Bracket

History

Chevrons (< >) were the earliest type of bracket to appear in written English. Desiderius Erasmus coined the term lunula to refer to the rounded parentheses (), recalling the shape of the crescent moon.

Names for various bracket symbols

Some of the following names are regional or contextual.

Typography

The characters ‹ › and « », known as guillemets or angular quote brackets, are actually quotation mark glyphs used in several European languages. Which one of each pair is the opening quote mark and which is the closing quote varies between languages.

In English, typographers generally prefer to not 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 set in italics too.

Parentheses

Parentheses /pəˈrɛnθᵻsiːz/ (singular, parenthesis /pəˈrɛnθᵻsᵻs/) (also called simply brackets, or round brackets, curved brackets, oval brackets, or, colloquially, parens /pəˈrɛnz/) contain 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.

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", .

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.)

Parentheses are included in the syntaxes of many computer programming languages. Typically needed to denote an argument; to tell the compiler what data type the Method/Function needs to look for first in order to initialise. In some cases, such as in LISP, parentheses are a fundamental construct of the language.

Parentheses in mathematics signify a different precedence of operators. For example: 2 + 3 × 4 equals 14, since the multiplication is done before the addition. However, (2 + 3) × 4 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:

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.

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 also represent intervals; (0, 5), for example, is the interval between 0 and 5, not including 0 or 5.

Parentheses may also be used to represent a binomial coefficient, and in chemistry to denote a polyatomic ion.

Since 2014, antisemites have used triple parentheses around the names of people to denote them as Jewish.

Lowercase latin letters used as indexes, rather than bullets or numbers, followed by unpaired parenthesis, are used in ordered lists especially in:

1. educational testing,
2. technical writing and diagrams,
3. market research, and
4. elections

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").

Usage in journalism

Square brackets—also called crotchets or simply brackets (US)—are mainly used to insert explanatory material or to mark where a passage was omitted from an original material by someone other than the original author, or to mark modifications in quotations.

A bracketed ellipsis […] is often used to indicate omitted material: "I'd like to thank [several unimportant people] for their tolerance..." Bracketed comments inserted into a quote indicate when the original has been modified for clarity: "I appreciate it [the honor], but I must refuse", and "the future of psionics [see definition] is in doubt". Or one can quote the original statement "I hate to do laundry" with a (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 text is omitted for succinctness, for example, when referring to a verbose original: "To the extent that policymakers and elite opinion in general have made use of economic analysis at all, they have, as the saying goes, done so the way a drunkard uses a lamppost: for support, not illumination", it can be quoted succinctly as: "[P]olicymakers (...) made use of economic analysis (...) the way a drunkard uses a lamppost: for support, not illumination." When nested parentheses are needed, brackets are used as a substitute for the inner pair of parentheses within the outer pair. 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.

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].

Usage in proofreading

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

Usage in scientific fields

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

Brackets can also be used in chemistry to represent the concentration of a chemical substance or to denote distributed charge in a complex ion.

Brackets are used in many computer programming languages, especially those derived or inspired by the C language, to indicate array indexing operators. In this context, the opening bracket is often pronounced as "sub", indicating a subscript.

Other uses

In linguistics, phonetic transcriptions are generally enclosed within brackets, 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.

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.

Curly brackets

Curly brackets- { } —also called braces in the US (or, colloquially, squiggly brackets)—are used in specialized ways in poetry and music (to mark repeats or joined lines). The musical terms for this mark joining staves are accolade and "brace", and connect two or more lines of music that are played simultaneously. In mathematics they delimit sets, and in writing, they may be used similarly, "Select your animal {goat, sheep, cow, horse} and follow me". In many programming languages, they enclose groups of statements. Such languages (C being one of the best-known examples) are therefore called curly brace languages.

General usage in North American English favours the term brace rather than curly bracket.

In classical mechanics, curly brackets are often also used to denote the Poisson bracket between two quantities.

Angle brackets

Chevrons ⟨ ⟩, similar to the commonly used less-than (<) and greater-than symbol (>), are often used to enclose highlighted material.

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

The inner product of two vectors is commonly written as ⟨ a , b ⟩ , but the notation ( a , b ) is also used.

In mathematical physics, especially quantum mechanics, it is common to write the inner product between elements as ⟨ a | b ⟩ , as a short version of ⟨ a | ⋅ | b ⟩ , or ⟨ a | O ^ | b ⟩ , where O ^ 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, chevrons indicate graphemes (i.e., written letters) or orthography, as in "The English word /kæt/ is spelled ⟨cat⟩."

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

In textual criticism, and hence in many editions of pre-modern works, chevrons denote sections of the text which are illegible or otherwise lost; the editor will often insert their own reconstruction where possible within them.

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

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.

Single and double pairs of comparison operators (<<, >>) (meaning much smaller than and much greater than) are sometimes used instead of guillemets («, ») (used as quotation marks in many languages) when the proper characters are not available.

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. Of course, since no foreign language is actually written, this is only notionally translated.

Chevron-like symbols are part of standard Chinese, 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. See also non-English usage of quotation marks.

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

In East Asian punctuation, angle brackets are used as quotation marks.

Lenticular brackets

Some East Asian languages use lenticular brackets 【 】, a combination of brackets and parentheses called 方頭括號 in Chinese and sumitsuki in Japanese. They are used for inference in Chinese and used in titles and headings in Japanese.

Floor and ceiling corners

The floor corner brackets ⌊ and ⌋, the ceiling corner brackets ⌈ and ⌉ are used to denote the integer floor and ceiling functions.

Quine corners and half brackets

The Quine corners ⌜ and ⌝ have at least two uses in mathematical logic: either as quasi-quotation, a generalization of quotation marks, or to denote the Gödel 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. 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.

Double brackets

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. The brackets stand for a function that maps a linguistic expression to its “denotation” or semantic value. Double brackets may also refer to the mathematical floor function.

Brackets with quills

Known as "spike parentheses" (Swedish: piggparenteser) ⁅ and ⁆ are used in Swedish dictionaries.

Computing

The various bracket characters are frequently used in many computer 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 arrays.

Mathematics

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.

Accounting

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.

Law

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.

When quoted material is in any way altered, the alterations are enclosed in brackets within the quotation. For example: Plaintiff asserts his cause is just, stating, "[m]y causes is [sic] just." Although in the original quoted sentence the word "my" was capitalized, it has been modified in the quotation and the change signalled with brackets. Similarly, where the quotation contained a grammatical error, the quoting author signalled that the error was in the original with "[sic]" (Latin for 'thus'). (California Style Manual, section 4:59 (4th ed.))

Sports

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.

Encoding in digital media

Representations of various kinds of brackets in Unicode and HTML are given below.

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

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.