Brackets
( ) [ ] { } ⟨ ⟩
Brackets (BE)
or
parentheses (AE)
or
round brackets (BE)[1]
Brackets (AE)
or
square brackets (BE)[1]
Braces (BE&AE)
or
curly brackets (BE)[1]
Angle brackets (BE&AE)[1]
or
chevrons [2]

A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings.[3] They come in four main pairs of shapes, as given in the box to the right, which also gives their names, that vary between British and American English.[1] "Brackets", without further qualification, are in British English the () marks and in American English the [] marks.[1][3]

Other minor bracket shapes exist, such as (for example) slash or diagonal brackets used by linguists to enclose phonemes.[4]

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",[5] respectively, depending on the directionality of the context.

In casual writing and in technical fields such as computing or linguistic analysis of grammar, brackets nest, with segments of bracketed material containing embedded within them other further bracketed sub-segments.[3] The number of opening brackets matches the number of closing brackets in such cases.[3]

Various forms of brackets are used in mathematics, with specific mathematical meanings, often for denoting specific mathematical functions and subformulas.

## History

Angle brackets or chevrons ⟨ ⟩ were the earliest type of bracket to appear in written English. Erasmus coined the term lunula to refer to the round brackets or parentheses ( ) recalling the shape of the crescent moon (Latin: luna).[6]

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 the IBM 7030 Stretch.[7]

In 1961, ASCII contained 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 in italics, even when the enclosed text is italic.[8] However, in other languages like German, if brackets enclose text in italics, they are usually also set in italics.[9]

## Parentheses or (round) brackets

 Several terms redirect here. For other uses, see Parenthesis (disambiguation), Paren (disambiguation), Parens (disambiguation), and ( ) (disambiguation).
 For technical reasons, ":)" redirects here. For the keyboard symbols, see List of emoticons.
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Parenthesis
( )
Parentheses (AE)
or
brackets (BE)
or
round brackets (BE)[1]
In Unicode
General purpose
(half-width)[10]
• U+0028 ( LEFT PARENTHESIS (&lpar;)
• U+0029 ) RIGHT PARENTHESIS (&rpar;)
General purpose
(full-width East Asian)[11]
• U+FF08 FULLWIDTH LEFT PARENTHESIS
• U+FF09 FULLWIDTH RIGHT PARENTHESIS
Arabic script
(Quranic quotations)[12]
• U+FD3E ORNATE LEFT PARENTHESIS
• U+FD3F ﴿ ORNATE RIGHT PARENTHESIS
• U+2E28 LEFT DOUBLE PARENTHESIS
• U+2E29 RIGHT DOUBLE PARENTHESIS
• 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 (&OverParenthesis;)
• U+23DD BOTTOM PARENTHESIS (&UnderParenthesis;)
• 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 (&lopar;)
• U+2986 RIGHT WHITE PARENTHESIS (&ropar;)
Phonetic punctuation[19]
• U+2E59 TOP HALF LEFT PARENTHESIS
• U+2E5A TOP HALF RIGHT PARENTHESIS
• U+2E5B BOTTOM HALF LEFT PARENTHESIS
• U+2E5C BOTTOM HALF RIGHT PARENTHESIS
• 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 ) are parentheses /pəˈrɛnθɪsz/ (singular parenthesis /pəˈrɛnθɪsɪs/) in American English, and either round brackets or simply brackets in British English.[1][4] They are also known as "parens" /pəˈrɛnz/, "circle brackets", or "smooth brackets".[21]

In careful or formal writing, "parentheses" is also used in British English.[citation needed]

### Uses of ( )

Parentheses contain adjunctive material that serves to clarify (in the manner of a gloss) or is aside from the main point.[22]

A comma before or after the material can also be used, though if the sentence contains commas for other purposes, visual confusion may result. A dash before 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 for gender-neutral language, especially in languages with grammatical 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 author William Faulkner (see Absalom, Absalom! and 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 is currently used in alternatives, such as "parenthesis)(parentheses". Examples of this usage can be seen in editions of Fowler'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

A parenthesis in rhetoric and linguistics refers 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").[23] 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.[24] (In non-specialist usage, the term "parenthetical phrase" is more widely understood.[25])

In phonetics, parentheses are used for indistinguishable[26] or unidentified utterances. They are also seen for silent articulation (mouthing),[27] where the expected phonetic transcription is derived from lip-reading, and with periods to indicate silent pauses, for example (…) or (2 sec).

#### Enumerations

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

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

#### Accounting

Traditionally in accounting, 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, and Bracket (mathematics)

Parentheses are used in mathematical notation to indicate grouping, often inducing a different order of operations. For example: in the usual order of algebraic operations, 4 × 3 + 2 equals 14, since the multiplication is done before the addition. 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:

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

#### Parentheses in programming languages

Parentheses are included in the syntaxes of many 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. In some cases, such as in LISP, parentheses are a fundamental construct of the language. They are also often used for scoping functions and operators and for arrays. In syntax diagrams they are used for grouping, such as in extended 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 the subgenus when giving the scientific name of an animal species or subspecies, the subgenus's name is provided in parentheses between the genus name and the specific epithet.[28] For instance, Polyphylla (Xerasiobia) alba is a way to cite the species Polyphylla alba while also mentioning that it is in the subgenus Xerasiobia.[29] 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 subgenus Xerasiobia within the genus Polyphylla.[30] Parentheses are similarly used to cite a subgenus with the name of a prokaryotic species, although the International Code of Nomenclature of Prokaryotes (ICNP) requires the use of the abbreviation "subgen." as well, e.g., Acetobacter (subgen. Gluconoacetobacter) liquefaciens.[31]

#### Chemistry

Parentheses are used in chemistry to 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 by Berzelius, 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).[32][33]

In chemical nomenclature, parentheses are used to distinguish structural features and multipliers for clarity, for example in the polymer poly(methyl methacrylate).[34]

## Square brackets

Square brackets
[ ]
In Unicode
General purpose
(half-width)[10]
• U+005B [ LEFT SQUARE BRACKET (&lbrack;, &lsqb;)
• U+005D ] RIGHT SQUARE BRACKET (&rbrack;, &rsqb;)
General purpose
(full-width East Asian)[11]
• U+FF3B FULLWIDTH LEFT SQUARE BRACKET
• U+FF3D FULLWIDTH RIGHT SQUARE BRACKET
• U+2045 LEFT SQUARE BRACKET WITH QUILL
• U+2046 RIGHT SQUARE BRACKET WITH QUILL
Technical/Mathematical[15][16][17][18]
• 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 (&OverBracket;, &tbrk;)
• U+23B5 BOTTOM SQUARE BRACKET (&bbrk;, &UnderBracket;)
• U+23B6 BOTTOM SQUARE BRACKET OVER TOP SQUARE BRACKET (&bbrktbrk;)
• U+27E6 MATHEMATICAL LEFT WHITE SQUARE BRACKET (&LeftDoubleBracket;, &lobrk;)
• U+27E7 MATHEMATICAL RIGHT WHITE SQUARE BRACKET (&RightDoubleBracket;, &robrk;)
• U+298B LEFT SQUARE BRACKET WITH UNDERBAR (&lbrke;)
• U+298C RIGHT SQUARE BRACKET WITH UNDERBAR (&rbrke;)
• U+298D LEFT SQUARE BRACKET WITH TICK IN TOP CORNER (&lbrkslu;)
• U+2990 RIGHT SQUARE BRACKET WITH TICK IN TOP CORNER (&rbrkslu;)
• U+298E RIGHT SQUARE BRACKET WITH TICK IN BOTTOM CORNER (&rbrksld;)
• U+298F LEFT SQUARE BRACKET WITH TICK IN BOTTOM CORNER (&lbrksld;)
Phonetic punctuation[19]
• 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)[35]
• U+301A LEFT WHITE SQUARE BRACKET
• U+301B RIGHT WHITE SQUARE BRACKET

[ and ] are square brackets in both British and American English, but are also more simply brackets in the latter.[1][3] An older name for these brackets is "crotchets".[36]

### 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.[37] 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 an annotation.[38] 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 bracketed ellipsis, [...], is often used to indicate omitted material: "I'd like to thank [several unimportant people] for their tolerance [...]"[39] 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 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", 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.[40] 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".[41]

In translated works, brackets are used to signify the same word or phrase in the original language to avoid ambiguity.[42] For example: He is trained in the way of the open hand [karate].

Style and usage guides originating in the news industry of the twentieth century, such as the AP Stylebook, recommend against the use of square brackets because "They cannot be transmitted over news wires."[43] However, this guidance has little relevance outside of the technological constraints of the industry and era.

In linguistics, phonetic transcriptions are generally enclosed within square brackets,[44] whereas phonemic transcriptions typically use paired slashes, according to International Phonetic Alphabet rules. Pipes (| |) are often used to indicate a morphophonemic rather than phonemic representation. Other conventions are double slashes (⫽ ⫽), double pipes (‖ ‖) and curly brackets ({ }).

In lexicography, square brackets usually surround the section of a dictionary entry which contains the etymology of the word the entry defines.

Brackets (called move-left symbols or move right symbols) are added to the sides of text in proofreading to indicate changes in indentation:

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

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 of law reports 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

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 the Solicitors Journal which 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 certain case law decisions.

#### Square brackets in mathematics

 Main article: Glossary of mathematical symbols § Square brackets

Brackets are used in mathematics in a variety of notations, including standard notations for commutators, the floor function, the Lie bracket, equivalence classes, the Iverson bracket, and matrices.

Square brackets may be used exclusively or in combination with parentheses to represent intervals as interval notation.[45] For example, [0,5] represents the set of real numbers from 0 to 5 inclusive. Both parentheses and brackets are used to denote a half-open interval; [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.[46][47] The endpoint adjoining the square bracket is known as closed, whereas the endpoint adjoining the parenthesis is known as open.[45]

In group theory and ring theory, brackets denote the commutator. In group theory, the commutator [g, h] is commonly defined as g −1h −1gh. In ring theory, the commutator [a, b] is defined as abba.

#### Chemistry

Square brackets can also be used in chemistry to represent the concentration of a chemical substance in solution and to denote charge a Lewis structure of an ion (particularly distributed charge in a complex ion), repeating chemical units (particularly in polymers) and transition state structures, among other uses.

#### Square brackets in programming languages

Brackets are used in many computer programming languages, primarily for array indexing. 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. In syntax diagrams they are used for optional portions, such as in extended Backus–Naur form.

### Double brackets ⟦ ⟧

Double brackets (or white square brackets or Scott brackets), ⟦ ⟧, are used to indicate the semantic evaluation function in formal semantics for natural language and denotational semantics for programming languages.[48][49] In the Wolfram Language, double brackets, either as iterated single brackets ([[) or ligatures (〚) are used for list indexing.[50]

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 denote intervals of integers or, less often, the floor function. In papyrology, following the Leiden Conventions, they are used to enclose text that has been deleted in antiquity.[51]

### Brackets with quills ⁅ ⁆

Known as "spike parentheses" (Swedish: piggparenteser), ⁅ and ⁆ are used in Swedish bilingual dictionaries to enclose supplemental constructions.[52]

## Curly brackets

Curly brackets
{ }
In Unicode
General
(half width)[10]
• U+007B { LEFT CURLY BRACKET (&lbrace;, &lcub;)
• U+007D } RIGHT CURLY BRACKET (&rbrace;, &rcub;)
General
(full-width East Asian)[11]
• U+FF5B FULLWIDTH LEFT CURLY BRACKET
• U+FF5D FULLWIDTH RIGHT CURLY BRACKET
Technical/Mathematical
(half-width)[15][16][17][18]
• 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 (&lmoust;, &lmoustache;)
• U+23B1 UPPER RIGHT OR LOWER LEFT CURLY BRACKET SECTION (&rmoust;, &rmoustache;)
• U+23DE TOP CURLY BRACKET (&OverBrace;)
• U+23DF BOTTOM CURLY BRACKET (&UnderBrace;)
• U+2774 MEDIUM LEFT CURLY BRACKET ORNAMENT
• U+2775 MEDIUM RIGHT CURLY BRACKET ORNAMENT

{ and } are braces in both American and British English, and also curly brackets in the latter.[1][3]

### Uses of { }

Curly brackets are used by text editors to mark editorial insertions[53] or interpolations.[54]

Braces used to be used to connect multiple lines of poetry, such as triplets in a poem of rhyming couplets,[55] although this usage had gone out of fashion by the 19th century.[56][57]

Another older use in prose was to eliminate duplication in lists and tables.[57] Two examples here from Charles Hutton's 19th century table table of weights and measures in his A Course of Mathematics:

 The standard of … ⎧ Length is a Yard. ⎪ Surface is a Square Yard, the .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄4840 of an Acre. ⎨ ⎰ Solidity is a Cubic Yard. ⎪ ⎱ Capacity is a Gallon. ⎩ Weight is a Pound.
 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 the International 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.[60]

#### Chemistry

The use of braces in chemistry is an old notation that has long since been superseded by subscripted numbers.[32] The chemical formula for water, H2O, was represented as ${\displaystyle \left.{\stackrel {H}{H))\right\}O}$.[32]

#### Curly brackets in programming languages

In many programming languages, curly brackets enclose groups of statements and create a local scope. Such languages (C, C#, C++ and many others) are therefore called curly bracket languages.[61] They are also used to define structures and enumerated type in these languages.

In various Unix shells, they enclose a group of strings that are used in a process known as brace 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.[62] The mechanism originated in the C shell and the string generation mechanism is a simple interpolation that can occur anywhere in a command line and takes no account of existing filenames.[63]

In syntax diagrams they are used for repetition, such as in extended Backus–Naur form.

In the Z formal specification language, braces define a set.

#### Curly brackets in mathematics

 Main article: Glossary of mathematical symbols § Braces

In mathematics they delimit sets, in what is called set notation.[64] Braces enclose either a literal list of set elements, or a rule that defines the set elements.[64] For example:

• S = {a, b} defines a set S containing a and b.[64]
• S = {x | x > 0} defines a set S containing elements (implied to be numbers) x0, x1, and so on where every xn satisfies the rule that it is greater than zero.[64]

They are often also used to denote the Poisson bracket between two quantities.

In ring theory, braces denote the anticommutator where {a, b} is defined as ab + ba.

## Angle brackets

 "Angle bracket" redirects here. For a mechanical part used for joining, see Angle bracket (fastener).
Angle brackets
⟨ ⟩ ⟪ ⟫ < >
Angle brackets (BE&AE)[1] Angle brackets (BE&AE)[1] less-than and greater-than
In Unicode
Less/greater than
(half width)[10]
• U+003C < LESS-THAN SIGN (&lt;, &LT;)
• U+003E > GREATER-THAN SIGN (&gt;, &GT;)
Less/greater than
(full-width East Asian)[11]
• U+FF1C FULLWIDTH LESS-THAN SIGN
• U+FF1E FULLWIDTH GREATER-THAN SIGN
Technical/Mathematical
(half-width)[15][16][17][18]
• U+2329 LEFT-POINTING ANGLE BRACKET[a]
• U+232A RIGHT-POINTING ANGLE BRACKET[a]
• U+27E8 MATHEMATICAL LEFT ANGLE BRACKET (&lang;, &langle;, &LeftAngleBracket;)[a]
• U+27E9 MATHEMATICAL RIGHT ANGLE BRACKET (&rang;, &rangle;, &RightAngleBracket;)[a]
• 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)[35]
• U+3008 LEFT ANGLE BRACKET
• U+3009 RIGHT ANGLE BRACKET
• U+300A LEFT DOUBLE ANGLE BRACKET
• U+300B RIGHT DOUBLE ANGLE BRACKET
• 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 are angle brackets in both American and British English.[1][3] In computer slang, they are known as "brokets".[65]

Strictly speaking they are distinct from V-shaped chevrons, as they have (where the typography permits it) a broader span than chevrons,[66] although when printed often no visual distinction is made.[4]

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, in HTML 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,[15] 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.

1. ^ a b c d &lang; and &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 XML Entity Definitions for Characters (Archived 27 January 2013 at the Wayback Machine).

### Shape

Angle brackets are larger than less-than and greater-than signs, which in turn are larger than guillemets.

### Uses of ⟨ ⟩

Angle brackets are infrequently used to denote words that are thought instead of spoken, such as:

⟨What an unusual flower!⟩

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 their own reconstruction where possible within them.[67]

In comic 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 only notionally translated.[citation needed]

In linguistics, angle brackets identify graphemes (e.g., letters of an alphabet) or orthography, as in "The English word /kæt/ is spelled ⟨cat⟩."[68][69][67]

In epigraphy, they may be used for mechanical transliterations of a text into the Latin script.[69]

In East Asian punctuation, angle brackets are used as quotation marks. Chevron-like symbols are part of standard Chinese, Japanese and – less frequently – Korean punctuation, where they generally enclose the titles of books, as: 〈 ... 〉 or 《 ... 》 for traditional vertical printing — written in vertical lines — and as 〈 ... 〉 or 《 ... 》 for horizontal printing — in horizontal.

#### Angle brackets in mathematics

 Main article: Glossary of mathematical symbols § ⟨⟩

Angle brackets (or 'chevrons') are used in group theory to write group presentations, and to denote the subgroup generated by a collection of elements. In set theory, chevrons or parentheses are used to denote ordered pairs[70] and other tuples, 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:

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

In mathematical physics, especially quantum mechanics, it is common to write the inner product between elements as a|b, as a short version of a|·|b, or a|Ô|b, where Ô is an operator. This is known as Dirac notation or bra–ket notation, to note vectors from the dual spaces of the Bra A| and the Ket |B. But there are other notations used.

In continuum mechanics, chevrons may be used as Macaulay brackets.

#### Angle brackets in programming languages

In C++ chevrons (actually less-than and greater-than) are used to surround arguments to templates. They are also used to surround the names of header files; this usage was inherited from and is also found in C.

In the Z formal specification language, chevrons define a sequence.

In HTML, 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【】

Some East Asian languages use lenticular brackets , a combination of square brackets and round brackets called 方頭括號 (fāngtóu kuòhào) in Chinese and 隅付き括弧 (sumitsuki kakko) in Japanese. They are used in titles and headings in both Chinese[71] and 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[15]
• U+2308 LEFT CEILING (&lceil;, &LeftCeiling;)
• U+2309 RIGHT CEILING (&rceil;, &RightCeiling;)
• U+230A LEFT FLOOR (&LeftFloor;, &lfloor;)
• U+230B RIGHT FLOOR (&rfloor;, &RightFloor;)

The floor corner brackets and , the ceiling corner brackets and (U+2308, U+2309) are used to denote the integer floor and ceiling functions.

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

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 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 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.[72] 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.[15]

## Unicode

Representations of various kinds of brackets in Unicode and their respective HTML 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[15] U+231C TOP LEFT CORNER (&ulcorn;, &ulcorner;) quasi-quotation
editorial notation
U+231D TOP RIGHT CORNER (&urcorn;, &urcorner;)
U+231E BOTTOM LEFT CORNER (&dlcorn;, &llcorner;) editorial notation
U+231F BOTTOM RIGHT CORNER (&drcorn;, &lrcorner;)
Fullwidth parentheses[11] U+FF5F FULLWIDTH LEFT WHITE PARENTHESIS ｟...｠
U+FF60 FULLWIDTH RIGHT WHITE PARENTHESIS
Technical/mathematical
(specialized)[15][16][17][18]
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 (&loang;) ⟬white tortoise shell brackets⟭
U+27ED MATHEMATICAL RIGHT WHITE TORTOISE SHELL BRACKET (&roang;)
U+2987 Z NOTATION LEFT IMAGE BRACKET RS
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 (&lparlt;) inequality sign brackets⦔
U+2994 RIGHT ARC GREATER-THAN BRACKET (&rpargt;)
U+2995 DOUBLE LEFT ARC GREATER-THAN BRACKET (&gtlPar;) ⦕inequality sign brackets⦖
U+2996 DOUBLE RIGHT ARC LESS-THAN BRACKET (&ltrPar;)
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[14] 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[73] 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[20] U+2772 LIGHT LEFT TORTOISE SHELL BRACKET ORNAMENT (&lbbrk;) ❲light tortoise shell bracket ornament❳
U+2773 LIGHT RIGHT TORTOISE SHELL BRACKET ORNAMENT (&rbbrk;)
N'Ko[14] U+2E1C LEFT LOW PARAPHRASE BRACKET ⸜ߒߞߏ⸝
U+2E1D RIGHT LOW PARAPHRASE BRACKET
Ogham[74] U+169B OGHAM FEATHER MARK ᚛ᚑᚌᚐᚋ᚜
U+169C OGHAM REVERSED FEATHER MARK
Old Hungarian U+2E42 DOUBLE LOW-REVERSED-9 QUOTATION MARK
Tibetan[75] 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 Testament editorial marks[14] 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[13][14] U+2E26 LEFT SIDEWAYS U BRACKET ⸦crux⸧
U+2E27 RIGHT SIDEWAYS U BRACKET
Quotation
(East-Asian texts)[35]
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[a]
Quotation
(halfwidth East-Asian texts)[15][11]
U+FF62 HALFWIDTH LEFT CORNER BRACKET ｢ｶﾀｶﾅ｣
U+FF63 HALFWIDTH RIGHT CORNER BRACKET
Quotation
(fullwidth East-Asian texts)[35]
U+300C LEFT CORNER BRACKET 「表題」
U+300D RIGHT CORNER BRACKET
U+300E LEFT WHITE CORNER BRACKET 『表題』
U+300F RIGHT WHITE CORNER BRACKET
U+3010 LEFT BLACK LENTICULAR BRACKET 【表題】
U+3011 RIGHT BLACK LENTICULAR BRACKET
Vertical bracket presentation forms[76][77][b] U+FE17 PRESENTATION FORM FOR VERTICAL LEFT WHITE LENTICULAR BRACKET ︗...︘
U+FE18 PRESENTATION FORM FOR VERTICAL RIGHT WHITE LENTICULAR BRACKET[c]
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
1. ^ This is fullwidth version of U+2033 DOUBLE PRIME. In vertical texts, U+301F LOW DOUBLE PRIME QUOTATION MARK is preferred.
2. ^ These characters are not used in typical documents. Instead the respective horizontal characters are used and the character that is rendered depends on the writing direction.
3. ^ The original name of this character is "Presentation Form For Vertical Right White Lenticular Brakcet [sic]". Since Unicode character names cannot be changed, this character has the corrected name as an alias.

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### Sources

• McArthur, Thomas Burns; McArthur, Roshan (2005). "Brackets". Concise Oxford Companion to the English Language. Oxford University Press. ISBN 9780192806376.
• Pointon, Graham; Clark, Stewart (2014). "Punctuation Guide". Words: A User's Guide. Routledge. ISBN 9781317864295.
• Peters, Pam (2007). The Cambridge Guide to Australian English Usage (2nd ed.). Cambridge University Press. ISBN 9781139465212.
• 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. ISBN 9781565923478.
• Sobell, Mark G.; Seebach, Peter (2005). A Practical Guide to UNIX for Mac OS X Users. Prentice Hall Professional. ISBN 9780321629982.
• Biggs, Norman (2002). "Set notation". Discrete Mathematics. OUP Oxford. ISBN 9780198507178.
• Ihde, Aaron J. (1984). The Development of Modern Chemistry. Dover Books on Chemistry. Courier Corporation. ISBN 9780486642352.
• Achatz, Thomas; Anderson, John G. (2005). McKenzie, Kathleen (ed.). Technical Shop Mathematics. Industrial Press. ISBN 9780831130862.
• 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. ISBN 0198112475.
• 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.
• Media related to Brackets at Wikimedia Commons
• The dictionary definition of bracket at Wiktionary