Language

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

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

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

Parentheses are used in mathematical notation to indicate grouping, often inducing a different order of operations. For example: in the usual order of algebraic operations, 4 × 3 + 2 equals 14, since the multiplication is done before the addition. However, 4 × (3 + 2) equals 20, because the parentheses override normal precedence, causing the addition to be done first. Some authors follow the convention in mathematical equations that, when parentheses have one level of nesting, the inner pair are parentheses and the outer pair are square brackets. Example:

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

Parentheses in programming languages

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

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

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).^[32][33]

In chemical nomenclature, parentheses are used to distinguish structural features and multipliers for clarity, for example in the polymer poly(methyl methacrylate).^[34]

Proofreading

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

Square brackets are used to denote parts of the text that need to be checked when preparing drafts prior to finalizing a document.

Law

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

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.^[45] 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.^[45]

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

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

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.

Music

In music, they are known as "accolades" or "braces", and connect two or more lines (staves) of music that are played simultaneously.^[58]

Chemistry

The use of braces in chemistry is an old notation that has long since been superseded by subscripted numbers.^[32] The chemical formula for water, H₂O, was represented as ${\displaystyle \left.{\stackrel {H}{H}}\right\}O}$.^[32]

Curly brackets in programming languages

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.^[60] 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.^[61]

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

In mathematics they delimit sets, in what is called set notation.^[62] Braces enclose either a literal list of set elements, or a rule that defines the set elements.^[62] For example:

• S = {a, b} defines a set S containing a and b.^[62]
• S = {x | x > 0} defines a set S containing elements (implied to be numbers) x₀, x₁, and so on where every x_n satisfies the rule that it is greater than zero.^[62]

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 .

Look up curly bracket, brace, { }, {, }, squiggly, or accolade in Wiktionary, the free dictionary.

Angle brackets in mathematics

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

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:

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

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

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.

Look up angle bracket, ⟨ ⟩, < >, broket, pointy bracket, or diamond bracket in Wiktionary, the free dictionary.