This article is about the family of punctuation marks. For other uses, see Bracket (disambiguation)
Brackets
( ) [ ] { } ⟨ ⟩ Brackets (BE)or
or
or
or
or
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. They come in four main pairs of shapes, as given in the box to the right, which also gives their names, that vary between British and American English. "Brackets", without further qualification, are in British English the (…) marks and in American English the […] marks.
Other minor bracket shapes exist, such as (for example) slash or diagonal brackets used by linguists to enclose phonemes.
Brackets are typically deployed in symmetric pairs, and an individual bracket may be identified as a 'left' or 'right' bracket or, alternatively, an "opening bracket" or "closing bracket",[5] respectively, depending on the directionality of the context.
In casual writing and in technical fields such as computing or linguistic analysis of grammar, brackets nest, with segments of bracketed material containing embedded within them other further bracketed sub-segments. The number of opening brackets matches the number of closing brackets in such cases.
Various forms of brackets are used in mathematics, with specific mathematical meanings, often for denoting specific mathematical functions and subformulas.
History
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Angle brackets or chevrons ⟨ ⟩ were the earliest type of bracket to appear in written English. Erasmus coined the term lunula to refer to the round brackets or parentheses ( ) recalling the shape of the crescent moon (Latin: luna).[6]
Most typewriters only had the left and right parentheses. Square brackets appeared with some teleprinters.
Braces (curly brackets) first became part of a character set with the 8-bit code of the IBM 7030 Stretch.[7]
In 1961, ASCII contained parentheses, square, and curly brackets, and also less-than and greater-than signs that could be used as angle brackets.
Typography
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In English, typographers mostly prefer not to set brackets in italics, even when the enclosed text is italic.[8] However, in other languages like German, if brackets enclose text in italics, they are usually also set in italics.[9]
Parentheses or (round) brackets
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Parenthesis
( ) Parentheses (AE)or
or
U+0028
(
LEFT PARENTHESIS
((
)U+0029
)
RIGHT PARENTHESIS
()
)U+FF08
（
FULLWIDTH LEFT PARENTHESIS
U+FF09
）
FULLWIDTH RIGHT PARENTHESIS
(Quranic quotations)
U+FD3E
﴾
ORNATE LEFT PARENTHESIS
U+FD3F
﴿
ORNATE RIGHT PARENTHESIS
U+2E28
⸨
LEFT DOUBLE PARENTHESIS
U+2E29
⸩
RIGHT DOUBLE PARENTHESIS
Technical
U+207D
⁽
SUPERSCRIPT LEFT PARENTHESIS
U+207E
⁾
SUPERSCRIPT RIGHT PARENTHESIS
U+208D
₍
SUBSCRIPT LEFT PARENTHESIS
U+208E
₎
SUBSCRIPT RIGHT PARENTHESIS
U+239B
⎛
LEFT PARENTHESIS UPPER HOOK
U+239C
⎜
LEFT PARENTHESIS EXTENSION
U+239D
⎝
LEFT PARENTHESIS LOWER HOOK
U+239E
⎞
RIGHT PARENTHESIS UPPER HOOK
U+239F
⎟
RIGHT PARENTHESIS EXTENSION
U+23A0
⎠
RIGHT PARENTHESIS LOWER HOOK
U+23DC
⏜
TOP PARENTHESIS
(⏜
)U+23DD
⏝
BOTTOM PARENTHESIS
(⏝
)U+27EE
⟮
MATHEMATICAL LEFT FLATTENED PARENTHESIS
U+27EF
⟯
MATHEMATICAL RIGHT FLATTENED PARENTHESIS
U+2983
⦃
LEFT WHITE CURLY BRACKET
U+2984
⦄
RIGHT WHITE CURLY BRACKET
U+2985
⦅
LEFT WHITE PARENTHESIS
(⦅
)U+2986
⦆
RIGHT WHITE PARENTHESIS
(⦆
)Phonetic punctuation
U+2E59
⹙
TOP HALF LEFT PARENTHESIS
U+2E5A
⹚
TOP HALF RIGHT PARENTHESIS
U+2E5B
⹛
BOTTOM HALF LEFT PARENTHESIS
U+2E5C
⹜
BOTTOM HALF RIGHT PARENTHESIS
U+2768
❨
MEDIUM FLATTENED LEFT PARENTHESIS ORNAMENT
U+2769
❩
MEDIUM FLATTENED RIGHT PARENTHESIS ORNAMENT
U+276A
❪
MEDIUM FLATTENED LEFT PARENTHESIS ORNAMENT
U+276B
❫
MEDIUM FLATTENED RIGHT PARENTHESIS ORNAMENT
( and ) are parentheses (singular parenthesis ) in American English, and either round brackets or simply brackets in British English. They are also known as "parens" , "circle brackets", or "smooth brackets".[21]
In careful or formal writing, "parentheses" is also used in British English.[citation needed]
Uses of ( )
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Parentheses contain adjunctive material that serves to clarify (in the manner of a gloss) or is aside from the main point.[22]
A comma before or after the material can also be used, though if the sentence contains commas for other purposes, visual confusion may result. A dash before and after the material is also sometimes used.
Parentheses may be used in formal writing to add supplementary information, such as "Senator John McCain (R - Arizona) spoke at length". They can also indicate shorthand for "either singular or plural" for nouns, e.g. "the claim(s)". It can also be used for gender-neutral language, especially in languages with grammatical gender, e.g. "(s)he agreed with his/her physician" (the slash in the second instance, as one alternative is replacing the other, not adding to it).
Parenthetical phrases have been used extensively in informal writing and stream of consciousness literature. Examples include the southern American author William Faulkner (see Absalom, Absalom! and the Quentin section of The Sound and the Fury) as well as poet E. E. Cummings.
Parentheses have historically been used where the dash is currently used in alternatives, such as "parenthesis)(parentheses". Examples of this usage can be seen in editions of Fowler's Dictionary of Modern English Usage.
Parentheses may be nested (generally with one set (such as this) inside another set). This is not commonly used in formal writing (though sometimes other brackets [especially square brackets] will be used for one or more inner set of parentheses [in other words, secondary {or even tertiary} phrases can be found within the main parenthetical sentence]).
Language
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A parenthesis in rhetoric and linguistics refers to the entire bracketed text, not just to the enclosing marks used (so all the text in this set of round brackets may be described as "a parenthesis").[23] Taking as an example the sentence "Mrs Pennyfarthing (What? Yes, that was her name!) was my landlady.", the explanatory phrase between the parentheses is itself called a parenthesis. Again, the parenthesis implies that the meaning and flow of the bracketed phrase is supplemental to the rest of the text and the whole would be unchanged were the parenthesized sentences removed. The term refers to the syntax rather than the enclosure method: the same clause in the form "Mrs Pennyfarthing – What? Yes, that was her name! – was my landlady" is also a parenthesis.[24] (In non-specialist usage, the term "parenthetical phrase" is more widely understood.[25])
In phonetics, parentheses are used for indistinguishable[26] or unidentified utterances. They are also seen for silent articulation (mouthing),[27] where the expected phonetic transcription is derived from lip-reading, and with periods to indicate silent pauses, for example (…) or (2 sec).
Enumerations
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An unpaired right parenthesis is often used as part of a label in an ordered list, such as this one:
a) educational testing,
b) technical writing and diagrams,
c) market research, and
d) elections.
Accounting
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]
Traditionally in accounting, contra amounts are placed in parentheses. A debit balance account in a series of credit balances will have parenthesis and vice versa.
Parentheses in mathematics
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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:
[ 4 × ( 3 + 2 ) ] 2 = 400. {\displaystyle [4\times (3+2)]^{2}=400.}
Parentheses in programming languages
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Parentheses are included in the syntaxes of many programming languages. Typically needed to denote an argument; to tell the compiler what data type the Method/Function needs to look for first in order to initialise. In some cases, such as in LISP, parentheses are a fundamental construct of the language. They are also often used for scoping functions and operators and for arrays. In syntax diagrams they are used for grouping, such as in extended Backus–Naur form.
In Mathematica and the Wolfram language, parentheses are used to indicate grouping – for example, with pure anonymous functions.
Taxonomy
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If it is desired to include the subgenus when giving the scientific name of an animal species or subspecies, the subgenus's name is provided in parentheses between the genus name and the specific epithet.[28] For instance, Polyphylla (Xerasiobia) alba is a way to cite the species Polyphylla alba while also mentioning that it is in the subgenus Xerasiobia.[29] There is also a convention of citing a subgenus by enclosing it in parentheses after its genus, e.g., Polyphylla (Xerasiobia) is a way to refer to the subgenus Xerasiobia within the genus Polyphylla.[30] Parentheses are similarly used to cite a subgenus with the name of a prokaryotic species, although the International Code of Nomenclature of Prokaryotes (ICNP) requires the use of the abbreviation "subgen." as well, e.g., Acetobacter (subgen. Gluconoacetobacter) liquefaciens.[31]
Chemistry
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Parentheses are used in chemistry to denote a repeated substructure within a molecule, e.g. HC(CH3)3 (isobutane) or, similarly, to indicate the stoichiometry of ionic compounds with such substructures: e.g. Ca(NO3)2 (calcium nitrate).
This is a notation that was pioneered by Berzelius, who wanted chemical formulae to more resemble algebraic notation, with brackets enclosing groups that could be multiplied (e.g. in 3(AlO2 + 2SO3) the 3 multiplies everything within the parentheses).
In chemical nomenclature, parentheses are used to distinguish structural features and multipliers for clarity, for example in the polymer poly(methyl methacrylate).[34]
Square brackets
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Square brackets
[ ] In UnicodeU+005B
[
LEFT SQUARE BRACKET
([, [
)U+005D
]
RIGHT SQUARE BRACKET
(], ]
)U+FF3B
［
FULLWIDTH LEFT SQUARE BRACKET
U+FF3D
］
FULLWIDTH RIGHT SQUARE BRACKET
U+2045
⁅
LEFT SQUARE BRACKET WITH QUILL
U+2046
⁆
RIGHT SQUARE BRACKET WITH QUILL
Technical/Mathematical
U+23A1
⎡
LEFT SQUARE BRACKET UPPER CORNER
U+23A2
⎢
LEFT SQUARE BRACKET EXTENSION
U+23A3
⎣
LEFT SQUARE BRACKET LOWER CORNER
U+23A4
⎤
RIGHT SQUARE BRACKET UPPER CORNER
U+23A5
⎥
RIGHT SQUARE BRACKET EXTENSION
U+23A6
⎦
RIGHT SQUARE BRACKET LOWER CORNER
U+23B4
⎴
TOP SQUARE BRACKET
(⎴, ⎴
)U+23B5
⎵
BOTTOM SQUARE BRACKET
(⎵, ⎵
)U+23B6
⎶
BOTTOM SQUARE BRACKET OVER TOP SQUARE BRACKET
(⎶
)U+27E6
⟦
MATHEMATICAL LEFT WHITE SQUARE BRACKET
(⟦, ⟦
)U+27E7
⟧
MATHEMATICAL RIGHT WHITE SQUARE BRACKET
(⟧, ⟧
)U+298B
⦋
LEFT SQUARE BRACKET WITH UNDERBAR
(⦋
)U+298C
⦌
RIGHT SQUARE BRACKET WITH UNDERBAR
(⦌
)U+298D
⦍
LEFT SQUARE BRACKET WITH TICK IN TOP CORNER
(⦍
)U+2990
⦐
RIGHT SQUARE BRACKET WITH TICK IN TOP CORNER
(⦐
)U+298E
⦎
RIGHT SQUARE BRACKET WITH TICK IN BOTTOM CORNER
(⦎
)U+298F
⦏
LEFT SQUARE BRACKET WITH TICK IN BOTTOM CORNER
(⦏
)Phonetic punctuation
U+2E55
⹕
LEFT SQUARE BRACKET WITH STROKE
U+2E56
⹖
RIGHT SQUARE BRACKET WITH STROKE
U+2E57
⹗
LEFT SQUARE BRACKET WITH DOUBLE STROKE
U+2E58
⹘
RIGHT SQUARE BRACKET WITH DOUBLE STROKE
Quotation(East-Asian texts)
U+301A
〚
LEFT WHITE SQUARE BRACKET
U+301B
〛
RIGHT WHITE SQUARE BRACKET
[ and ] are square brackets in both British and American English, but are also more simply brackets in the latter. An older name for these brackets is "crotchets".[36]
Uses of [ ]
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Square brackets are often used to insert explanatory material or to mark where a [word or] passage was omitted from an original material by someone other than the original author, or to mark modifications in quotations.[37] In transcribed interviews, sounds, responses and reactions that are not words but that can be described are set off in square brackets — "... [laughs] ...".
When quoted material is in any way altered, the alterations are enclosed in square brackets within the quotation to show that the quotation is not exactly as given, or to add an annotation.[38] For example: The Plaintiff asserted his cause is just, stating,
In the original quoted sentence, the word "my" was capitalized: it has been modified in the quotation given and the change signalled with brackets. Similarly, where the quotation contained a grammatical error (is/are), the quoting author signalled that the error was in the original with "[sic]" (Latin for 'thus').
A bracketed ellipsis, [...], is often used to indicate omitted material: "I'd like to thank [several unimportant people] for their tolerance [...]"[39] Bracketed comments inserted into a quote indicate where the original has been modified for clarity: "I appreciate it [the honor], but I must refuse", and "the future of psionics [see definition] is in doubt". Or one can quote the original statement "I hate to do laundry" with a (sometimes grammatical) modification inserted: He "hate[s] to do laundry".
Additionally, a small letter can be replaced by a capital one, when the beginning of the original printed text is being quoted in another piece of text or when the original text has been omitted for succinctness— for example, when referring to a verbose original: "To the extent that policymakers and elite opinion in general have made use of economic analysis at all, they have, as the saying goes, done so the way a drunkard uses a lamppost: for support, not illumination", can be quoted succinctly as: "[P]olicymakers [...] have made use of economic analysis [...] the way a drunkard uses a lamppost: for support, not illumination." When nested parentheses are needed, brackets are sometimes used as a substitute for the inner pair of parentheses within the outer pair.[40] When deeper levels of nesting are needed, convention is to alternate between parentheses and brackets at each level.
Alternatively, empty square brackets can also indicate omitted material, usually single letter only. The original, "Reading is also a process and it also changes you." can be rewritten in a quote as: It has been suggested that reading can "also change[] you".[41]
In translated works, brackets are used to signify the same word or phrase in the original language to avoid ambiguity.[42] For example: He is trained in the way of the open hand [karate].
Style and usage guides originating in the news industry of the twentieth century, such as the AP Stylebook, recommend against the use of square brackets because "They cannot be transmitted over news wires."[43] However, this guidance has little relevance outside of the technological constraints of the industry and era.
In linguistics, phonetic transcriptions are generally enclosed within square brackets,[44] whereas phonemic transcriptions typically use paired slashes, according to International Phonetic Alphabet rules. Pipes (| |) are often used to indicate a morphophonemic rather than phonemic representation. Other conventions are double slashes (// //), double pipes (|| ||) and curly brackets ({ }).
In lexicography, square brackets usually surround the section of a dictionary entry which contains the etymology of the word the entry defines.
Proofreading
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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 upSquare brackets are used to denote parts of the text that need to be checked when preparing drafts prior to finalizing a document.
Law
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Square brackets are used in some countries in the citation of law reports to identify parallel citations to non-official reporters. For example:
Chronicle Pub. Co. v Superior Court (1998) 54 Cal.2d 548, [7 Cal.Rptr. 109]
In some other countries (such as England and Wales), square brackets are used to indicate that the year is part of the citation and parentheses are used to indicate the year the judgment was given. For example:
National Coal Board v England [1954] AC 403
This case is in the 1954 volume of the Appeal Cases reports, although the decision may have been given in 1953 or earlier. Compare with:
(1954) 98 Sol Jo 176
This citation reports a decision from 1954, in volume 98 of the Solicitors Journal which may be published in 1955 or later.
They often denote points that have not yet been agreed to in legal drafts and the year in which a report was made for certain case law decisions.
Square brackets in mathematics
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Brackets are used in mathematics in a variety of notations, including standard notations for commutators, the floor function, the Lie bracket, equivalence classes, the Iverson bracket, and matrices.
Square brackets may be used exclusively or in combination with parentheses to represent intervals as interval notation. For example, [0,5] represents the set of real numbers from 0 to 5 inclusive. Both parentheses and brackets are used to denote a half-open interval; [5, 12) would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth, but 12.0 is not included. In some European countries, the notation [5, 12[ is also used.[citation needed] The endpoint adjoining the square bracket is known as closed, whereas the endpoint adjoining the parenthesis is known as open.
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 .
Chemistry
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Square brackets can also be used in chemistry to represent the concentration of a chemical substance in solution and to denote charge a Lewis structure of an ion (particularly distributed charge in a complex ion), repeating chemical units (particularly in polymers) and transition state structures, among other uses.
Square brackets in programming languages
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Brackets are used in many computer programming languages, primarily for array indexing. But they are also used to denote general tuples, sets and other structures, just as in mathematics. There may be several other uses as well, depending on the language at hand. In syntax diagrams they are used for optional portions, such as in extended Backus–Naur form.
Double brackets ⟦ ⟧
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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.[46][47] In the Wolfram Language, double brackets, either as iterated single brackets ([[) or ligatures (〚) are used for list indexing.[48]
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.[49]
Brackets with quills ⁅ ⁆
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Known as "spike parentheses" (Swedish: piggparenteser), ⁅ and ⁆ are used in Swedish bilingual dictionaries to enclose supplemental constructions.[50]
Curly brackets
[
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]
Curly brackets
{ } In UnicodeU+007B
{
LEFT CURLY BRACKET
({, {
)U+007D
}
RIGHT CURLY BRACKET
(}, }
)U+FF5B
｛
FULLWIDTH LEFT CURLY BRACKET
U+FF5D
｝
FULLWIDTH RIGHT CURLY BRACKET
Technical/Mathematical(half-width)
U+23A7
⎧
LEFT CURLY BRACKET UPPER HOOK
U+23A8
⎨
LEFT CURLY BRACKET MIDDLE PIECE
U+23A9
⎩
LEFT CURLY BRACKET LOWER HOOK
U+23AB
⎫
RIGHT CURLY BRACKET UPPER HOOK
U+23AC
⎬
RIGHT CURLY BRACKET MIDDLE PIECE
U+23AD
⎭
RIGHT CURLY BRACKET LOWER HOOK
U+23AA
⎪
CURLY BRACKET EXTENSION
U+23B0
⎰
UPPER LEFT OR LOWER RIGHT CURLY BRACKET SECTION
(⎰, ⎰
)U+23B1
⎱
UPPER RIGHT OR LOWER LEFT CURLY BRACKET SECTION
(⎱, ⎱
)U+23DE
⏞
TOP CURLY BRACKET
(⏞
)U+23DF
⏟
BOTTOM CURLY BRACKET
(⏟
)U+2774
❴
MEDIUM LEFT CURLY BRACKET ORNAMENT
U+2775
❵
MEDIUM RIGHT CURLY BRACKET ORNAMENT
{ and } are braces in both American and British English, and also curly brackets in the latter.
Uses of { }
[
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]
An example of curly brackets used to group sentences togetherCurly brackets are used by text editors to mark editorial insertions[51] or interpolations.[52]
Braces used to be used to connect multiple lines of poetry, such as triplets in a poem of rhyming couplets,[53] although this usage had gone out of fashion by the 19th century.
Another older use in prose was to eliminate duplication in lists and tables. Two examples here from Charles Hutton's 19th century table table of weights and measures in his A Course of Mathematics:
In this kingdom The standard of … ⎧ Length is a Yard. ⎪ Surface is a Square Yard, the1
⁄4840
of an Acre. ⎨ ⎰ Solidity is a Cubic Yard. ⎪ ⎱ Capacity is a Gallon. ⎩ Weight is a Pound.
Imperial measure of CAPACITY for coals, culm, lime, fish, potatoes, fruit,– and other goods commonly sold by heaped measure: 2 Gallons = 1 Peck = 764 ⎱ Cubic Inches, nearly 8 Gallons = 1 Bushel = 28131
⁄2
⎰ 3 Bushels = 1 Sack = 48
⁄9
⎱ Cubic Feet, nearly 12 Sacks = 1 Chald. = 582
⁄3
⎰
As an extension to the International Phonetic Alphabet (IPA), braces are used for prosodic notation.
Music
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]
In music, they are known as "accolades" or "braces", and connect two or more lines (staves) of music that are played simultaneously.[58]
Chemistry
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The use of braces in chemistry is an old notation that has long since been superseded by subscripted numbers. The chemical formula for water, H2O, was represented as H H } O {\displaystyle \left.{\stackrel {H}{H}}\right\}O} .
Curly brackets in programming languages
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]
In many programming languages, curly brackets enclose groups of statements and create a local scope. Such languages (C, C#, C++ and many others) are therefore called curly bracket languages.[59] They are also used to define structures and enumerated type in these languages.
In various Unix shells, they enclose a group of strings that are used in a process known as brace expansion, where each successive string in the group is interpolated at that point in the command line to generate the command-line's final form. The mechanism originated in the C shell and the string generation mechanism is a simple interpolation that can occur anywhere in a command line and takes no account of existing filenames.
In syntax diagrams they are used for repetition, such as in extended Backus–Naur form.
In the Z formal specification language, braces define a set.
Curly brackets in mathematics
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]
In mathematics they delimit sets, in what is called set notation. Braces enclose either a literal list of set elements, or a rule that defines the set elements. For example:
S = {
a
,b
} defines a setS
containinga
andb
.S = {
x
|x
> 0} defines a setS
containing elements (implied to be numbers)x
0,x
1, and so on where everyx
n
satisfies the rule that it is greater than zero.They are often also used to denote the Poisson bracket between two quantities.
In ring theory, braces denote the anticommutator where {a, b} is defined as a b + b a .
Angle brackets
[
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]
"Angle bracket" redirects here. For a mechanical part used for joining, see Angle bracket (fastener)
Angle brackets
⟨ ⟩ ⟪ ⟫ < > Angle brackets (BE&AE) Angle brackets (BE&AE) less-than and greater-than In UnicodeU+003C
<
LESS-THAN SIGN
(<, <
)U+003E
>
GREATER-THAN SIGN
(>, >
)U+FF1C
＜
FULLWIDTH LESS-THAN SIGN
U+FF1E
＞
FULLWIDTH GREATER-THAN SIGN
Technical/Mathematical(half-width)
U+2329
〈
LEFT-POINTING ANGLE BRACKET
[a]U+232A
〉
RIGHT-POINTING ANGLE BRACKET
[a]U+27E8
⟨
MATHEMATICAL LEFT ANGLE BRACKET
(⟨, ⟨, ⟨
)[a]U+27E9
⟩
MATHEMATICAL RIGHT ANGLE BRACKET
(⟩, ⟩, ⟩
)[a]U+27EA
⟪
MATHEMATICAL LEFT DOUBLE ANGLE BRACKET
(⟪
)U+27EB
⟫
MATHEMATICAL RIGHT DOUBLE ANGLE BRACKET
(⟫
)U+2991
⦑
LEFT ANGLE BRACKET WITH DOT
(⦑
)U+2992
⦒
RIGHT ANGLE BRACKET WITH DOT
(⦒
)U+29FC
⧼
LEFT-POINTING CURVED ANGLE BRACKET
U+29FD
⧽
RIGHT-POINTING CURVED ANGLE BRACKET
Quotation(fullwidth East-Asian texts)
U+3008
〈
LEFT ANGLE BRACKET
U+3009
〉
RIGHT ANGLE BRACKET
U+300A
《
LEFT DOUBLE ANGLE BRACKET
U+300B
》
RIGHT DOUBLE ANGLE BRACKET
U+276C
❬
MEDIUM LEFT-POINTING ANGLE BRACKET ORNAMENT
U+276D
❭
MEDIUM RIGHT-POINTING ANGLE BRACKET ORNAMENT
U+2770
❰
HEAVY LEFT-POINTING ANGLE BRACKET ORNAMENT
U+2771
❱
HEAVY RIGHT-POINTING ANGLE BRACKET ORNAMENT
U+276E
❮
HEAVY LEFT-POINTING ANGLE QUOTATION MARK ORNAMENT
U+276F
❯
HEAVY RIGHT-POINTING ANGLE QUOTATION MARK ORNAMENT
⟨ and ⟩ are angle brackets in both American and British English. They are also known as "pointy brackets", "triangular brackets", "diamond brackets", "tuples", "guillemets", "left and right carets", "broken brackets",[citation needed] or (in computer slang) "brokets".[63]
Strictly speaking they are distinct from V-shaped chevrons, as they have (where the typography permits it) a broader span than chevrons, although when printed often no visual distinction is made.
The ASCII less-than and greater-than characters <> are often used for angle brackets. In most cases only those characters are accepted by computer programs, and the Unicode angle brackets are not recognized (for instance, in HTML tags). The characters for "single" guillemets ‹› are also often used, and sometimes normal guillemets «» when nested angle brackets are needed.
The angle brackets or chevrons at U+27E8 and U+27E9 are for mathematical use and Western languages, whereas U+3008 and U+3009 are for East Asian languages. The chevrons at U+2329 and U+232A are deprecated in favour of the U+3008 and U+3009 East Asian angle brackets. Unicode discourages their use for mathematics and in Western texts,[15] because they are canonically equivalent to the CJK code points U+300x and thus likely to render as double-width symbols. The less-than and greater-than symbols are often used as replacements for chevrons.
Shape
[
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Angle brackets are larger than less-than and greater-than signs, which in turn are larger than guillemets.
Angle brackets, less-than/greater-than signs and single guillemets in fonts Cambria, DejaVu Serif, Andron Mega Corpus, Andika and Everson MonoUses of ⟨ ⟩
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Angle brackets are infrequently used to denote words that are thought instead of spoken, such as:
⟨What an unusual flower!⟩
In textual criticism, and hence in many editions of pre-modern works, chevrons denote sections of the text which are illegible or otherwise lost; the editor will often insert their own reconstruction where possible within them.[65]
In comic books, chevrons are often used to mark dialogue that has been translated notionally from another language; in other words, if a character is speaking another language, instead of writing in the other language and providing a translation, one writes the translated text within chevrons. Since no foreign language is actually written, this is only notionally translated.[citation needed]
In linguistics, angle brackets identify graphemes (e.g., letters of an alphabet) or orthography, as in "The English word /kæt/ is spelled ⟨cat⟩."[66][67][65] (See IPA Brackets and transcription delimiters.)
In epigraphy, they may be used for mechanical transliterations of a text into the Latin script.[67]
In East Asian punctuation, angle brackets are used as quotation marks. Chevron-like symbols are part of standard Chinese, Japanese and – less frequently – Korean punctuation, where they generally enclose the titles of books, as: ︿... ﹀ or ︽...︾ for traditional vertical printing — written in vertical lines — and as 〈 ... 〉 or 《 ... 》 for horizontal printing — in horizontal.
Angle brackets in mathematics
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Angle brackets (or 'chevrons') are used in group theory to write group presentations, and to denote the subgroup generated by a collection of elements. In set theory, chevrons or parentheses are used to denote ordered pairs[68] and other tuples, whereas curly brackets are used for unordered sets.
Physics and mechanics
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In physical sciences and statistical mechanics, angle brackets are used to denote an average (expected value) over time or over another continuous parameter. For example:
⟨ V ( t ) 2 ⟩ = lim T → ∞ 1 T ∫ − T 2 T 2 V ( t ) 2 d t . {\displaystyle \left\langle V(t)^{2}\right\rangle =\lim _{T\to \infty }{\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}V(t)^{2}\,{\rm {d}}t.}
In mathematical physics, especially quantum mechanics, it is common to write the inner product between elements as ⟨a|b⟩, as a short version of ⟨a|·|b⟩, or ⟨a|Ô|b⟩, where Ô is an operator. This is known as Dirac notation or bra–ket notation, to note vectors from the dual spaces of the Bra ⟨A| and the Ket |B⟩. But there are other notations used.
In continuum mechanics, chevrons may be used as Macaulay brackets.
Angle brackets in programming languages
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In C++ chevrons (actually less-than and greater-than) are used to surround arguments to templates. They are also used to surround the names of header files; this usage was inherited from and is also found in C.
In the Z formal specification language, chevrons define a sequence.
In HTML, chevrons (actually 'greater than' and 'less than' symbols) are used to bracket meta text. For example <b> denotes that the following text should be displayed as bold. Pairs of meta text tags are required – much as brackets themselves are usually in pairs. The end of the bold text segment would be indicated by </b>. This use is sometimes extended as an informal mechanism for communicating mood or tone in digital formats such as messaging, for example adding "<sighs>" at the end of a sentence.
Other brackets
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Lenticular brackets【】
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Some East Asian languages use lenticular brackets 【 】, a combination of square brackets and round brackets called 方頭括號 (fāngtóu kuòhào) in Chinese and 隅付き括弧 (sumitsuki kakko) in Japanese. They are used in titles and headings in both Chinese[69] and Japanese. On the Internet, they are used to emphasize a text. In Japanese, they are most frequently seen in dictionaries for quoting Chinese characters and Sino-Japanese loanwords.
Floor ⌊ ⌋ and ceiling ⌈ ⌉ corner brackets
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Floor and ceiling
⌈ceiling⌉ ⌊floor⌋ In UnicodeU+2308
⌈
LEFT CEILING
(⌈, ⌈
)U+2309
⌉
RIGHT CEILING
(⌉, ⌉
)U+230A
⌊
LEFT FLOOR
(⌊, ⌊
)U+230B
⌋
RIGHT FLOOR
(⌋, ⌋
)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 ⸢ ⸣
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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.[70] For example, Callimachus Iambus 1.2 reads: ἐκ τῶν ὅκου βοῦν κολλύ⸤βου π⸥ιπρήσκουσιν. A hole in the papyrus has obliterated βου π, but these letters are supplied by an ancient commentary on the poem. Second intermittent sources can be between ⸢ and ⸣. Quine corners are sometimes used instead of half brackets.[15]
Unicode
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Representations of various kinds of brackets in Unicode and HTML, that are not in the infoboxes in preceding sections, are given below.
Unicode and HTML encodings for various bracket characters Uses Unicode SGML/HTML/XML entities Sample Quine corners[15] U+231C Top left corner ⌜ ⌜quasi-quotation⌝⏠
tortoise shell brackets
⏡
R
⦇S
⦈ U+2988 Z notation right image bracket ⦈ U+2989 Z notation left binding bracket ⦉ ⦉x
:ℤ⦊ U+298A Z notation right binding bracket ⦊ U+2993 Left arc less-than bracket ⦓ ⦓inequality sign brackets⦔ U+2994 Right arc greater-than bracket ⦔ U+2995 Double left arc greater-than bracket ⦕ ⦕inequality sign brackets⦖ U+2996 Double right arc less-than bracket ⦖ U+2997 Left black tortoise shell bracket ⦗ ⦗black tortoise shell brackets⦘ U+2998 Right black tortoise shell bracket ⦘ U+29D8 Left wiggly fence ⧘ ⧘...⧙ U+29D9 Right wiggly fence ⧙ U+29DA Left double wiggly fence ⧚ ⧚...⧛ U+29DB Right double wiggly fence ⧛ Half brackets[14] U+2E22 Top left half bracket ⸢ ⸢editorial notation⸣ U+2E23 Top right half bracket ⸣ U+2E24 Bottom left half bracket ⸤ ⸤editorial notation⸥ U+2E25 Bottom right half bracket ⸥ Dingbats[20] U+2772 Light left tortoise shell bracket ornament ❲ ❲light tortoise shell bracket ornament❳ U+2773 Light right tortoise shell bracket ornament ❳ N'Ko[14] U+2E1C Left low paraphrase bracket ⸜⸜ߒߞߏ⸝
U+2E1D Right low paraphrase bracket ⸝ Ogham[71] U+169B Ogham feather mark ᚛ ᚛ᚑᚌᚐᚋ᚜ U+169C Ogham reversed feather mark ᚜ Old Hungarian U+2E42 Double low-reversed-9 quotation mark ⹂ ⹂ Tibetan[72] U+0F3A Tibetan mark gug rtags gyon ༺ ༺དབུ་ཅན་༻ U+0F3B Tibetan mark gug rtags gyas ༻ U+0F3C Tibetan mark ang khang gyon ༼ ༼༡༢༣༽ U+0F3D Tibetan mark ang khang gyas ༽ New Testament editorial marks[14] U+2E02 Left substitution bracket ⸂ ⸂...⸃ U+2E03 Right substitution bracket ⸃ U+2E04 Left dotted substitution bracket ⸄ ⸄...⸅ U+2E05 Right dotted substitution bracket ⸅ U+2E09 Left transposition bracket ⸉ ⸉...⸊ U+2E0A Right transposition bracket ⸊ U+2E0C Left raised omission bracket ⸌ ⸌...⸍ U+2E0D Right raised omission bracket ⸍ Medieval studies[13][14] U+2E26 Left sideways u bracket ⸦ ⸦crux⸧ U+2E27 Right sideways u bracket ⸧ QuotationThis is fullwidth version of U+2033 DOUBLE PRIME. In vertical texts, U+301F LOW DOUBLE PRIME QUOTATION MARK is preferred.
See also
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References
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Sources
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Bibliography
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Welcome to Brighterly’s fun-filled world of mathematics, where we unravel the mysteries of numbers and equations to make learning a joyous journey. Today, we’re exploring a fascinating aspect of math – brackets. These seemingly simple symbols – parentheses (), curly brackets {}, and square brackets [] – are silent powerhouses, steering the course of calculations and leading us to accurate results.
Brackets are like the traffic signals of math. They guide us on when to stop and solve a particular part of a complex problem, ensuring a smooth ride through the maze of numbers and operations. Without them, we could get lost in the labyrinth of mathematical expressions, leading to confusion and incorrect solutions. Brackets, in their different forms, serve as our pathfinders, directing us on the right path to follow in the order of operations.
At Brighterly, we believe that understanding the function and importance of brackets is an essential step in your child’s mathematical journey. With our engaging and easy-to-understand approach, we’ll ensure that your child not only learns the concepts but also enjoys the process. So, let’s embark on this exciting voyage of discovery together!
Understanding brackets in math is crucial for any young mathematician. You’ve likely seen these mysterious symbols in your math assignments, and you might be wondering what they mean. Brackets are symbols used in mathematical expressions to denote the order of operations. They tell us which part of a mathematical equation or expression should be solved first. You can think of them as mathematical road signs, guiding us through complex numerical landscapes.
In mathematics, there are three main types of brackets: parentheses (), curly brackets {}, and square brackets []. Each type has its unique use and place in mathematical expressions. They are used to group numbers and operations to clarify the sequence in which the operations should be performed. Using brackets can significantly alter the result of a mathematical expression, so understanding their function is crucial.
Parentheses are the most commonly used type of brackets in mathematics. They are denoted by the symbols (). When parentheses are used in a math problem, it means that the operation inside the parentheses should be performed first. If there are multiple operations inside the parentheses, you should follow the regular order of operations (BIDMAS/BODMAS) within them.
Parentheses are used in math to group certain parts of an equation together. This grouping tells us to do the operations inside the parentheses first, before moving on to the rest of the equation. It’s like telling us, “Hey, pay attention to this part first!” For example, in the equation 5 * (3 + 2), we add 3 + 2 together before multiplying by 5, because the parentheses tell us to do so.
Curly brackets, also known as braces, are denoted by {}. They are used to denote a set or a list of numbers or objects. For example, {1, 2, 3, 4, 5} represents a set of the first five natural numbers. Curly brackets are also used to indicate that operations inside them should be performed first, though they are used less frequently than parentheses for this purpose.
Square brackets are the third type of brackets, denoted by []. They are used in math to denote closed intervals in set theory and calculus. For example, [0, 1] represents all numbers from 0 to 1, inclusive. Square brackets can also indicate the order of operations, like parentheses and curly brackets.
The order of operations for brackets, often remembered by the acronym BIDMAS/BODMAS or PEMDAS, instructs us to solve bracketed operations first. The order is Brackets (or Parentheses), Indices (or Exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). Within brackets, this order is maintained. For example, in (3 + 2 * 4), we do 2 * 4 first, then add 3 because of the BIDMAS/BODMAS rule.
Square brackets are used to denote a closed interval in mathematics. For example, [3, 5] means all the numbers between 3 and 5, including 3 and 5. They’re also used in mathematical functions and sequences. When used in the context of the order of operations, they work similarly to parentheses.
Curly brackets or braces {} are primarily used in mathematics to denote a set of numbers or objects. For example, {2, 4, 6, 8} represents a set of even numbers. In the context of the order of operations, they function similarly to parentheses and square brackets.
In the grand scheme of mathematics, brackets play a role akin to the unsung heroes. They may seem like minor additions to an equation, but their impact on the outcome is immense. They bring clarity to operations, add structure to intricate equations, and define sets and intervals. They are the invisible forces that direct the flow of calculations, ensuring we reach the correct result.
Mastering the use of brackets in math is like acquiring a superpower – it opens up a new realm of problem-solving skills. It is the foundation on which many advanced mathematical concepts are built. The understanding of brackets and their correct usage equips learners with the ability to dissect complex problems and solve them step by step.
At Brighterly, we strive to illuminate the path of learning for our young scholars. We understand that every concept, no matter how small, serves as a stepping stone towards mathematical proficiency. That’s why we take pride in simplifying complex concepts, like brackets, and making learning an enjoyable experience.
Remember, every mathematical journey begins with a single step. Understanding brackets might be one of the first steps, but it’s a significant stride towards a brighter mathematical future. So keep learning, stay curious, and remember, the world of Brighterly is always here to guide you on your journey.
Brackets in math are used to denote the order of operations. They serve as clear indicators dictating which part of a complex equation or expression needs to be calculated first. By encapsulating certain parts of an equation, they effectively create a microcosm within the larger mathematical universe of the equation, which must be solved before any further calculations can be made. This mechanism is crucial in avoiding ambiguity and ensuring that mathematical equations are universally understood and can be calculated in the same way, no matter who is performing the calculations.
Parentheses (), curly brackets {}, and square brackets [] each have unique uses in mathematics. Parentheses are the most common type of brackets used to signify priority in the order of operations. When you see a part of an equation within parentheses, it should be your first stop in the calculation journey. On the other hand, curly brackets are often used to denote a set of numbers or objects. They encapsulate elements that belong together due to shared properties or conditions. Square brackets, meanwhile, are typically used to signify closed intervals on a number line. This means they include all the numbers between and including the specified endpoints. Understanding the nuances of these different brackets helps us to navigate and interpret complex mathematical information accurately.
The order of operations is a set of rules in mathematics that dictates the sequence in which operations should be performed to ensure consistent and correct results. It is often remembered by the acronym BIDMAS/BODMAS or PEMDAS. This stands for: Brackets (or Parentheses) first, then Indices (or Exponents), followed by Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right). This rule ensures that everyone interprets and solves mathematical equations in the same way. Even within brackets, this order of operations is maintained. It’s like a roadmap guiding us through the terrain of mathematical operations, ensuring we arrive at the correct destination: the right answer.
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