Bracket

Parentheses /pəˈrɛnθɪsiːz/ (singular, parenthesis /pəˈrɛnθɪsɪs/) (also called simply brackets, or round brackets, curves, curved brackets, oval brackets, stalls or, colloquially, parens /pəˈrɛnz/) contain material that serves to clarify (in the manner of a gloss) or is aside from the main point.[9] 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 the parenthetical.

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 "Sen. 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).[10]

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 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 [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]).

Any punctuation inside parentheses or other brackets is independent of the rest of the text: "Mrs. Pennyfarthing (What? Yes, that was her name!) was my landlady." In this usage, the explanatory text in the parentheses is a parenthesis. Parenthesized text is usually short and within a single sentence. Where several sentences of supplemental material are used in parentheses the final full stop would be within the parentheses, 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").[11]

Lower-case Latin letters used as indexes, rather than bullets or numbers, followed by an unpaired parenthesis, are used in ordered lists especially in:
a) educational testing,
b) technical writing and diagrams,
c) market research, and
d) elections.

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:

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

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.[12]

Parentheses are also used to set apart the arguments in mathematical functions. For example, f(x) is the function f applied to the variable x. In coordinate systems parentheses are used to denote a set of coordinates; so in the Cartesian coordinate system (4, 7) may represent the point located at 4 on the x-axis and 7 on the y-axis.

Parentheses may be used to represent a binomial coefficient, and also matrices.

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 for arrays. In syntax diagrams they are used for grouping eg in Extended Backus–Naur form.

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).

They can be used in various fields as notation to indicate the amount of uncertainty in a numerical quantity. For example:[13]

1234.56789(11)

is equivalent to:

1234.56789 ± 0.00011

e.g. the value of the Boltzmann constant could be quoted as 1.38064852(79)×10⁻²³ J⋅K⁻¹

Many online Roleplayers use double parentheses to connotate out-of-character (OOC) messages that one may send another.

Square brackets—also called crotchets or simply brackets (US)—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.[14]

A bracketed ellipsis, [...], is often used to indicate omitted material: "I'd like to thank [several unimportant people] for their tolerance [...]"[15] 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.[16] When deeper levels of nesting are needed, convention is to alternate between parentheses and brackets at each level.

Alternatively, empty square brackets can also indicate omitted material, usually single letter only. The original, "Reading is also a process and it also changes you." can be rewritten in a quote as: It has been suggested that reading can "also change[] you".

The bracketed expression "[sic]" is used after a quote or reprinted text to indicate the passage appears exactly as in the original source, where it may otherwise appear that a mistake has been made in reproduction. Sic, however, also commonly appears within parentheses as (sic), along with many other Latin abbreviations such as (e.g.) and (i.e.).

In translated works, brackets are used to signify the same word or phrase in the original language to avoid ambiguity.[17] For example: He is trained in the way of the open hand [karate].

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.
Center	]Paradise Lost[
Move up

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 also represent closed intervals; ${\displaystyle [0,5]}$ for example, represents the set of real numbers from 0 to 5 inclusive.

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.

Brackets are used in many computer programming languages, primarily to force the order of evaluation and for parameter lists and 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 eg in Extended Backus–Naur form.

In linguistics, phonetic transcriptions are generally enclosed within square brackets,[18] often using the International Phonetic Alphabet, whereas phonemic 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, square brackets usually surround the section of a dictionary entry which contains the etymology of the word the entry defines.

Square brackets are used to denote parts of the text that need to be checked when preparing drafts prior to finalizing a document. They often denote points that have not yet been agreed to in legal drafts and the year in which a report was made for certain case law decisions.

Look up curly bracket in Wiktionary, the free dictionary.

An example of curly brackets used to group sentences together

Curly brackets { and }, also known as curly braces (UK and US), flower brackets (India) and squiggly brackets (colloquially), 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).[19] More commonly, they are used to indicate a group of lines that should be taken together, as in when referring to several lines of poetry that should be repeated.

In music, they are known as "accolades" or "braces", and connect two or more lines (staves) of music that are played simultaneously.[20]

In mathematics they delimit sets, and in writing, they may be used similarly, "Select your animal {goat, sheep, cow, horse} and follow me". Curly brackets are often also used to denote the Poisson bracket between two quantities.

In many programming languages, they enclose groups of statements and create a local scope. Such languages (C, C#, C++ and many others) are therefore called curly bracket languages.[21] They are used for enumerated type, eg in C. In syntax diagrams they are used for repetition eg in Extended Backus–Naur form.

As an extension to the International Phonetic Alphabet, braces are used for prosodic notation.

Look up angle bracket or chevron in Wiktionary, the free dictionary.

Angle brackets ⟨ ⟩, (often substituted informally by less-than < and greater-than signs > or by "single" guillemets ‹ ›), are often used to enclose highlighted material.

In physical sciences, angle brackets are used to denote an average over time or over another continuous parameter. For example,

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

The inner product of two vectors is commonly written as ⟨a,b⟩, but there are other notations used.

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.

In set theory, chevrons or parentheses are used to denote ordered pairs and other tuples, whereas curly brackets are used for unordered sets.

In linguistics, angle brackets identify graphemes (i.e., letters of an alphabet) or orthography, as in "The English word /kæt/ is spelled ⟨cat⟩."[22][23][24]

In epigraphy, they may be used for mechanical transliterations of a text into the Latin script.[23]

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.[24]

In HTML, chevrons (greater 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 usage is sometimes extended as a mechanism for communicating mood, or tone, in digital formats such as messaging, for example adding “<sighs>” at the end of a sentence.

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

⟨ What an unusual flower! ⟩

The mathematical or logical symbols for greater-than > and less-than < are inequality symbols; when either symbol is bisected by a vertical line, it represents "not greater than" or "not less than," respectively. These symbols are not punctuation marks when used, as intended, to represent an inequality. However, as true chevrons are not present on computer keyboards, the available less-than and greater-than symbols are often used instead. They are loosely referred to as angle[d] brackets or chevrons in this case, but more properly—and less confusingly—as pointy brackets (see the Names section above).

Single and double pairs of comparison operators <<, >> (meaning much smaller than and much greater than, although the single symbols ≪ and ≫ should be used for that purpose) are sometimes used as a fallback instead of guillemets «, » (used as quotation marks in many languages) when the proper characters are not available on the keyboard nor in the input editor. Similarly, early Internet messaging conventions developed to use the greater-than sign >, available in the ASCII character set, to mark quoted lines. This format, known as Usenet quoting, is used by email clients when operating in plain text mode.

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.

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

In East Asian punctuation, angle brackets are used as quotation marks. Chevron-like symbols are part of standard Chinese, Japanese 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.

Look up 【 】 in Wiktionary, the free dictionary.

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 すみ付き (sumitsuki) in Japanese. They are used for inference in Chinese and used in titles and headings in Japanese.

Look up ⌊ ⌋ or ⌈ ⌉ in Wiktionary, the free dictionary.

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

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.[25] 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.[26]

Double brackets (or white square brackets), ⟦ ⟧, are used to indicate the semantic evaluation function in formal semantics for natural language and denotational semantics for programming languages.[27][28] 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.[29]

Known as "spike parentheses" (Swedish: piggparenteser), ⁅ and ⁆, are used in Swedish bilingual dictionaries to enclose supplemental constructions.[30]

The various bracket characters are frequently used in many programming languages as operators or for other syntax markup. For instance, in C-like languages, { and } are often used to delimit a code block, and the parameters of method calls are generally enclosed by ( and ).

In C, C++, Java and other C-derived languages—as well as in Scheme-influenced languages that have adopted C/Java syntax, such as JavaScript—the "{}" symbols are referred to as "braces" or "curly braces" and never as brackets. Since the term "brace" is documented in the definitive programming specifications for these languages, it is preferable to use the correct term brace so there is no confusion between the brace (used to denote compound statements) and the bracket, used to denote other concepts, such as array indices.[31][32]

In addition to the use of parentheses to specify the order of operations, both parentheses and brackets are used to denote an interval, also referred to as a half-open range. The notation [a,c) is used to indicate an interval from a to c that is inclusive of a but exclusive of c. That is, [5, 12) would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In some European countries, the notation [5, 12[ is also used for this. The endpoint adjoining the bracket is known as closed, whereas the endpoint adjoining the parenthesis is known as open. If both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever +∞ or −∞ is used as an endpoint, it is normally considered open and adjoined to a parenthesis. See Interval (mathematics) for a more complete treatment.

In quantum mechanics, chevrons are also used as part of Dirac's formalism, bra–ket notation, to note vectors from the dual spaces of the Bra ⟨A| and the Ket |B⟩. Mathematicians will also commonly write ⟨a, b⟩ for the inner product of two vectors. In statistical mechanics, chevrons denote ensemble or time average. Chevrons are used in group theory to write group presentations, and to denote the subgroup generated by a collection of elements. Note that obtuse angled chevrons are not always (and even not by all users) distinguished from a pair of less-than and greater-than signs <>, which are sometimes used as a typographic approximation of chevrons.

In group theory and ring theory, brackets denote the commutator. In group theory, the commutator [g, h] is commonly defined as g⁻¹ h⁻¹ g h . In ring theory, the commutator [a, b] is defined as a b − b a . Furthermore, in ring theory, braces denote the anticommutator where {a, b} is defined as a b + b a . The bracket is also used to denote the Lie derivative, or more generally the Lie bracket in any Lie algebra.

Various notations, like the vinculum have a similar effect to brackets in specifying order of operations, or otherwise grouping several characters together for a common purpose.

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

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

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.[33] 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').

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, is in the 1954 volume of the Appeal Cases reports although the decision may have been given in 1953 or earlier, whereas (1954) 98 Sol Jo 176 reports a decision from 1954, in volume 98 of the Solicitor's Journal which may be published in 1955 or later.

Tournament brackets, the diagrammatic representation of the series of games played during a tournament usually leading to a single winner, are so named for their resemblance to brackets or braces.