Bracket (character)

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Punctuation marks
Comma, comma	,
Semicolon, semicolon	;
Colon, colon	:
Point	.
Ellipsis	...
Focus	·
bullet point	•
Question mark	?
Exclamation, exclamation, call signs	!
Apostrophe, apostrophe	'
- - Hyphen ; Hyphen ; Supplementary line
Indent ; Up line	-
quotation marks "" »«  /  «» ‚'› ‹  /  ‹ ›
Slashes / \
Brackets () []

Brackets are characters or symbols that are usually inserted in pairs before and after parts of a text. This border, known as brackets, delimits the content of the parts or changes their function.

In the written language, brackets serve as punctuation marks for structuring the syntactic form (see also parenthesis ). Generous use of brackets is considered bad style in German typesetting , dashes or the resolution of box sentences are mostly preferred. In other languages, e.g. B. in English, brackets are used more often.

In mathematics , brackets express, among other things, the precedence of one arithmetic operation to be carried out over others in the arithmetic order. For example, the result is equal to 5, because the calculation inside the brackets is carried out first, but it is equal to 3, since in this case the work proceeds from left to right. In advanced mathematics , parentheses serve many other purposes as well, most notably to denote the arguments of a function . Curly, square and angle brackets usually have a special meaning in mathematics. ${\displaystyle 10-(6-1)}$ ${\displaystyle 10-6-1}$

Similarly, in many programming languages , parentheses are used to group several types of program elements.

In the natural sciences, brackets are not only used for mathematical arithmetic operations. In chemistry, square brackets are used to identify concentrations . In addition, there are also round brackets when natural constants cannot be measured precisely but are estimated. To do this, another number is appended to the value of the constant in brackets - see CODATA .

Brackets in grammar and typography

Several types of brackets are commonly used as punctuation marks, which are almost exclusively used in pairs (i.e. as opening and closing brackets); the English names differ in British (BE) and American (AE) English :

A space is always placed before an opening bracket and after a closing bracket (unless it is followed by a punctuation mark, as is the case here, or the bracket denotes alternatives as in colleagues ). After an opening bracket and before a closing bracket, however, not. (A sentence point only comes before a closing bracket if a complete sentence - as here - is in brackets.)

Round brackets

()

(…): ( Greek / English: parentheses [AE] or round brackets [BE]): the usual brackets as used in the running text to separate parts of sentences and to summarize them. Unicode : U + 0028 and U + 0029

Square brackets

[]

[…]: (English: brackets [AE] or square brackets [BE]): Are u. a. used when something is to be bracketed within a bracket expression or to indicate omissions and insertions in quotations . In linguistics , phones are usually put in square brackets. ; Examples: [ baɪ̯ˌʃpiːlə ] ( IPA - phonetics ); “[AE]” and “[BE]” in this paragraph, “[ sic ]” and “[…]”. Unicode: U + 005B and U + 005D

Curly brackets (accolades)

{}

{…}: Also called braces or nose clips (English: braces [AE] or curly brackets [BE], French : accolades ): Are rarely used to summarize several lines. For example, they have a special meaning in dictionaries . Unicode: U + 007B and U + 007D

Angle brackets

⟨⟩

⟨…⟩: Also called "angle brackets" (English: angle brackets ; Unicode: U + 27E8 and U + 27E9, or: see below 〈…〉 in the section on CJK brackets ). They are rarely used. In dictionaries they have a special meaning, for example the (etymological) origin of a word is put in angle brackets, and more rarely style specifications in dictionaries. In linguistics, graphemes and grapheme chains are put in angle brackets. Since these characters are missing in the ASCII character set, the ASCII characters " Less than " <and " Greater than "> (Unicode: U + 003C and U + 003E; HTML : <and >) are often used instead . The latter are often used in electronic data processing to differentiate between name and email address - for example: Max Mustermann <[email protected]>

CJK brackets

Other types of brackets are used in the CJK scripts; the Unicode character standard contains the additional encodings for this.

〈〉	《》	「」
3008/3009	300A / 300B	300C / 300D
『』	【】	〔〕
300E / 300F	3010/3011	3014/3015
〖〗	〘〙	〚〛
3016/3017	3018/3019	301A / 301B

Brackets in the International Phonetic Alphabet

Square brackets

[]

The International Phonetic Alphabet (IPA) differentiates between the left square bracket "[" and the right square bracket "]".

In the IPA the characters "[" and "]" indicate the beginning and the end of the phonetic transcription ; they have the IPA numbers 901or 902( HTML entity [ = [ and ] = ]).

Curly braces

{}

The curly / curly brackets in the International Phonetic Alphabet indicate the beginning or the end of prosodic notation ; (HTML entities { = {and } = }).

Parentheses in math

In mathematics , brackets are also mostly used in pairs, with opening and closing brackets being mirror-symmetrical to each other. However, there are exceptions, for example with interval brackets and individual, non-paired brackets are sometimes used.

Grouping brackets in terms

Brackets group sub-terms and can thus change the ranking and order of the calculation or are only used to visually combine sub-terms. Round brackets are usually used here:

${\displaystyle a+b\cdot c}$	Since the multiplication has priority (" point calculation before line calculation "), this means that it is calculated first and added to the result. ${\displaystyle b\cdot c}$ ${\displaystyle a}$
${\displaystyle (a+b)\cdot c}$	The brackets indicate that the total should be calculated first and then multiplied by. ${\displaystyle a+b}$ ${\displaystyle c}$

In the case of complex terms or if special sub-terms are to be identified, these can be enclosed in square brackets.

Example:

${\displaystyle \left[(a+b)^{2}-(a+c)^{2}\right]^{2}-\left[(a+b)^{2}+(n^{2}-1)\right]^{2}}$

instead of

${\displaystyle \left((a+b)^{2}-(a+c)^{2}\right)^{2}-\left((a+b)^{2}+(n^{2}-1)\right)^{2}}$

The typographical size of a bracket is usually adapted to its hierarchical position, as in the last example.

Quantity brackets

When specifying quantities , curly brackets are usually used:

${\displaystyle M:=\{1,2^{2},3^{3},4^{4},\ldots ,n^{n},\ldots \}\cup \{x\mid x^{2}<2^{x}\}}$

Interval brackets

Different notations exist for intervals . The two most common are in the case of an open interval and a half-open interval : ${\displaystyle A=\{x\mid a<x<b\}}$ ${\displaystyle B=\{x\mid a\leq x<b\}}$

• ${\displaystyle A=\left]a;b\right[\quad B=\left[a;b\right[}$
• ${\displaystyle A=(a;b)\quad B=[a;b)}$

Instead of a semicolon, a comma is often used to separate the interval limits, if confusion with the decimal point is impossible.

Function arguments

Normally, the arguments of functions are put in round brackets, occasionally also in pointed brackets to make it easier to distinguish between grouped brackets ( f is a function, g a variable):

${\displaystyle f\left\langle g+h\right\rangle +g(h+j)}$ instead of ${\displaystyle f(g+h)+g(h+j)}$

Such a notation is mainly found where different functions appear in complex bracketed terms, for example in statistics:

${\displaystyle \operatorname {cov} \left\langle X_{1},X_{2}\right\rangle =E\left\langle (X_{1}-E\left\langle X_{1}\right\rangle )(X_{2}-E\left\langle X_{2}\right\rangle )\right\rangle }$

And brackets

If multiple statements are vertically grouped in a large brace, which means that this and be -linked. Example:

${\displaystyle \left\{{\begin{matrix}x\geq 3\\x\leq y\end{matrix}}\right\}}$is synonymous with . ${\displaystyle (x\geq 3)\;\wedge \;(x\leq y)}$

Special operators

Other brackets also used in pairs are special operators or functions:

• ${\displaystyle \left\lfloor x\right\rfloor }$(sometimes also ) denotes the largest whole number less than or equal to (" Gaussian bracket ") ${\displaystyle [x]}$ ${\displaystyle x}$
• ${\displaystyle \textstyle \left\lceil x\right\rceil }$ denotes the smallest integer greater than or equal to ${\displaystyle x}$
• ${\displaystyle \left|x\right|}$denotes the amount of ${\displaystyle x}$
• ${\displaystyle \langle x\rangle }$ is a notation for the mean or expected value of a quantity
• ${\displaystyle \textstyle {\binom {n}{k}}}$may include a binomial coefficients denote (when and are an integer ) or a matrix , this matrix may be a vector representing ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle n\geq k}$
• ${\displaystyle \textstyle s_{n,k}=\left[{n \atop k}\right]}$denotes the Stirling numbers of the first kind
• ${\displaystyle \textstyle S_{n,k}=\left\{\!{n \atop k}\!\right\}}$refers to the Stirling numbers of the second kind
• ${\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle }$is a scalar product or Cantor's pairing function from the individual vectors and ; the Bra-Ket notation is derived from this . ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {y} }$
• ${\displaystyle [A,B]=AB-BA}$is the commutator of the two operators and ${\displaystyle A}$ ${\displaystyle B}$
• ${\displaystyle [{\hat {A}},{\hat {B}}]_{+}={\hat {A}}{\hat {B}}+{\hat {B}}{\hat {A}}}$, the anti-commutator of two operators in the mathematical formalism of quantum mechanics . An alternative notation is . ${\displaystyle \{{\hat {A}},{\hat {B}}\}}$
• ${\displaystyle \textstyle \left\{F,G\right\}=\sum _{i=1}^{n}{\left({\frac {\partial F}{\partial q_{i}}}{\frac {\partial G}{\partial p_{i}}}-{\frac {\partial F}{\partial p_{i}}}{\frac {\partial G}{\partial q_{i}}}\right)}}$is the Poisson bracket , a bilinear differential operator in Hamiltonian mechanics .
• ${\displaystyle [F(x)]_{a}^{b}}$is the short form of the integral , see also integral calculus . ${\displaystyle \textstyle \int _{a}^{b}f(x)\,\mathrm {d} x}$
• ${\displaystyle (x)_{n},(x)^{n}}$denotes the falling or factorial . But risk of confusion with the Pochhammer symbol , which, depending on the author, is also represented as , or . ${\displaystyle x^{(n)}}$ ${\displaystyle \left(x_{n}\right)}$ ${\displaystyle (x,n)}$

Derivatives

Higher derivatives are often marked with an exponent in parentheses instead of derivative bars, which improves readability:

${\displaystyle f^{(4)}=f''''\,}$

This notation is also used when the number of derivatives is to be expressed in terms of a variable or term:

${\displaystyle f^{(n+1)}=f^{(n)}+f^{(n-1)}\,}$

Round brackets in statistical tables

According to DIN 55301 (design of statistical tables), round brackets, which enclose a value (number) in a table compartment, stand for "informational value limited, since the number is statistically uncertain" as a value-supplementary symbol, including quality indicators (as opposed to value-substituting symbols) in tables of official statistics .

Brackets in programming languages

Brackets have different meanings in different programming languages . However, certain meanings are fairly common:

Round brackets:

• Determination of the calculation order in terms (as in mathematics)
• Function arguments
• Type conversion operator (e.g. in C and C ++ )
• Index access to arrays (e.g. in BASIC )
• List creation (e.g. in LISP and related languages)
• Comment limits (e.g. in Forth )
• double parentheses are used by gcc for attributes.

Square brackets:

• Index access to arrays (e.g. in C and related languages)
• List operations (e.g. in Python , Logo and some others)
• Syntactic element for introducing a lambda expression; possibly contains the catch clause of the lambda expression (in C ++ 11)
• double square brackets are used for attributes in C ++ 11.

Curly brackets (also: "curved brackets"):

• Block boundaries (e.g. in C, C ++, Java , JavaScript , PHP , LilyPond )
• Array and structure literals (C, C ++, Java)
• Comments ( Pascal / Delphi )
• Precompiler ( Pascal / Delphi )
• Hash access ( Perl )
• Hash literals ( Ruby )
• Dictionary ( Python )

Angle brackets (only the ASCII characters <and>):

• Template arguments (e.g. in C ++, Java 1.5 or higher)
• Tag delimiters (e.g. in SGML , HTML , XML )

Brackets as graphic elements

:-)
(-:

• Brackets, especially the round bracket, are used in emoticons . Depicting a laughing mouth, depending on your orientation.
• Pointed brackets as the tips of arrows, supplemented by horizontal lines or equal signs as double lines. For marking text or representing a process, such as a causal chain.

See also : ((()))

swell

1. ↑ Duden: brackets
2. ↑ Stahel, Werner A .: Statistical data analysis. An introduction for scientists. 5th edition. Wiesbaden: Vieweg 2008.
3. ↑ Guidelines for the design of statistical tables for network programming, Working Group Publications of the State Statistical Offices, Wiesbaden 1997, 41 pages, here: page 36.
4. ↑ 5. Data Structures - Python 3.8.1 documentation. Retrieved January 19, 2020 .