PARENTHESIS UPPER HOOK U+239F⎟RIGHT PARENTHESIS EXTENSION U+23A0⎠RIGHT PARENTHESIS LOWER HOOK U+23DC⏜TOP PARENTHESIS(⏜) U+23DD⏝BOTTOM PARENTHESIS(⏝) U+27EE⟮MATHEMATICAL LEFT FLATTENED PARENTHESIS U+27EF⟯MATHEMATICAL RIGHT FLATTENED PARENTHESIS U+2983⦃LEFT WHITE CURLY BRACKET U+2984⦄RIGHT WHITE CURLY BRACKET U+2985⦅LEFT WHITE PARENTHESIS(⦅) U+2986⦆RIGHT WHITE PARENTHESIS(⦆) Phonetic punctuation U+2E59⹙TOP HALF LEFT PARENTHESIS U+2E5A⹚TOP HALF RIGHT PARENTHESIS U+2E5B⹛BOTTOM HALF LEFT PARENTHESIS U+2E5C⹜BOTTOM HALF RIGHT PARENTHESIS Dingbats U+2768❨MEDIUM LEFT PARENTHESIS ORNAMENT U+2769❩MEDIUM RIGHT PARENTHESIS ORNAMENT U+276A❪MEDIUM FLATTENED LEFT PARENTHESIS ORNAMENT U+276B❫MEDIUM FLATTENED RIGHT PARENTHESIS ORNAMENT

(and)areparentheses/pəˈrɛnθɪsiːz/(singularparenthesis/pəˈrɛnθɪsɪs/) in American English, and eitherround bracketsor simplybracketsin British English. They are also known as "parens"/pəˈrɛnz/,"circle brackets", or "smooth brackets".

In formal writing, "parentheses" is also used in British English.

Uses of ( )

Parentheses containadjunctivematerial that serves to clarify (in the manner of agloss) or is aside from the main point.

A comma before or after the material can also be used, though if the sentence contains commas for other purposes, visual confusion may result. Adashbefore and after the material is also sometimes used.

Parentheses may be used in formal writing to add supplementary information, such as "Senator John McCain (R- Arizona) spoke at length ". They can also indicate shorthand for"either singular or plural"for nouns, e.g." the claim(s) ". It can also be used forgender-neutral language,especially in languages withgrammatical gender,e.g. "(s)he agreed with his/her physician" (the slash in the second instance, as one alternative is replacing the other, not adding to it).

Parenthetical phrases have been used extensively in informal writing and stream of consciousness literature. Examples include the southern American authorWilliam Faulkner(seeAbsalom, Absalom!andthe Quentin section ofThe Sound and the Fury) as well as poetE. E. Cummings.

Parentheses have historically been used where theem dashis currently used in alternatives, such as "parenthesis)(parentheses". Examples of this usage can be seen in editions ofFowler's Dictionary of Modern English Usage.

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]).

Language

Aparenthesis in rhetoricandlinguisticsrefers to the entire bracketed text, not just to the enclosing marks used (so all the text in this set of round brackets may be described as "a parenthesis" ).Taking as an example the sentence "Mrs. Pennyfarthing (What? Yes, that was her name!) was my landlady.", the explanatory phrase between the parentheses is itself called a parenthesis. Again, the parenthesis implies that the meaning and flow of the bracketed phrase is supplemental to the rest of the text and the whole would be unchanged were the parenthesized sentences removed. The term refers to the syntax rather than the enclosure method: the same clause in the form "Mrs. Pennyfarthing – What? Yes, that was her name! – was my landlady" is also a parenthesis.(In non-specialist usage, the term "parenthetical phrase" is more widely understood.)

Inphonetics,parentheses are used for indistinguishableor unidentified utterances. They are also seen for silent articulation (mouthing),where the expected phonetic transcription is derived from lip-reading, and with periods to indicate silent pauses, for example(…)or(2sec).

Enumerations

An unpaired right parenthesis is often used as part of a label in an ordered list, such as this one:

a)educationaltesting,
b) technical writing and diagrams,
c)market research,and
d)elections.

Accounting

Traditionally inaccounting,contra amounts are placed in parentheses. A debit balance account in a series of credit balances will have parenthesis and vice versa.

Parentheses in mathematics

Main articles:Glossary of mathematical symbols § Parentheses,andBracket (mathematics)

Parentheses are used inmathematical notationto indicate grouping, often inducing a differentorder of operations.For example: in the usual order of algebraic operations,4 × 3 + 2equals 14, since themultiplicationis done before theaddition.However,4 × (3 + 2)equals 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:

$[4\times (3+2)]^{2}=400.$

Parentheses in programming languages

Parentheses are included in the syntaxes of manyprogramming 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. In some cases, such as inLISP,parentheses are a fundamental construct of the language. They are also often used for scoping functions and operators and for arrays. Insyntax diagramsthey are used for grouping, such as inextended Backus–Naur form.

In Mathematica and the Wolfram language, parentheses are used to indicate grouping – for example, with pure anonymous functions.

Taxonomy

If it is desired to include thesubgenuswhen giving thescientific nameof an animal species orsubspecies,the subgenus's name is provided in parentheses between thegenus nameand thespecific epithet.For instance,Polyphylla(Xerasiobia)albais a way to cite the speciesPolyphylla albawhile also mentioning that it is in the subgenusXerasiobia.There is also a convention of citing a subgenus by enclosing it in parentheses after its genus, e.g.,Polyphylla(Xerasiobia) is a way to refer to the subgenusXerasiobiawithin the genusPolyphylla.Parentheses are similarly used to cite a subgenus with the name of aprokaryoticspecies, although theInternational Code of Nomenclature of Prokaryotes(ICNP) requires the use of the abbreviation "subgen." as well, e.g.,Acetobacter(subgen.Gluconoacetobacter)liquefaciens.

Chemistry

Parentheses are used inchemistryto denote a repeated substructure within a molecule, e.g. HC(CH₃)₃(isobutane) or, similarly, to indicate the stoichiometry of ionic compounds with such substructures: e.g. Ca(NO₃)₂(calcium nitrate).

This is a notation that was pioneered byBerzelius,who wanted chemical formulae to more resemble algebraic notation, with brackets enclosing groups that could be multiplied (e.g. in 3(AlO₂+ 2SO₃) the 3 multiplies everything within the parentheses).

Inchemical nomenclature,parentheses are used to distinguish structural features and multipliers for clarity, for example in the polymerpoly(methyl methacrylate).

Square brackets

Square brackets
[ ]
In Unicode
General purpose (half-width) U+005B[LEFT SQUARE BRACKET([, [) U+005D]RIGHT SQUARE BRACKET(], ]) General purpose (full-width East Asian) U+FF3B［FULLWIDTH LEFT SQUARE BRACKET U+FF3D］FULLWIDTH RIGHT SQUARE BRACKET Mediaeval studies U+2045⁅LEFT SQUARE BRACKET WITH QUILL U+2046⁆RIGHT SQUARE BRACKET WITH QUILL Technical/Mathematical U+23A1⎡LEFT SQUARE BRACKET UPPER CORNER U+23A2⎢LEFT SQUARE BRACKET EXTENSION U+23A3⎣LEFT SQUARE BRACKET LOWER CORNER U+23A4⎤RIGHT SQUARE BRACKET UPPER CORNER U+23A5⎥RIGHT SQUARE BRACKET EXTENSION U+23A6⎦RIGHT SQUARE BRACKET LOWER CORNER U+23B4⎴TOP SQUARE BRACKET(⎴, ⎴) U+23B5⎵BOTTOM SQUARE BRACKET(⎵, ⎵) U+23B6⎶BOTTOM SQUARE BRACKET OVER TOP SQUARE BRACKET(⎶) U+27E6⟦MATHEMATICAL LEFT WHITE SQUARE BRACKET(⟦, ⟦) U+27E7⟧MATHEMATICAL RIGHT WHITE SQUARE BRACKET(⟧, ⟧) U+298B⦋LEFT SQUARE BRACKET WITH UNDERBAR(⦋) U+298C⦌RIGHT SQUARE BRACKET WITH UNDERBAR(⦌) U+298D⦍LEFT SQUARE BRACKET WITH TICK IN TOP CORNER(⦍) U+2990⦐RIGHT SQUARE BRACKET WITH TICK IN TOP CORNER(⦐) U+298E⦎RIGHT SQUARE BRACKET WITH TICK IN BOTTOM CORNER(⦎) U+298F⦏LEFT SQUARE BRACKET WITH TICK IN BOTTOM CORNER(⦏) Phonetic punctuation U+2E55⹕LEFT SQUARE BRACKET WITH STROKE U+2E56⹖RIGHT SQUARE BRACKET WITH STROKE U+2E57⹗LEFT SQUARE BRACKET WITH DOUBLE STROKE U+2E58⹘RIGHT SQUARE BRACKET WITH DOUBLE STROKE Quotation (East-Asian texts) U+301A〚LEFT WHITE SQUARE BRACKET U+301B〛RIGHT WHITE SQUARE BRACKET

[and]aresquare bracketsin both British and American English, but are also more simplybracketsin the latter. An older name for these brackets is "crotchets".

Uses of [ ]

Square brackets are often used to insert explanatory material or to mark where a [word or] passage was omitted from an original material by someone other than the original author, or to mark modifications in quotations.In transcribed interviews, sounds, responses and reactions that are not words but that can be described are set off in square brackets — "... [laughs]...".

When quoted material is in any way altered, the alterations are enclosed in square brackets within the quotation to show that the quotation is not exactly as given, or to add anannotation.For example:The Plaintiff asserted his cause is just, stating,

[m]y causes is[sic]just.

In the original quoted sentence, the word "my" was capitalized: it has been modified in the quotation given and the change signalled with brackets. Similarly, where the quotation contained a grammatical error (is/are), the quoting author signalled that the error was in the original with "[sic] "(Latin for 'thus').

A bracketedellipsis,[...], is often used to indicate omitted material: "I'd like to thank [several unimportant people] for their tolerance [...]" Bracketed comments inserted into a quote indicate where 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 (sometimes grammatical) modification inserted: He "hate[s] to do laundry".

Additionally, a small letter can be replaced by a capital one, when the beginning of the original printed text is being quoted in another piece of text or when the original text has been omitted for succinctness— for example, when referring to averboseoriginal: "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", can be quoted succinctly as: "[P]olicymakers [...] have made use of economic analysis [...] the way a drunkard uses a lamppost: for support, not illumination." When nested parentheses are needed, brackets are sometimes used as a substitute for the inner pair of parentheses within the outer pair.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".

In translated works, brackets are used to signify the same word or phrase in the original language to avoid ambiguity. For example:He is trained in the way of the open hand [karate].

Style and usage guidesoriginating in thenews industry of the twentieth century,such as theAP Stylebook,recommend against the use of square brackets because "They cannot be transmitted overnews wires."However, this guidance has little relevance outside of the technological constraints of the industry and era.

In linguistics,phonetic transcriptionsare generally enclosed within square brackets,whereasphonemictranscriptions typically use pairedslashes,according toInternational Phonetic Alphabetrules. Pipes (| |) are often used to indicate amorphophonemicrather than phonemic representation. Other conventions are double slashes (⫽ ⫽), double pipes (‖ ‖) and curly brackets ({ }).

Inlexicography,square brackets usually surround the section of a dictionary entry which contains theetymologyof the word the entry defines.

Proofreading

Brackets (calledmove-left symbolsormove right symbols) are added to the sides of text inproofreadingto indicate changes in indentation:

Move left	[To Fate I sue, of other means bereft, the only refuge for the wretched left.
Center	]Paradise Lost[
Move up

Square brackets are used to denote parts of the text that need to be checked when preparing drafts prior to finalizing a document.

Law

Square brackets are used in some countries in the citation oflaw reportsto identify parallel citations to non-official reporters. For example:

Chronicle Pub. Co. vSuperior Court(1998) 54 Cal.2d 548, [7 Cal.Rptr. 109]

In some other countries (such asEngland 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

This case is in the 1954 volume of the Appeal Cases reports, although the decision may have been given in 1953 or earlier. Compare with:

(1954) 98 Sol Jo 176

This citation reports a decision from 1954, in volume 98 of theSolicitors Journalwhich may be published in 1955 or later.

They often denote points that have not yet been agreed to in legal drafts and the year in which a report was made for certaincase lawdecisions.

Square brackets in mathematics

Main article:Glossary of mathematical symbols § Square brackets

Brackets are used inmathematicsin a variety of notations, including standard notations forcommutators,thefloor function,theLie bracket,equivalence classes,theIverson bracket,andmatrices.

Square brackets may be used exclusively or in combination with parentheses to representintervalsasinterval notation.For example,[0,5]represents the set of real numbers from 0 to 5 inclusive. Both parentheses and brackets are used to denote ahalf-openinterval;[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, but 12.0 is not included. In some European countries, the notation[5, 12[is also used.The endpoint adjoining the square bracket is known asclosed,whereas the endpoint adjoining the parenthesis is known asopen.

Ingroup theoryandring theory,brackets denote thecommutator.In group theory, the commutator[g, h]is commonly defined asg⁻¹ h⁻¹ g h .In ring theory, the commutator[a, b]is defined asa b−b a .

Chemistry

Square brackets can also be used inchemistryto represent theconcentrationof achemical substancein solution and to denote charge a Lewis structure of an ion (particularly distributed charge in acomplex ion), repeating chemical units (particularly in polymers) and transition state structures, among other uses.

Square brackets in programming languages

Brackets are used in many computerprogramming languages,primarily forarrayindexing. But they are also used to denote general tuples, sets and other structures, just as in mathematics. There may be several other uses as well, depending on the language at hand. Insyntax diagramsthey are used for optional portions, such as inextended Backus–Naur form.

Double brackets ⟦ ⟧

Double brackets (or white square brackets orScottbrackets), ⟦ ⟧, are used to indicate thesemantic evaluation functioninformal semanticsfor natural language anddenotational semanticsfor programming languages.In theWolfram Language,double brackets, either as iterated single brackets ([[) or ligatures (〚) are used forlist indexing.

The brackets stand for a function that maps a linguistic expression to its "denotation" or semantic value. In mathematics, double brackets may also be used to denoteintervals of integersor, less often, thefloor function.In papyrology, following theLeiden Conventions,they are used to enclose text that has been deleted in antiquity.

Brackets with quills ⁅ ⁆

Known as "spike parentheses" (Swedish:piggparenteser),⁅and⁆are used in Swedishbilingual dictionariesto enclose supplemental constructions.

Curly brackets

Curly brackets
{ }
In Unicode
General (half width) U+007B{LEFT CURLY BRACKET({, {) U+007D}RIGHT CURLY BRACKET(}, }) General (full-width East Asian) U+FF5B｛FULLWIDTH LEFT CURLY BRACKET U+FF5D｝FULLWIDTH RIGHT CURLY BRACKET Technical/Mathematical (half-width) U+23A7⎧LEFT CURLY BRACKET UPPER HOOK U+23A8⎨LEFT CURLY BRACKET MIDDLE PIECE U+23A9⎩LEFT CURLY BRACKET LOWER HOOK U+23AB⎫RIGHT CURLY BRACKET UPPER HOOK U+23AC⎬RIGHT CURLY BRACKET MIDDLE PIECE U+23AD⎭RIGHT CURLY BRACKET LOWER HOOK U+23AA⎪CURLY BRACKET EXTENSION U+23B0⎰UPPER LEFT OR LOWER RIGHT CURLY BRACKET SECTION(⎰, ⎰) U+23B1⎱UPPER RIGHT OR LOWER LEFT CURLY BRACKET SECTION(⎱, ⎱) U+23DE⏞TOP CURLY BRACKET(⏞) U+23DF⏟BOTTOM CURLY BRACKET(⏟) Dingbats U+2774❴MEDIUM LEFT CURLY BRACKET ORNAMENT U+2775❵MEDIUM RIGHT CURLY BRACKET ORNAMENT

{and}arebracesin both American and British English, and alsocurly bracketsin the latter.

Uses of { }

An example of curly brackets used to group sentences together

Curly brackets are used by text editors to mark editorial insertionsor interpolations.

Braces used to be used to connect multiple lines of poetry, such as triplets in a poem of rhyming couplets,although this usage had gone out of fashion by the 19th century.

Another older use in prose was to eliminate duplication in lists and tables. Two examples here fromCharles Hutton's 19th century table table of weights and measures in hisA Course of Mathematics:

In this kingdom

The standard of…	⎧	Length is a Yard.
	⎪	Surface is a Square Yard, the1⁄4840of an Acre.
	⎨	⎰ Solidity is a Cubic Yard.
	⎪	⎱ Capacity is a Gallon.
	⎩	Weight is a Pound.

Imperial measure of CAPACITY for coals, culm, lime, fish, potatoes, fruit,– and other goods commonly sold byheaped measure:

2 Gallons	= 1 Peck	= 764	⎱	Cubic Inches, nearly
8 Gallons	= 1 Bushel	= 28131⁄2	⎰
3 Bushels	= 1 Sack	= 48⁄9	⎱	Cubic Feet, nearly
12 Sacks	= 1 Chald.	= 582⁄3	⎰

As an extension to theInternational Phonetic Alphabet(IPA),braces are used for prosodic notation.

Music

In music, they are known as "accolades"or"braces",and connect two or more lines (staves) of music that are played simultaneously.

Chemistry

The use of braces in chemistry is an old notation that has long since been superseded by subscripted numbers. The chemical formula for water, H₂O, was represented as$\left.{{H} \atop {H}}\right\}O$.

Curly brackets in programming languages

See also:Bracing style

In many programming languages, curly brackets enclose groups ofstatementsand create a localscope.Such languages (C,C#, C++ and many others) are therefore calledcurly bracket languages.They are also used to define structures andenumerated typein these languages.

In variousUnix shells,they enclose a group of strings that are used in a process known asbrace expansion,where each successive string in the group is interpolated at that point in the command line to generate the command-line's final form. The mechanism originated in theC shelland the string generation mechanism is a simple interpolation that can occur anywhere in a command line and takes no account of existing filenames.

Insyntax diagramsthey are used for repetition, such as inextended Backus–Naur form.

In theZformal specificationlanguage, braces define a set.

Curly brackets in mathematics

Main article:Glossary of mathematical symbols § Braces

Inmathematicsthey delimitsets,in what is calledset notation. Braces enclose either a literal list of set elements, or a rule that defines the set elements. For example:

• S = {a, b}defines a setScontaininga andb.
• S = {x| x> 0}defines a setScontaining elements (implied to be numbers)x₀,x₁, and so on where everyx_nsatisfies the rule that it is greater than zero.

They are often also used to denote thePoisson bracketbetween two quantities.

Inring theory,braces denote theanticommutatorwhere{a, b}is defined asa b+b a .

Angle brackets

"Angle bracket" redirects here. For a mechanical part used for joining, seeAngle bracket (fastener).

Angle brackets
⟨ ⟩ ⟪ ⟫ < > Angle brackets (BE&AE) Angle brackets (BE&AE) less-than and greater-than
In Unicode
Less/greater than (half width) U+003C<LESS-THAN SIGN(&lt;, &LT;) U+003E>GREATER-THAN SIGN(&gt;, &GT;) Less/greater than (full-width East Asian) U+FF1C＜FULLWIDTH LESS-THAN SIGN U+FF1E＞FULLWIDTH GREATER-THAN SIGN Technical/Mathematical (half-width) U+2329〈LEFT-POINTING ANGLE BRACKET U+232A〉RIGHT-POINTING ANGLE BRACKET U+27E8⟨MATHEMATICAL LEFT ANGLE BRACKET(&lang;, &langle;, &LeftAngleBracket;) U+27E9⟩MATHEMATICAL RIGHT ANGLE BRACKET(&rang;, &rangle;, &RightAngleBracket;) U+27EA⟪MATHEMATICAL LEFT DOUBLE ANGLE BRACKET(&Lang;) U+27EB⟫MATHEMATICAL RIGHT DOUBLE ANGLE BRACKET(&Rang;) U+2991⦑LEFT ANGLE BRACKET WITH DOT(&langd;) U+2992⦒RIGHT ANGLE BRACKET WITH DOT(&rangd;) U+29FC⧼LEFT-POINTING CURVED ANGLE BRACKET U+29FD⧽RIGHT-POINTING CURVED ANGLE BRACKET Quotation (fullwidth East-Asian texts) U+3008〈LEFT ANGLE BRACKET U+3009〉RIGHT ANGLE BRACKET U+300A《LEFT DOUBLE ANGLE BRACKET U+300B》RIGHT DOUBLE ANGLE BRACKET Dingbats U+276C❬MEDIUM LEFT-POINTING ANGLE BRACKET ORNAMENT U+276D❭MEDIUM RIGHT-POINTING ANGLE BRACKET ORNAMENT U+2770❰HEAVY LEFT-POINTING ANGLE BRACKET ORNAMENT U+2771❱HEAVY RIGHT-POINTING ANGLE BRACKET ORNAMENT U+276E❮HEAVY LEFT-POINTING ANGLE QUOTATION MARK ORNAMENT U+276F❯HEAVY RIGHT-POINTING ANGLE QUOTATION MARK ORNAMENT

⟨and⟩areangle bracketsin both American and British English.In computer slang, they are known as "brokets".

Strictly speaking they are distinct from V-shapedchevrons,as they have (where the typography permits it) a broader span than chevrons,although when printed often no visual distinction is made.

The ASCII less-than and greater-than characters<>are often used for angle brackets. In most cases only those characters are accepted by computer programs, and the Unicode angle brackets are not recognized (for instance, inHTML tags). The characters for "single"guillemets‹›are also often used, and sometimes normal guillemets«»when nested angle brackets are needed.

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,because they are canonically equivalent to the CJK code points U+300x and thus likely to render as double-width symbols. Theless-thanandgreater-thansymbols are often used as replacements for chevrons.

Shape

Angle brackets are larger thanless-thanandgreater-than signs,which in turn are larger thanguillemets.

Angle brackets, less-than/greater-than signs and singleguillemetsin fontsCambria,DejaVuSerif,AndronMega Corpus,AndikaandEverson Mono

Uses of ⟨ ⟩

Angle brackets are infrequently used to denotewords that are thoughtinstead of spoken, such as:

⟨What an unusual flower!⟩

Intextual 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 their own reconstruction where possible within them.

Incomic books,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. Since no foreign language is actually written, this is onlynotionallytranslated.

Inlinguistics,angle brackets identifygraphemes(e.g.,letters of an alphabet) ororthography,as in "The English word/kæt/is spelled⟨cat⟩."(SeeIPA Brackets and transcription delimiters.)

Inepigraphy,they may be used for mechanical transliterations of a text into the Latin script.

InEast Asian punctuation,angle brackets are used asquotation marks.Chevron-like symbols are part of standardChinese,Japaneseand – less frequently –Koreanpunctuation, where they generally enclose the titles of books, as:〈... 〉or《... 》for traditionalvertical printing— written in vertical lines — and as 〈... 〉 or 《... 》 forhorizontalprinting — in horizontal.

Angle brackets in mathematics

Main article:Glossary of mathematical symbols § ⟨⟩

Angle brackets (or 'chevrons') are used ingroup theoryto writegroup presentations,and to denote thesubgroup generatedby a collection of elements. Inset theory,chevrons or parentheses are used to denoteordered pairsand othertuples,whereas curly brackets are used for unordered sets.

Physics and mechanics

In physical sciences and statistical mechanics, angle brackets are used to denote an average (expected value) over time or over another continuous parameter. For example:

$\left\langle V(t)^{2}\right\rangle =\lim _{T\to \infty }{\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}V(t)^{2}\,{\rm {d}}t.$

In mathematical physics, especiallyquantum mechanics,it is common to write theinner productbetween elements as⟨a|b⟩,as a short version of⟨a|·|b⟩,or⟨a|Ô|b⟩,whereÔis anoperator.This is known asDirac notationorbra–ket notation,to note vectors from thedual spacesof the Bra⟨A|and the Ket|B⟩.But there areother notationsused.

Incontinuum mechanics,chevrons may be used asMacaulay brackets.

Angle brackets in programming languages

InC++chevrons (actually less-than and greater-than) are used to surround arguments totemplates.They are also used to surround the names ofheader files;this usage was inherited from and is also found inC.

In theZformal specificationlanguage, chevrons define a sequence.

InHTML,chevrons (actually 'greater than' and 'less than' symbols) are used to bracket meta text. For example<b>denotes that the following text should be displayed as bold. Pairs of meta text tags are required – much as brackets themselves are usually in pairs. The end of the bold text segment would be indicated by</b>.This use is sometimes extended as an informal mechanism for communicating mood or tone in digital formats such as messaging, for example adding "<sighs>" at the end of a sentence.

Other brackets

Lenticular brackets【】

SomeEast Asianlanguages use lenticular brackets【】,a combination of square brackets and round brackets calledPhương đầu quát hào(fāngtóu kuòhào) inChineseandNgung phó き quát hồ(sumitsuki kakko) inJapanese.They are used in titles and headings in both Chineseand Japanese. On the Internet, they are used to emphasize a text. In Japanese, they are most frequently seen in dictionaries for quoting Chinese characters and Sino-Japanese loanwords.

Floor ⌊ ⌋ and ceiling ⌈ ⌉ corner brackets

Floor and ceiling
⌈ceiling⌉ ⌊floor⌋
In Unicode
Floor and ceiling functions U+2308⌈LEFT CEILING(⌈, ⌈) U+2309⌉RIGHT CEILING(⌉, ⌉) U+230A⌊LEFT FLOOR(⌊, ⌊) U+230B⌋RIGHT FLOOR(⌋, ⌋)

The floor corner brackets⌊and⌋,the ceiling corner brackets⌈and⌉(U+2308, U+2309) are used to denote the integerfloor and ceiling functions.

Quine corners ⌜⌝ and half brackets ⸤ ⸥ or ⸢ ⸣

The Quine corners⌜and⌝have at least two uses inmathematical logic:either asquasi-quotation,a generalization of quotation marks, or to denote theGödel numberof the enclosedexpression.

Half brackets are used in English to mark added text, such as in translations: "Bill saw ⸤her⸥".

In editions ofpapyrologicaltexts, 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.For example,CallimachusIambus1.2 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.

Unicode

Representations of various kinds of brackets inUnicodeand their respectiveHTML entities,that are not in the infoboxes in preceding sections, are given below.

Unicode and HTML encodings for various bracket characters

Uses	Unicode/HTML	Sample
Quine corners	U+231C⌜TOP LEFT CORNER(⌜, ⌜)	⌜quasi-quotation⌝ ⌜editorial notation⌝
	U+231D⌝TOP RIGHT CORNER(⌝, ⌝)
	U+231E⌞BOTTOM LEFT CORNER(⌞, ⌞)	⌞editorial notation⌟
	U+231F⌟BOTTOM RIGHT CORNER(⌟, ⌟)
Fullwidth parentheses	U+FF5F｟FULLWIDTH LEFT WHITE PARENTHESIS	｟...｠
	U+FF60｠FULLWIDTH RIGHT WHITE PARENTHESIS
Technical/mathematical (specialized)	U+23B8⎸LEFT VERTICAL BOX LINE	⎸boxed text⎹
	U+23B9⎹RIGHT VERTICAL BOX LINE
	U+23E0⏠TOP TORTOISE SHELL BRACKET	⏠ tortoise shell brackets ⏡
	U+23E1⏡BOTTOM TORTOISE SHELL BRACKET
	U+27C5⟅LEFT S-SHAPED BAG DELIMITER	⟅...⟆
	U+27C6⟆RIGHT S-SHAPED BAG DELIMITER
	U+27D3⟓LOWER RIGHT CORNER WITH DOT	⟓pullback...pushout⟔
	U+27D4⟔UPPER LEFT CORNER WITH DOT
	U+27EC⟬MATHEMATICAL LEFT WHITE TORTOISE SHELL BRACKET(⟬)	⟬white tortoise shell brackets⟭
	U+27ED⟭MATHEMATICAL RIGHT WHITE TORTOISE SHELL BRACKET(⟭)
	U+2987⦇Z NOTATION LEFT IMAGE BRACKET	R⦇S⦈
	U+2988⦈Z NOTATION RIGHT IMAGE BRACKET
	U+2989⦉Z NOTATION LEFT BINDING BRACKET	⦉x:ℤ⦊
	U+298A⦊Z NOTATION RIGHT BINDING BRACKET
	U+2993⦓LEFT ARC LESS-THAN BRACKET(⦓)	⦓inequalitysign brackets⦔
	U+2994⦔RIGHT ARC GREATER-THAN BRACKET(⦔)
	U+2995⦕DOUBLE LEFT ARC GREATER-THAN BRACKET(⦕)	⦕inequality sign brackets⦖
	U+2996⦖DOUBLE RIGHT ARC LESS-THAN BRACKET(⦖)
	U+2997⦗LEFT BLACK TORTOISE SHELL BRACKET	⦗black tortoise shell brackets⦘
	U+2998⦘RIGHT BLACK TORTOISE SHELL BRACKET
	U+29D8⧘LEFT WIGGLY FENCE	⧘...⧙
	U+29D9⧙RIGHT WIGGLY FENCE
	U+29DA⧚LEFT DOUBLE WIGGLY FENCE	⧚...⧛
	U+29DB⧛RIGHT DOUBLE WIGGLY FENCE
Half brackets	U+2E22⸢TOP LEFT HALF BRACKET	⸢editorial notation⸣
	U+2E23⸣TOP RIGHT HALF BRACKET
	U+2E24⸤BOTTOM LEFT HALF BRACKET	⸤editorial notation⸥
	U+2E25⸥BOTTOM RIGHT HALF BRACKET
Compatibility variants for CNS 11643	U+FE59﹙SMALL LEFT PARENTHESIS	﹙...﹚
	U+FE5A﹚SMALL RIGHT PARENTHESIS
	U+FE5B﹛SMALL LEFT CURLY BRACKET	﹛...﹜
	U+FE5C﹜SMALL RIGHT CURLY BRACKET
	U+FE5D﹝SMALL LEFT TORTOISE SHELL BRACKET	﹝...﹞
	U+FE5E﹞SMALL RIGHT TORTOISE SHELL BRACKET
Dingbats	U+2772❲LIGHT LEFT TORTOISE SHELL BRACKET ORNAMENT(❲)	❲light tortoise shell bracket ornament❳
	U+2773❳LIGHT RIGHT TORTOISE SHELL BRACKET ORNAMENT(❳)
N'Ko	U+2E1C⸜LEFT LOW PARAPHRASE BRACKET	⸜ߒߞߏ⸝‎
	U+2E1D⸝RIGHT LOW PARAPHRASE BRACKET
Ogham	U+169B᚛OGHAM FEATHER MARK	᚛ᚑᚌᚐᚋ᚜
	U+169C᚜OGHAM REVERSED FEATHER MARK
Old Hungarian	U+2E42⹂DOUBLE LOW-REVERSED-9 QUOTATION MARK	⹂
Tibetan	U+0F3A༺TIBETAN MARK GUG RTAGS GYON	༺དབུ་ཅན་༻
	U+0F3B༻TIBETAN MARK GUG RTAGS GYAS
	U+0F3C༼TIBETAN MARK ANG KHANG GYON	༼༡༢༣༽
	U+0F3D༽TIBETAN MARK ANG KHANG GYAS
New Testamenteditorial marks	U+2E02⸂LEFT SUBSTITUTION BRACKET	⸂...⸃
	U+2E03⸃RIGHT SUBSTITUTION BRACKET
	U+2E04⸄LEFT DOTTED SUBSTITUTION BRACKET	⸄...⸅
	U+2E05⸅RIGHT DOTTED SUBSTITUTION BRACKET
	U+2E09⸉LEFT TRANSPOSITION BRACKET	⸉...⸊
	U+2E0A⸊RIGHT TRANSPOSITION BRACKET
	U+2E0C⸌LEFT RAISED OMISSION BRACKET	⸌...⸍
	U+2E0D⸍RIGHT RAISED OMISSION BRACKET
Medieval studies	U+2E26⸦LEFT SIDEWAYS U BRACKET	⸦crux⸧
	U+2E27⸧RIGHT SIDEWAYS U BRACKET
Quotation (East-Asian texts)	U+3014〔LEFT TORTOISE SHELL BRACKET	〔...〕
	U+3015〕RIGHT TORTOISE SHELL BRACKET
	U+3016〖LEFT WHITE LENTICULAR BRACKET	〖...〗
	U+3017〗RIGHT WHITE LENTICULAR BRACKET
	U+3018〘LEFT WHITE TORTOISE SHELL BRACKET	〘...〙
	U+3019〙RIGHT WHITE TORTOISE SHELL BRACKET
	U+301D〝REVERSED DOUBLE PRIME QUOTATION MARK	〝...〞
	U+301E〞DOUBLE PRIME QUOTATION MARK
Quotation (halfwidthEast-Asian texts)	U+FF62｢HALFWIDTH LEFT CORNER BRACKET	｢ｶﾀｶﾅ｣
	U+FF63｣HALFWIDTH RIGHT CORNER BRACKET
Quotation (fullwidth East-Asian texts)	U+300C“LEFT CORNER BRACKET	“Biểu đề”
	U+300D”RIGHT CORNER BRACKET
	U+300E『LEFT WHITE CORNER BRACKET	『 biểu đề 』
	U+300F』RIGHT WHITE CORNER BRACKET
	U+3010【LEFT BLACK LENTICULAR BRACKET	【 biểu đề 】
	U+3011】RIGHT BLACK LENTICULAR BRACKET
Vertical bracket presentation forms	U+FE17︗PRESENTATION FORM FOR VERTICAL LEFT WHITE LENTICULAR BRACKET	︗...︘
	U+FE18︘PRESENTATION FORM FOR VERTICAL RIGHT WHITE LENTICULAR BRACKET
	U+FE35︵PRESENTATION FORM FOR VERTICAL LEFT PARENTHESIS	︵...︶
	U+FE36︶PRESENTATION FORM FOR VERTICAL RIGHT PARENTHESIS
	U+FE37︷PRESENTATION FORM FOR VERTICAL LEFT CURLY BRACKET	︷...︸
	U+FE38︸PRESENTATION FORM FOR VERTICAL RIGHT CURLY BRACKET
	U+FE39︹PRESENTATION FORM FOR VERTICAL LEFT TORTOISE SHELL BRACKET	︹...︺
	U+FE3A︺PRESENTATION FORM FOR VERTICAL RIGHT TORTOISE SHELL BRACKET
	U+FE3B︻PRESENTATION FORM FOR VERTICAL LEFT BLACK LENTICULAR BRACKET	︻...︼
	U+FE3C︼PRESENTATION FORM FOR VERTICAL RIGHT BLACK LENTICULAR BRACKET
	U+FE3D︽PRESENTATION FORM FOR VERTICAL LEFT DOUBLE ANGLE BRACKET	︽...︾
	U+FE3E︾PRESENTATION FORM FOR VERTICAL RIGHT DOUBLE ANGLE BRACKET
	U+FE3F︿PRESENTATION FORM FOR VERTICAL LEFT ANGLE BRACKET	︿...﹀
	U+FE40﹀PRESENTATION FORM FOR VERTICAL RIGHT ANGLE BRACKET
	U+FE41﹁PRESENTATION FORM FOR VERTICAL LEFT CORNER BRACKET	﹁...﹂
	U+FE42﹂PRESENTATION FORM FOR VERTICAL RIGHT CORNER BRACKET
	U+FE43﹃PRESENTATION FORM FOR VERTICAL LEFT WHITE CORNER BRACKET	﹃...﹄
	U+FE44﹄PRESENTATION FORM FOR VERTICAL RIGHT WHITE CORNER BRACKET
	U+FE47﹇PRESENTATION FORM FOR VERTICAL LEFT SQUARE BRACKET	﹇...﹈
	U+FE48﹈PRESENTATION FORM FOR VERTICAL RIGHT SQUARE BRACKET

See also

• Bracket (mathematics)
• International variation in quotation marks
• Emoticon
• Japanese typographic symbols
• Order of operations
• Triple parentheses

References

Sources

• McArthur, Thomas Burns;McArthur, Roshan (2005). "Brackets".Concise Oxford Companion to the English Language.Oxford University Press.ISBN9780192806376.
• Pointon, Graham; Clark, Stewart (2014). "Punctuation Guide".Words: A User's Guide.Routledge.ISBN9781317864295.
• Peters, Pam(2007).The Cambridge Guide to Australian English Usage(2nd ed.). Cambridge University Press.ISBN9781139465212.
• Watts, Henry(1877). "Notation".A Dictionary of Chemistry and the Allied Branches of Other Sciences.Vol. 4. Longmans, Green, and Company.
• Newham, Cameron; Rosenblatt, Bill (1998).Learning the Bash Shell.O'Reilly Media.ISBN9781565923478.
• Sobell, Mark G.; Seebach, Peter (2005).A Practical Guide to UNIX for Mac OS X Users.Prentice Hall Professional.ISBN9780321629982.
• Biggs, Norman (2002). "Set notation".Discrete Mathematics.OUP Oxford.ISBN9780198507178.
• Ihde, Aaron J. (1984).The Development of Modern Chemistry.Dover Books on Chemistry. Courier Corporation.ISBN9780486642352.
• Achatz, Thomas; Anderson, John G. (2005). McKenzie, Kathleen (ed.).Technical Shop Mathematics.Industrial Press.ISBN9780831130862.
• Wilson, John (1850).Treatise on English Punctuation(2nd ed.). Boston.{{cite book}}:CS1 maint: location missing publisher (link)
• Robertson, Joseph(1785).An Essay on Punctuation.London: J. Walter.
• Hutton, Charles(1836). Gregory, Olinthus (ed.).A Course of Mathematics.Vol. 1 (11th ed.). London: Longman, Rees.

Bibliography

• Lennard, John(1991).But I Digress: The Exploitation of Parentheses in English Printed Verse.Oxford: Clarendon Press.ISBN0198112475.
• Turnbull, Arthur T.; Baird, Russell N. (1964).The Graphics of Communication: Typography, Layout, Design.New York: Holt, Rinehart and Winston.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.

External links

• Media related toBracketsat Wikimedia Commons
• The dictionary definition ofbracketat Wiktionary