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A bracket is either of two tall fore- or back-facing
punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, whether read silently or aloud. An ...
marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'right' bracket or, alternatively, an "opening bracket" or "closing bracket", respectively, depending on the directionality of the context. Specific forms of the mark include parentheses (also called "rounded brackets"), square brackets, curly brackets (also called 'braces'), and angle brackets (also called 'chevrons'), as well as various less common pairs of symbols. As well as signifying the overall class of punctuation, the word "bracket" is commonly used to refer to a specific form of bracket, which varies from region to region. In most English-speaking countries, an unqualified word "bracket" refers to the parenthesis (round bracket); in the United States, the square bracket. Various forms of brackets are used in mathematics, with specific mathematical meanings, often for denoting specific mathematical functions and
subformula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be ...
s.


History

Angle brackets ("
Chevrons Chevron (often relating to V-shaped patterns) may refer to: Science and technology * Chevron (aerospace), sawtooth patterns on some jet engines * Chevron (anatomy), a bone * '' Eulithis testata'', a moth * Chevron (geology), a fold in rock l ...
") ⟨ ⟩ were the earliest type of bracket to appear in
written English English orthography is the writing system used to represent spoken English, allowing readers to connect the graphemes to sound and to meaning. It includes English's norms of spelling, hyphenation, capitalisation, word breaks, emphasis, and p ...
.
Desiderius Erasmus Roterodamus Desiderius Erasmus Roterodamus (; ; English: Erasmus of Rotterdam or Erasmus;''Erasmus'' was his baptismal name, given after St. Erasmus of Formiae. ''Desiderius'' was an adopted additional name, which he used from 1496. The ''Roterodamus'' wa ...
coined the term to refer to the rounded parentheses () recalling the shape of the crescent moon ( la, luna). Most typewriters only had the left and right parenthesis (and quotation marks). 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 The IBM 7030, also known as Stretch, was IBM's first transistorized supercomputer. It was the fastest computer in the world from 1961 until the first CDC 6600 became operational in 1964."Designed by Seymour Cray, the CDC 6600 was almost three ti ...
. In 1961,
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
contained parenthesis, 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. However, in other languages like German, if brackets enclose text in italics, they are usually also set in italics.


Parentheses

Parentheses (singular, parenthesis ) are also called "brackets" (UK, Ireland, Canada, West Indies, New Zealand, South Africa and Australia), "parens" , "round brackets", "circle brackets" or "smooth brackets".


Uses of ( )

Parentheses contain
adjunctive In linguistics, an adjunct is an optional, or ''structurally dispensable'', part of a sentence, clause, or phrase that, if removed or discarded, will not structurally affect the remainder of the sentence. Example: In the sentence ''John helped Bill ...
material that serves to clarify (in the manner of a gloss) or is aside from the main point. 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. That issue is fixed by using a pair of dashes instead, to
bracket 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. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
the
parenthetical 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. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
. In American usage, parentheses are usually considered separate from other brackets, and calling them "brackets" is unusual. 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! ''Absalom, Absalom!'' is a novel by the American author William Faulkner, first published in 1936. Taking place before, during, and after the American Civil War, it is a story about three families of the American South, with a focus on the life ...
'' and the Quentin section of ''The Sound and the Fury'') as well as poet E. E. Cummings. Parentheses have historically been used where the dash is currently used in alternatives, such as "parenthesis)(parentheses". Examples of this usage can be seen in editions of '' 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 specially square bracketswill be used for one or more inner set of parentheses
n other words, secondary phrases can be found within the main parenthetical sentence N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
. 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 use, the explanatory text in the parentheses is a 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 The full stop (Commonwealth English), period ( North American English), or full point , is a punctuation mark. It is used for several purposes, most often to mark the end of a declarative sentence (as distinguished from a question or exclamatio ...
would be within the parentheses, or simply omitted. 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. In more formal usage, " 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 "a parenthesis", "a parenthetical", or "a parenthetical phrase"). In linguistics, parentheses are used for indistinguishableIPA ''Handbook'' p. 175 or unidentified utterances. They are also seen for silent articulation (mouthing),IPA ''Handbook'' p. 191 where the expected phonetic transcription is derived from lip-reading, and with periods to indicate silent pauses, for example or .


Enumerations

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


Accounting

Traditionally in
accounting Accounting, also known as accountancy, is the measurement, processing, and communication of financial and non financial information about economic entities such as businesses and corporations. Accounting, which has been called the "language ...
, 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

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, equals 14, since the multiplication is done before the
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
. However, 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: : \times (3 + 2)2 = 400. 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. Various notations, like the vinculum, have a similar effect in specifying order of operations, or otherwise grouping several characters together for a common purpose. Parentheses are also be used for many other purposes in mathematics. Two common uses are for function arguments and tuples. They are used to set apart the
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
s in mathematical
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s. For example, is the function applied to the
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
. In
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s parentheses are used to denote a tuple of coordinates; so in the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
may represent the point located at 4 on the ''x''-axis and 7 on the ''y''-axis.


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 diagram Syntax diagrams (or railroad diagrams) are a way to represent a context-free grammar. They represent a graphical alternative to Backus–Naur form, EBNF, Augmented Backus–Naur form, and other text-based grammars as metalanguages. Early books us ...
s 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. For instance, ''Polyphylla'' (''Xerasiobia'') ''alba'' is a way to cite the species ''Polyphylla alba'' while also mentioning that it's in the subgenus ''Xerasiobia''. 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''. 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''. In some contexts, it is typical to cite the author's name alongside the taxon. In these contexts, parentheses mean that the author placed that species in a different genus from the one in that combination. The
International Code of Zoological Nomenclature The International Code of Zoological Nomenclature (ICZN) is a widely accepted convention in zoology that rules the formal scientific naming of organisms treated as animals. It is also informally known as the ICZN Code, for its publisher, the I ...
gives the example of '' Hymenolepis diminuta'' to indicate that
Karl Rudolphi Karl Asmund Rudolphi (14 July 1771 – 29 November 1832) was a Swedish-born German naturalist, who is credited with being the "father of helminthology". Life Rudolphi was born in Stockholm to German parents. He was awarded his PhD in 1793 an ...
did not consider this species to be in the genus '' Hymenolepis'' when he first described the species. The author citation in zoology also allows the possibility of citing whoever transferred the species to the new genus, as in, ''Methiolopsis geniculata'' . Parentheses are similarly used for
new combinations ''Combinatio nova'', abbreviated ''comb. nov.'' (sometimes ''n. comb.''), is Latin for "new combination". It is used in taxonomic biology literature when a new name is introduced based on a pre-existing name. The term should not to be confused wi ...
of prokaryotes as well; the ICNP provides the example: ''
Microbacterium oxydans ''Microbacterium oxydans'' is a Gram-positive bacterium from the genus ''Microbacterium'' which occurs in human clinical specimens. ''Microbacterium oxydans'' has the ability to degrade alginate and laminarin The molecule laminarin (also known ...
'' to indicate that Chatelain and Second first described the species in a different genus, namely ''
Brevibacterium ''Brevibacterium'' is a genus of bacteria of the order Micrococcales. They are Gram-positive soil Soil, also commonly referred to as earth or dirt, is a mixture of organic matter, minerals, gases, liquids, and organisms that together ...
'', but in 1999 Schumann ''et al.'' transferred it to its present genus. Author citations in botany also use parentheses in this way where the author (or abbreviation thereof) of the
basionym In the scientific name of organisms, basionym or basyonym means the original name on which a new name is based; the author citation of the new name should include the authors of the basionym in parentheses. The term "basionym" is used in both botan ...
is in parentheses followed by the author (or abbreviation thereof) of whoever created that particular combination; the
International Code of Nomenclature for algae, fungi, and plants The ''International Code of Nomenclature for algae, fungi, and plants'' (ICN) is the set of rules and recommendations dealing with the formal botanical names that are given to plants, fungi and a few other groups of organisms, all those "tra ...
provides the example ''Helianthemum aegyptiacum'' to indicate that Carl Linnaeus first described this species in a different genus, in this case '' Cistus'', but then Philip Miller transferred it to the genus '' Helianthemum''.


Chemistry and physics

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 Calcium nitrate, also called ''Norgessalpeter'' (Norwegian salpeter), is an inorganic compound with the formula Ca(NO3)2(H2O)x. The anhydrous compound, which is rarely encountered, absorbs moisture from the air to give the tetrahydrate. Both anhy ...
). They can be used in various fields as notation to indicate the amount of uncertainty in a numerical quantity. For example: : 1234.56789(11) is equivalent to: : e.g. the value of the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
could be quoted as J⋅K−1 .


Square brackets

Square brackets
and or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boolea ...
are also called simply "brackets" (US), as well as "crotchets", "closed brackets", or "hard brackets". Tournament brackets, the diagrammatic representation of the series of games played during a sports tournament usually leading to a single winner, are so named for their resemblance to brackets or braces.


Uses of /h2>

Square brackets are often used to insert explanatory material or to mark where a
ord or Ord or ORD may refer to: Places * Ord of Caithness, landform in north-east Scotland * Ord, Nebraska, USA * Ord, Northumberland, England * Muir of Ord, village in Highland, Scotland * Ord, Skye, a place near Tarskavaig * Ord River, Western Austr ...
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 — "... aughs...". 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 An annotation is extra information associated with a particular point in a document or other piece of information. It can be a note that includes a comment or explanation. Annotations are sometimes presented in the margin of book pages. For anno ...
. For example: ''The Plaintiff asserted his cause is just, stating,'' 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 everal unimportant peoplefor their tolerance .. Bracketed comments inserted into a quote indicate where the original has been modified for clarity: "I appreciate it
he honor He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana へ * He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...
but I must refuse", and "the future of psionics ee definitionis in doubt". Or one can quote the original statement "I hate to do laundry" with a (sometimes grammatical) modification inserted: He "hate 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 Verbosity or verboseness is speech or writing that uses more words than necessary. The opposite of verbosity is plain language. Some teachers, including the author of '' The Elements of Style'', warn against verbosity; similarly Mark Twain and E ...
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: " licymakers ..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 guide, Style and usage guides originating in the News media#History, news industry of the twentieth century, such as the ''
AP Stylebook The ''AP Stylebook'', also known by its full name ''The Associated Press Stylebook and Briefing on Media Law'', is an American English grammar style and usage guide created by American journalists working for or connected with the Associated P ...
'', recommend against the use of square brackets because "They cannot be transmitted over news wires." 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, often using the International Phonetic Alphabet#Brackets and transcription delimiters, whereas
phonemic In phonology and linguistics, a phoneme () is a unit of sound that can distinguish one word from another in a particular language. For example, in most dialects of English, with the notable exception of the West Midlands and the north-west ...
transcriptions typically use paired slashes. Pipes (, , ) are often used to indicate a morphophonemic rather than phonemic representation. Other conventions are double slashes (// //), double pipes (, , , , ) and curly brackets (). In
lexicography Lexicography is the study of lexicons, and is divided into two separate academic disciplines. It is the art of compiling dictionaries. * Practical lexicography is the art or craft of compiling, writing and editing dictionaries. * Theoretica ...
, square brackets usually surround the section of a dictionary entry which contains the etymology of the word the entry defines.


Proofreading

Brackets (called ''move-left symbols'' or ''move right symbols'') are added to the sides of text in proofreading to indicate changes in indentation: 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: 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: 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: This citation reports a decision from 1954, in volume 98 of the ''
Solicitors Journal ''Solicitors Journal'' is a monthly legal journal published in the United Kingdom by the International In-house Counsel Journal, Cambridge.
'' 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 Case law, also used interchangeably with common law, is law that is based on precedents, that is the judicial decisions from previous cases, rather than law based on constitutions, statutes, or regulations. Case law uses the detailed facts of a le ...
decisions.


Square brackets in mathematics

Brackets are used in mathematics in a variety of notations, including standard notations for
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
s, the floor function, the Lie bracket,
equivalence classes In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
, the Iverson bracket, and matrices. Square brackets may be used exclusively or in combination with parentheses to represent intervals. For example, represents the set of real numbers from 0 to 5 inclusive. Both parentheses and brackets are used to denote a ''half-open'' interval; 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 is also used. The endpoint adjoining the square bracket is known as ''closed'', whereas the endpoint adjoining the parenthesis is known as ''open''. In
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
and ring theory, brackets denote the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
. In group theory, the commutator is commonly defined as . In ring theory, the commutator is defined as .


Chemistry

Square brackets can also be used in chemistry to represent the concentration of a
chemical substance A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wit ...
in solution and to denote charge a Lewis structure of an ion (particularly distributed charge in a
complex ion A coordination complex consists of a central atom or ion, which is usually metallic and is called the ''coordination centre'', and a surrounding array of bound molecules or ions, that are in turn known as ''ligands'' or complexing agents. Many ...
), 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 An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
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 diagram Syntax diagrams (or railroad diagrams) are a way to represent a context-free grammar. They represent a graphical alternative to Backus–Naur form, EBNF, Augmented Backus–Naur form, and other text-based grammars as metalanguages. Early books us ...
s they are used for optional portions, such as in extended Backus–Naur form.


Curly brackets

Curly brackets are also known as "curly braces" or simply "braces" (UK and US), "definite brackets", "swirly brackets", "birdie brackets", "French brackets", "Scottish brackets", "squirrelly brackets", "gullwings", "seagulls", "squiggly brackets", "twirly brackets", "Tuborg brackets" (DK), "accolades" (NL), "pointy brackets", "fancy brackets", "M Braces", "moustache brackets", "squiggly parentheses", or "flower brackets" (India).


Uses of

Curly brackets are rarely used in prose and have no widely accepted use in formal writing, but may be used to mark words or sentences that should be taken as a group, to avoid confusion when other types of brackets are already in use, or for a special purpose specific to the publication (such as in a dictionary). More commonly, they are used to indicate a group of lines that should be taken together, such as in when referring to several lines of poetry that should be repeated. As an extension to the International Phonetic Alphabet (IPA), braces are used for prosodic notation.


Music

In music, they are known as "
accolade The accolade (also known as dubbing or adoubement) ( la, benedictio militis) was the central act in the rite of passage ceremonies conferring knighthood in the Middle Ages. From about 1852, the term ''accolade'' was used much more generally t ...
s" or " braces", and connect two or more lines (staves) of music that are played simultaneously.


Curly brackets in programming languages

In many programming languages, curly brackets enclose groups of statements and create a local
scope Scope or scopes may refer to: People with the surname * Jamie Scope (born 1986), English footballer * John T. Scopes (1900–1970), central figure in the Scopes Trial regarding the teaching of evolution Arts, media, and entertainment * Cinema ...
. Such languages ( C, C#, C++ and many others) are therefore called
curly bracket language This is a list of notable programming languages, grouped by type. There is no overarching classification scheme for programming languages. Thus, in many cases, a language is listed under multiple headings (in this regard, see " Multiparadigm lan ...
s. They are also used to define structures and enumerated type in these languages. In
syntax diagram Syntax diagrams (or railroad diagrams) are a way to represent a context-free grammar. They represent a graphical alternative to Backus–Naur form, EBNF, Augmented Backus–Naur form, and other text-based grammars as metalanguages. Early books us ...
s they are used for repetition, such as in extended Backus–Naur form. In the Z
formal specification In computer science, formal specifications are mathematically based techniques whose purpose are to help with the implementation of systems and software. They are used to describe a system, to analyze its behavior, and to aid in its design by verif ...
language, braces define a set.


Curly brackets in mathematics

In mathematics they delimit
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s and are often also used to denote the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
between two quantities. In ring theory, braces denote the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
where is defined as .


Angle brackets

"Angle brackets" ⟨ and ⟩ are also called "chevrons", "pointy brackets", "triangular brackets", "diamond brackets", "tuples", "
guillemets Guillemets (, also , , ) are a pair of punctuation marks in the form of sideways double chevrons, and , used as quotation marks in a number of languages. In some of these languages "single" guillemets, and , are used for a quotation inside an ...
", "left and right carets", "broken brackets", or "brokets". 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, 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.


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: : 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. In
comic book A comic book, also called comicbook, comic magazine or (in the United Kingdom and Ireland) simply comic, is a publication that consists of comics art in the form of sequential juxtaposed panels that represent individual scenes. Panels are oft ...
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. Since no foreign language is actually written, this is only ''notionally'' translated. In linguistics, angle brackets identify
grapheme In linguistics, a grapheme is the smallest functional unit of a writing system. The word ''grapheme'' is derived and the suffix ''-eme'' by analogy with ''phoneme'' and other names of emic units. The study of graphemes is called ''graphemics' ...
s (, letters of an alphabet) or orthography, as in "The English word is spelled ." In epigraphy, they may be used for mechanical transliterations of a text into the Latin script. In East Asian punctuation, angle brackets are used as quotation marks. Chevron-like symbols are part of standard
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of v ...
,
Japanese Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspor ...
and Korean punctuation, where they generally enclose the titles of books: ︿ and ﹀ or ︽ and ︾ for traditional vertical printing, and 〈 and 〉 or 《 and 》 for horizontal printing.


Angle brackets in mathematics

Angle brackets (or 'chevrons') are used in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
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 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 In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
'') over time or over another continuous parameter. For example: :\left\langle V(t)^2 \right\rangle = \lim_ \frac\int_^ V(t)^2\,t. In mathematical physics, especially quantum mechanics, it is common to write the inner product between elements as , as a short version of , or , where is an
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
. This is known as ''Dirac notation'' or ''
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathemat ...
'', to note vectors from the dual spaces of the Bra . But there are other notations used. In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, chevrons may be used as Macaulay brackets.


Angle brackets in programming languages

In
C++ C, or c, is the third letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''. History "C" ...
chevrons (actually less-than and greater-than) are used to surround arguments to templates. In the Z
formal specification In computer science, formal specifications are mathematically based techniques whose purpose are to help with the implementation of systems and software. They are used to describe a system, to analyze its behavior, and to aid in its design by verif ...
language chevrons define a sequence. In HTML, chevrons (actually 'greater than' and 'less than' symbols) are used to bracket meta text. For example 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 . 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 (''fāngtóu kuòhào'') in
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of v ...
and (''sumitsuki'') in
Japanese Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspor ...
. They are used in titles and headings in both Chinese and Japanese. In Japanese, they are most frequently seen in dictionaries for quoting Chinese characters and Sino-Japanese loanwords.


Floor ⌊ ⌋ and ceiling ⌈ ⌉ corner brackets

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 Quasi-quotation or Quine quotation is a linguistic device in formal languages that facilitates rigorous and terse formulation of general rules about linguistic expressions while properly observing the use–mention distinction. It was introduced ...
, a generalization of quotation marks, or to denote the Gödel number of the enclosed
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, ...
. 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. For example,
Callimachus Callimachus (; ) was an ancient Greek poet, scholar and librarian who was active in Alexandria during the 3rd century BC. A representative of Ancient Greek literature of the Hellenistic period, he wrote over 800 literary works in a wide variet ...
''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.


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. 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 The Leiden Conventions or Leiden system is an established set of rules, symbols, and brackets used to indicate the condition of an epigraphic or papyrological text in a modern edition. In previous centuries of classical scholarship, scholars who ...
, they are used to enclose text that has been deleted in antiquity.


Brackets with quills ⁅ ⁆

Known as "spike parentheses" ( sv, piggparenteser), and are used in Swedish
bilingual dictionaries A bilingual dictionary or translation dictionary is a specialized dictionary used to translate words or phrases from one language to another. Bilingual dictionaries can be ''unidirectional'', meaning that they list the meanings of words of one la ...
to enclose supplemental constructions.Examples may be found under the corresponding entry at :sv:Parentes.


Unicode

Representations of various kinds of brackets in Unicode and HTML are given below. &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 i
XML Entity Definitions for Characters
This is fullwidth version of U+2033 DOUBLE PRIME. In vertical texts, U+301F LOW DOUBLE PRIME QUOTATION MARK is preferred.
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. The ''less-than'' and ''greater-than'' symbols are often used as replacements for chevrons.


See also

*
Bracket (mathematics) In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets nbsp; braces and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of gro ...
*
International variation in quotation marks Quotation marks (also known as quotes, quote marks, speech marks, inverted commas, or talking marks) are punctuation marks used in pairs in various writing systems to set off direct speech, a quotation, or a phrase. The pair consists of an ...
* Emoticon * Japanese typographic symbols * Order of operations *
Triple parentheses Triple parentheses or triple brackets, also known as an (((echo))), are an antisemitic symbol that has been used to highlight the names of individuals assumed to be of a Jewish background, or organizations thought to be owned by Jews. The pract ...
*
Arrowhead An arrowhead or point is the usually sharpened and hardened tip of an arrow, which contributes a majority of the projectile mass and is responsible for impacting and penetrating a target, as well as to fulfill some special purposes such as sign ...
(combining and standalone characters similar to angle brackets or less-than and greater-than characters)


References


Bibliography

* * 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

* * {{navbox punctuation Punctuation Mathematical notation