Bracket

Brackets are tall punctuation marks used in matched pairs within text, to set apart or interject other text. Used unqualified, brackets refer to different types of brackets in different parts of the world and in different contexts.

List of types

( ) — parentheses, round brackets, soft brackets, or circle brackets
[ ] — square brackets, closed brackets, hard brackets, or brackets (US)
{ } — braces (UK and US), French brackets, curly brackets, definite brackets, swirly brackets, curly braces, birdie brackets, Scottish brackets, squirrelly brackets, gullwings, seagulls, squiggly brackets, sideways mustaches, Tuborg brackets (DK), accolades (NL) or fancy brackets
⟨ ⟩ — pointy brackets, angle brackets, triangular brackets, diamond brackets, tuples, or chevrons
< > — inequality signs, pointy brackets, or brackets. Sometimes referred to as angle brackets, in such cases as HTML markup. Occasionally known as broken brackets or brokets.^[1]
⸤ ⸥; ｢｣ — corner brackets

Characters ‹ › and « », known as guillemets or angular quote brackets, are actually quotation mark glyphs used in several European languages.

History

The chevron was the earliest type to appear in written English. Desiderius Erasmus coined the term lunula to refer to the rounded parentheses (), recalling the round shape of the moon.^[2]

Usage

In addition to referring to the class of all types of brackets, the unqualified word bracket is most commonly used to refer to a specific type of bracket. In modern American usage this is usually the square bracket and in modern British usage this is usually the parenthesis.

In American usage, parentheses are usually considered separate from other brackets, and calling them "brackets" at all is unusual even though they serve a similar function. 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 or a parenthetical).^[3]

According to early typographic practice, brackets are never set in italics, even when the surrounding characters are italic.^[4]

Types

Parentheses ( )

Parentheses /pəˈrɛnθ[invalid input: 'ɨ']siːz/ (singular, parenthesis /pəˈrɛnθ[invalid input: 'ɨ']s[invalid input: 'ɨ']s/) (also called simply brackets, or round brackets, curved brackets, oval brackets, or, colloquially, parens) contain material that could be omitted without destroying or altering the meaning of a sentence (in most writing, overuse of parentheses is usually a sign of a badly structured text)^{[according to whom?]}. 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.

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)" – or for "either masculine or feminine" in some languages with grammatical gender.^[5]

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—that is, in order to depict 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]).^[6]

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. 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.

Parentheses in mathematics signify a different precedence of operators. Normally, $2 + 3 \times 4$ would be 14, since the multiplication is done before the addition. On the other hand $(2 + 3) \times 4$ is 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:

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

A related convention is that when parentheses have two levels of nesting, braces are the outermost pair.

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.

In Chinese and Japanese, 【】, a combination of brackets and parentheses called 方頭括號 and sumitsuki, are used for inference in Chinese and used in titles and headings in Japanese.

Square brackets [ ]

Square brackets – also called 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.^[7]

A bracketed ellipsis […] is often used to indicate omitted material: "I'd like to thank [several unimportant people] for their tolerance..."^[8] 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 modification inserted in the middle of it: 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.^[9] 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.

In translated works, brackets are used to signify the same word or phrase in the original language to avoid ambiguity.^[10] For example: He is trained in the way of the open hand [karate].

In linguistics, phonetic transcriptions are generally enclosed within brackets,^[11] 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 ({ }).

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 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.

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.

From the top: brackets, braces, parentheses, angle brackets/chevrons, and inequality signs

Curly brackets or braces { }

Curly brackets – also properly called braces in the US – 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.^[12] 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 bracket languages. Some people use a brace to signify movement in a particular direction.^{[citation needed]}

Presumably due to the similarity of the words brace and bracket (although they do not share an etymology), many people mistakenly treat brace as a synonym for bracket. Therefore, when it is necessary to avoid any possibility of confusion, such as in computer programming, it may be best to use the term curly bracket rather than brace. However, general usage in North American English favours the latter form.^{[citation needed]} Indian programmers often use the name "flower bracket".^[13]

In classical mechanics, curly brackets are often also used to denote the Poisson bracket between two quantities. It is defined as follows:

\{f,g\}=\sum _{i=1}^{N}\left[{\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right]

Angle brackets or chevrons ⟨ ⟩

Chevrons ⟨ ⟩;^[14] 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,

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

The inner product of two vectors is commonly written as $\langle a,b\rangle$ , 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 $\langle a|b\rangle$ , as a short version of $\langle a|\cdot |b\rangle$ , or $\langle a|{\hat {O}}|b\rangle$ , where ${\hat {O}}$ 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 his own reconstruction where possible within them.

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

⟨ What a beautiful flower! ⟩

The mathematical or logical symbols for greater-than (>) and less-than (<) are inequality symbols, and are not punctuation marks when so used. Nevertheless, true chevrons are not available on a typical computer keyboard, but the less-than and greater-than symbols are, so they are often substituted. They are loosely referred to as angled brackets or chevrons in this case.^{[citation needed]}

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.^{[citation needed]}

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.

Corner and half brackets ｢｣, ⌊ ⌋

In East Asian punctuation, angle brackets are used as quotation marks. Half brackets are used in English to mark added text, such as in translations: "Bill saw ⌊her⌋".

The corner brackets ⌈ and ⌉ have at least two uses in mathematical logic: either as a generalization of quotation marks or to denote the gödel number of the enclosed expression. They are also used to denote the integer floor and ceiling functions.

In editions of papyrological texts, half brackets 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.^[15] 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.

Double brackets ⟦ ⟧

In formal semantics, double brackets, ⟦ ⟧, also called Strachey brackets, are used to indicate the semantic evaluation function. The CJK glyphs 〚〛 look identical except they have added width. They can be typeset in LaTeX with the package stmaryrd.

Computing

Encoding

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

Usage	Unicode		SGML/HTML/XML entities	Sample
General purpose	U+0028	Left parenthesis	( &lparen;	(parentheses)
	U+0029	Right parenthesis	) &rparen;	(parentheses)
	U+005B	Left square bracket	[	[sic]
	U+005D	Right square bracket	]	[sic]
Technical/mathematical (common)	U+003C	Less-than sign	< <	<HTML>
	U+003E	Greater-than sign	> >	<HTML>
	U+007B	Left curly bracket	{	{round, square, curly}
	U+007D	Right curly bracket	}	{round, square, curly}
Quotation (Western texts)	U+00AB	Left double guillemet	«	« words »
	U+00BB	Right double guillemet	»	« words »
	U+2039	Left single guillemet	‹	‹ x ›
	U+203A	Right single guillemet	›	‹ x ›
Floor and ceiling functions^[16]	U+2308	Left ceiling	⌈	⌈ceiling⌉
	U+2309	Right ceiling	⌉	⌈ceiling⌉
	U+230A	Left floor	⌊	⌊floor⌋
	U+230B	Right floor	⌋	⌊floor⌋
Technical/mathematical (specialized)^[16]	U+239B	Left parenthesis upper hook	⎛	⎛ ⎞ ⎜large parentheses⎟ ⎝ ⎠
	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+23A1	Left square bracket upper corner	⎡	⎡ ⎤ ⎢large square brackets⎥ ⎣ ⎦
	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+23A7	Left curly bracket upper hook	⎧	⎧ ⎫ ⎨large curly brackets⎬ ⎩ ⎭
	U+23A8	Left curly bracket middle piece	⎨
	U+23A9	Left curly bracket lower hook	⎩
	U+23AB	Right curly bracket upper hook	⎫
	U+23AC	Right curly bracket middle piece	⎬
	U+23AD	Right curly bracket lower hook	⎭
	U+23AA	Curly bracket extension	⎪	⎪
	U+23B0	Upper left or lower right curly bracket section	⎰	⎰ ⎱ ⎱ ⎰
	U+23B1	Upper right or lower left curly bracket section	⎱	⎰ ⎱ ⎱ ⎰
	U+23B4	Top square bracket	⎴	⎴horizontal square brackets⎵
	U+23B5	Bottom square bracket	⎵	⎴horizontal square brackets⎵
	U+23B6	Bottom square bracket over top square bracket	⎶	⎶
	U+23B8	Left vertical box line	⎸	⎸boxed text⎹
	U+23B9	Right vertical box line	⎹	⎸boxed text⎹
	U+23DC	Top parenthesis	⏜	⏜horizontal parentheses⏝
	U+23DD	Bottom parenthesis	⏝	⏜horizontal parentheses⏝
	U+23DE	Top curly bracket	⏞	⏞horizontal curly brackets⏟
	U+23DF	Bottom curly bracket	⏟	⏞horizontal curly brackets⏟
	U+23E0	Top tortoise shell bracket	⏠	⏠tortoise shell brackets⏡
	U+23E1	Bottom tortoise shell bracket	⏡	⏠tortoise shell brackets⏡
Technical mathematical symbols^[17]^[18]	U+27E6	Mathematical left white square bracket	⟦	⟦white square brackets⟧
	U+27E7	Mathematical right white square bracket	⟧	⟦white square brackets⟧
	U+27E8	Mathematical left angle bracket	⟨ &lang;^*	⟨ $a$ , $b$ ⟩
	U+27E9	Mathematical right angle bracket	⟩ &rang;^*	⟨ $a$ , $b$ ⟩
	U+27EA	Mathematical left double angle bracket	⟪	⟪ $A$ , $B$ ⟫
	U+27EB	Mathematical right double angle bracket	⟫	⟪ $A$ , $B$ ⟫
	U+27EC	Mathematical left white tortoise shell bracket	⟬	⟬white tortoise shell brackets⟭
	U+27ED	Mathematical right white tortoise shell bracket	⟭	⟬white tortoise shell brackets⟭
	U+27EE	Mathematical left flattened parenthesis	⟮	⟮flattened parentheses⟯
	U+27EF	Mathematical right flattened parenthesis	⟯	⟮flattened parentheses⟯
	U+2983	Left white curly bracket	⦃	⦃white curly brackets⦄
	U+2984	Right white curly bracket	⦄	⦃white curly brackets⦄
	U+2985	Left white parenthesis	⦅	⦅white/double parentheses⦆
	U+2986	Right white parenthesis	⦆	⦅white/double parentheses⦆
	U+2987	Z notation left image bracket	⦇	$R$ ⦇ $S$ ⦈
	U+2988	Z notation right image bracket	⦈	$R$ ⦇ $S$ ⦈
	U+2989	Z notation left binding bracket	⦉	$A$ ⦉ $B$ ⦊
	U+298A	Z notation right binding bracket	⦊	$A$ ⦉ $B$ ⦊
	U+298B	Left square bracket with underbar	⦋	⦋underlined square brackets⦌
	U+298C	Right square bracket with underbar	⦌	⦋underlined square brackets⦌
	U+298D	Left square bracket with tick in top corner	⦍	⦍ticked square brackets⦎
	U+298E	Right square bracket with tick in bottom corner	⦎	⦍ticked square brackets⦎
	U+298F	Left square bracket with tick in bottom corner	⦏	⦏ticked square brackets⦐
	U+2990	Right square bracket with tick in top corner	⦐	⦏ticked square brackets⦐
	U+2991	Left angle bracket with dot	⦑	⦑dotted angle brackets⦒
	U+2992	Right angle bracket with dot	⦒	⦑dotted angle brackets⦒
	U+2993	Left arc less-than bracket	⦓	⦓inequality sign brackets⦔
	U+2994	Right arc greater-than bracket	⦔	⦓inequality sign brackets⦔
	U+2995	Double left arc greater-than bracket	⦕	⦕inequality sign brackets⦖
	U+2996	Double right arc less-than bracket	⦖	⦕inequality sign brackets⦖
	U+2997	Left black tortoise shell bracket	⦗	⦗black tortoise shell brackets⦘
	U+2998	Right black tortoise shell bracket	⦘	⦗black tortoise shell brackets⦘
Quotation (halfwidth East-Asian texts)	U+2329	Left pointing angle bracket	〈 &lang;^*	〈deprecated〉
	U+232A	Right pointing angle bracket	〉 &rang;^*	〈deprecated〉
	U+FF62	Halfwidth left corner bracket	｢	｢ｶﾀｶﾅ｣
	U+FF63	Halfwidth right corner angle bracket	｣	｢ｶﾀｶﾅ｣
Quotation (fullwidth East-Asian texts)	U+3008	Left angle bracket	〈	〈한〉
	U+3009	Right angle bracket	〉	〈한〉
	U+300A	Left double angle bracket	《	《한한》
	U+300B	Right double angle bracket	》	《한한》
	U+300C	Left corner bracket	「	「白八櫨」
	U+300D	Right corner bracket	」	「白八櫨」
	U+300E	Left corner bracket	『	『カタカナ』
	U+300F	Right corner bracket	』	『カタカナ』
	U+3010	Left thick square bracket	【	【ひらがな】
	U+3011	Right thick square bracket	】	【ひらがな】
General purpose (fullwidth East-Asian)	U+FF08	Fullwidth left parenthesis	（	（Ｗｉｋｉ）
	U+FF09	Fullwidth right parenthesis	）	（Ｗｉｋｉ）
	U+FF3B	Fullwidth left square bracket	［	［ｓｉｃ］
	U+FF3D	Fullwidth right square bracket	］	［ｓｉｃ］
Technical/mathematical (fullwidth East-Asian)	U+FF1C	Fullwidth less-than sign	＜	＜ＨＴＭＬ＞
	U+FF1E	Fullwidth greater-than sign	＞	＜ＨＴＭＬ＞
	U+FF5B	Fullwidth left curly bracket	｛	｛１、２｝
	U+FF5D	Fullwidth right curly bracket	｝	｛１、２｝

^*&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.^{[citation needed]}

Braces (curly brackets) first became part of a character set with the 8-bit code of the IBM 7030 Stretch.^[19]

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,^[16] 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.

These various bracket characters are frequently used in many computer languages as operators or for other syntax markup. The more common uses follow.

Uses of "(" and ")"

Parentheses are often used to define the syntactic structure of expressions, overriding operator precedence: a*(b+c) has subexpressions a and b+c, whereas a*b+c has subexpressions a*b and c.
They are used for passing parameters or arguments to functions, especially in C and similar languages, and invoking a function or function-like construct: substring($val,10,1).
In Lisp, they open and close s-expressions and therefore function applications: (cons a b).
In many regular expression syntaxes, parentheses create a capturing group, allowing the matched portion inside to be retrieved by the user.
In Forth, they open and close comments in the code.
In Fortran-family and COBOL languages, they are also used for array references.
In the Perl programming language through Perl 5, they are used to define lists, static array-like structures; this idiom is extended to their use as containers of subroutine (function) arguments.
In the Perl 6 programming language, they define captures, a structure that defers contextual interpretation. This usage extends to ordinary parentheses as well. They are also used to indicate arguments to function calls and to declare signatures of formal parameters or other variables.
In Python they are used to disambiguate tuple literals (immutable ordered lists), which are usually formed by commas, in places where parentheses and commas would otherwise be a part of a function call.
In Tcl, they are used to enclose the name of an element of an associative array variable.
In joined brackets in a table going vertically downwards, a ")" refers to repetition of a term for the number of items towards the left of this joined list of brackets.

Uses of "[" and "]"

Square brackets are often used to refer to elements of an array or associative array, and sometimes to define the number of elements in an array, especially in C-like languages: queue[3].
In many languages, they may be used to define a literal anonymous array or list: [5, 10, 15].
In most regular expression syntaxes, brackets denote a character class: a set of possible characters to choose from.
In Forth, "[" causes the system to enter interpretation state and "]" causes the system to enter compilation state. For example, within a definition, [ 2 3 + ] literal causes the compiler to switch to the interpreter mode, calculate the expression 2 + 3, leave the result on stack and resume compilation. As a result, a literal constant "5" will be compiled into the definition, instead of the whole expression.
In Tcl, they enclose a sub-script to be evaluated and the result substituted.
In some of the CLI languages, most notably C# and C++, they are used to denote metadata attributes.
In C++11 they introduce Lambda expressions and hold an optional capture clause.
In x86 assembly implementations such as FASM, they are used to distinguish pointers from their data.
In Smalltalk, brackets are used to delineate "blocks" or "block closures", grouping of code that can be executed immediately or later via messages send such as "value" sent to the block. Blocks are full first class objects in Smalltalk.
In Objective-C, brackets are used to send a message to (i.e. call a method on) an object.
On Unix, "[" is a shorthand for the test command.
In JSON they are used to define an array (ordered sequence of comma-separated values).
In programming documentation and metalanguages (e.g. in descriptions of operator or command syntax), optional elements are enclosed in square brackets. For example, "echo [-n] [-e] <text>" means that the -n and -e parameters are optional.
Delimiting IPv6 addresses in URLs, for example: http://[2001:db8:3c4d:15::abcd:ef12]:8080.
In Python used to denote lists and tuples, also known as arrays.

Uses of "{" and "}"

Curly brackets are used in some programming languages to define the beginning and ending of blocks of code or data. Languages which use this convention are said to belong to the curly bracket family of programming languages.
They are used to represent certain type definitions or literal data values, such as a composite structure or associative array.
In mathematics, they enclose elements of sets and denote sets.
In Curl they are used to delimit expressions and statements (similar to Lisp's use of parenthesis).
In Pascal they define the beginning and ending of comments.
In most regular expression syntaxes, they are used as quantifiers, matching n repetitions of the previous group.
In Perl they are also used to refer to elements of an associative array.
In PHP they are used to determine structures.
In Tcl they enclose a string to be substituted without any internal substitutions being performed.
In Python and Ruby they are used for dictionaries (a mutable set of key: value pairs, separated by commas) and for sets.
In TeX/LaTeX they can be used for grouping parts sharing the same local format, wrap parameters, or definitions, depending on the local catcode value.
In JSON they are used to define an object (an unordered collection of key:value pairs).
In metalanguages (e.g. in descriptions of operator or command syntax), possible alternatives are enclosed in braces, if at least one is mandatory.
These are also used in music at the start of a stave.

Uses of "<" and ">"

These symbols are used in pairs as if they are brackets.

Greater-than and less-than signs are used to set apart URLs and e-mail addresses in text, such as "I found it on Example.com <http://www.example.com/>" and "This photo is copyrighted by John Smith <johnsmith@example.com>". This is also the computer-readable form for addresses in e-mail headers, specified by RFC 2822.
In documentation, they are often used to specify parameters or other user-specified information (e.g. "The command 'echo <text>' can be used to display <text>").
To enclose code tags in SGML, HTML, and XML (e.g. <div>).
To target children of a parent element in CSS (e.g. ul.main>li whereas all direct child selectors of the ul.main tag are targeted).
In the C++, C#, and Java programming languages, (among others) they delimit generic arguments.
In Perl through Perl 5 they are used to read a line from an input source.
In Perl 6 they combine quoting and associative array lookup.
In BNF, they are used to denote nonterminals (e.g. <name> ::= <first-name> <last-name>).
In ABAP they denote field symbols – placeholders or symbolic names for other fields, which can point to any data object.
To indicate an action or status (e.g. <Waves> or <Offline>), particularly in online, real-time text-based discussions (instant messaging, bulletin boards, etc.). (Here, asterisks can also be used to signify an action.)

When the signs are not used in pairs to delimit text (not acting as brackets):

They are used as less-than and greater-than relational operators, possibly in combination with other marks. In some languages the pair together as <> denotes an inequation ("not equal to").
When doubled as << or >> they may represent bit shift operators, or in C++, also stream input/output operators.
They indicate the redirection of input/output in various command shells.^[20]
Right-angle brackets are used in nested Usenet quoting and various e-mail formats, and as such are standard quotation mark glyphs.
A pair of right-angle brackets followed by the character's name and a colon are used in some production scripts and translated closed captioning to denote when there is a change of speaker. This is so a performer can easily scan for their lines when rehearsing a script.

Layout styles

In normal writing (prose) an opening bracket is rarely left hanging at the end of a line of text nor is a closing bracket permitted to start one. However, in computer code this is often done intentionally to aid readability. For example, a bracketed list of items separated by semicolons may be written with the brackets on separate lines, and the items, followed by the semicolon, each on one line.

A common error in programming is mismatching braces; accordingly, many IDEs have braces matching to highlight matching pairs.

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 Europe, 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, negative amounts are placed in parentheses.

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.

References

^ "broket". Catb.org. Retrieved 2013-02-13.
^ Truss, Lynne. Eats, Shoots & Leaves, 2003. p. 161. ISBN 1-59240-087-6.
^ "The Free Online Dictionary". Thefreedictionary.com. Retrieved 2013-02-13.
^ Robert Bringhurst, The Elements of Typographic Style, §5.3.2.
^ Slash (punctuation)#Gender-neutrality in Spanish and Portuguese
^ Fogarty, Mignon. "Parentheses, Brackets, and Braces". Quick and Dirty Tips. Retrieved 27 March 2011.
^ The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.104
^ The Columbia Guide to Standard American English
^ The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.102 and §6.106
^ The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.105
^ The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.107
^ Decodeunicode.org > U+007B LEFT CURLY BRACKET Retrieved on May 3, 2009
^ K R Venugopal, Rajkumar Buyya, T Ravishankar. Mastering C++, 1999. p. 34. ISBN 0-07-463454-2.
^ Some fonts don't display these characters correctly. Please refer to the image on the right instead.
^ M.L. West (1973) Textual Criticism and Editorial Technique (Stuttgart) 81.
^ ^a ^b ^c "Miscellaneous Technical" (PDF), The Unicode Standard, Version 6.1, 2012, retrieved 2012-02-01
^ "Miscellaneous Mathematical Symbols-A" (PDF), The Unicode Standard, Version 6.1, 2013, retrieved 2013-05-16
^ "Miscellaneous Mathematical Symbols-B" (PDF), The Unicode Standard, Version 6.1, 2013, retrieved 2013-05-16
^ Bob, Bemer. "The Great Curly Brace Trace Chase". Retrieved 2009-09-05.
^ Bryant, Randal E.; O'Hallaron, David. Computer Systems: A Programmer's Perspective, 2003. p. 794. ISBN 0-13-034074-X.

Bibliography

Lennard, John (1991). But I Digress: The Exploitation of Parentheses in English Printed Verse. Oxford: Clarendon Press. ISBN 0-19-811247-5.
Turnbull (1964). The Graphics of Communication. New York: Holt. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) 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.

[1] "broket". Catb.org. Retrieved 2013-02-13.

[2] Truss, Lynne. Eats, Shoots & Leaves, 2003. p. 161. ISBN 1-59240-087-6.

[3] "The Free Online Dictionary". Thefreedictionary.com. Retrieved 2013-02-13.

[4] Robert Bringhurst, The Elements of Typographic Style, §5.3.2.

[5] Slash (punctuation)#Gender-neutrality in Spanish and Portuguese

[6] Fogarty, Mignon. "Parentheses, Brackets, and Braces". Quick and Dirty Tips. Retrieved 27 March 2011.

[7] The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.104

[8] The Columbia Guide to Standard American English

[9] The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.102 and §6.106

[10] The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.105

[11] The Chicago Manual of Style, 15th ed., The University of Chicago Press, 2003, §6.107

[12] Decodeunicode.org > U+007B LEFT CURLY BRACKET Retrieved on May 3, 2009

[13] K R Venugopal, Rajkumar Buyya, T Ravishankar. Mastering C++, 1999. p. 34. ISBN 0-07-463454-2.

[14] Some fonts don't display these characters correctly. Please refer to the image on the right instead.

[15] M.L. West (1973) Textual Criticism and Editorial Technique (Stuttgart) 81.

[Unicode_Miscellaneous_Technical-16] "Miscellaneous Technical" (PDF), The Unicode Standard, Version 6.1, 2012, retrieved 2012-02-01

[Unicode_Miscellaneous_Mathematical_Symbols-A-17] "Miscellaneous Mathematical Symbols-A" (PDF), The Unicode Standard, Version 6.1, 2013, retrieved 2013-05-16

[Unicode_Miscellaneous_Mathematical_Symbols-B-18] "Miscellaneous Mathematical Symbols-B" (PDF), The Unicode Standard, Version 6.1, 2013, retrieved 2013-05-16

[19] Bob, Bemer. "The Great Curly Brace Trace Chase". Retrieved 2009-09-05.

[20] Bryant, Randal E.; O'Hallaron, David. Computer Systems: A Programmer's Perspective, 2003. p. 794. ISBN 0-13-034074-X.

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