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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
bracket 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. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
s of various typographical forms, such as parentheses ( ),
square brackets 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. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
nbsp; In word processing and digital typesetting, a non-breaking space, , also called NBSP, required space, hard space, or fixed space (though it is not of fixed width), is a space character that prevents an automatic line break at its position. In ...
braces and
angle brackets 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. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
⟨ ⟩, are frequently used in
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...
. Generally, such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Sometimes, for the clarity of reading, different kinds of brackets are used to express the same meaning of precedence in a single expression with deep nesting of sub-expressions. Historically, other notations, such as the vinculum, were similarly used for grouping. In present-day use, these notations all have specific meanings. The earliest use of brackets to indicate aggregation (i.e. grouping) was suggested in 1608 by Christopher Clavius, and in 1629 by
Albert Girard Albert Girard () (11 October 1595 in Saint-Mihiel, France − 8 December 1632 in Leiden, The Netherlands) was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra" and g ...
.


Symbols for representing angle brackets

A variety of different symbols are used to represent angle brackets. In e-mail and other
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
text, it is common to use the less-than (<) and greater-than (>) signs to represent angle brackets, because ASCII does not include angle brackets.
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
has pairs of dedicated characters; other than less-than and greater-than symbols, these include: * and * and * and * and * and , which are deprecated In
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
the markup is \langle and \rangle: \langle\ \rangle. Non-mathematical angled brackets include: * and , used in East-Asian text quotation * and , which are dingbats There are additional dingbats with increased line thickness, and some angle quotation marks and deprecated characters.


Algebra

In elementary algebra, parentheses ( ) are used to specify the
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For exampl ...
. Terms inside the bracket are evaluated first; hence 2×(3 + 4) is 14, is 2 and (2×3) + 4 is 10. This notation is extended to cover more general
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
involving variables: for example . Square brackets are also often used in place of a second set of parentheses when they are nested—so as to provide a visual distinction. In mathematical expressions in general, parentheses are also used to indicate grouping (i.e., which parts belong together) when necessary to avoid ambiguities and improve clarity. For example, in the formula (\varepsilon \eta)_X = \varepsilon_\eta_X, used in the definition of composition of two natural transformations, the parentheses around \varepsilon \eta serve to indicate that the indexing by ''X'' is applied to the composition \varepsilon \eta, and not just its last component \eta.


Functions

The arguments to a function are frequently surrounded by brackets: f(x) . When there is little chance of ambiguity, it is common to omit the parentheses around the argument altogether (e.g., \sin x).


Coordinates and vectors

In the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, brackets are used to specify the coordinates of a point. For example, (2,3) denotes the point with ''x''-coordinate 2 and ''y''-coordinate 3. The
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of two vectors is commonly written as \langle a, b\rangle, but the notation (''a'', ''b'') is also used.


Intervals

Both parentheses, ( ), and square brackets, can also be used to denote an interval. The notation finite number of 9s), but 12.0 is not included. In some European countries, the notation [5,12[ is also used for this, and wherever comma is used as decimal separator">finite set">finite number of 9s), but 12.0 is not included. In some European countries, the notation [5,12[ is also used for this, and wherever comma is used as decimal separator, semicolon might be used as a separator to avoid ambiguity (e.g., (0 ; 1)). The endpoint adjoining the square bracket is known as ''closed'', while 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
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
or negative infinity is used as an endpoint (in the case of intervals on the real number line), it is always considered ''open'' and adjoined to a parenthesis. The endpoint can be closed when considering intervals on the
extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
. A common convention in
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
is to define /math> as the set of positive integer numbers less or equal than n. That is, /math> would correspond to the set \.


Sets and groups

Braces are used to identify the elements of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
. For example, denotes a set of three elements ''a'', ''b'' and ''c''. Angle brackets ⟨ ⟩ are used in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
to specify group presentations, and to denote the subgroup or
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
generated by a collection of elements.


Matrices

An explicitly given
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is commonly written between large round or square brackets: :\begin 1 & -1 \\ 2 & 3 \end \quad\quad\begin c & d \end


Derivatives

The notation :f^(x) stands for the ''n''-th derivative of function ''f'', applied to argument ''x''. So, for example, if f(x) = \exp(\lambda x), then f^(x) = \lambda^n\exp(\lambda x). This is to be contrasted with f^n(x) = f(f(\ldots(f(x))\ldots)), the ''n''-fold application of ''f'' to argument ''x''.


Falling and rising factorial

The notation (x)_n is used to denote the ''
falling factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
'', an ''n''-th degree
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
defined by :(x)_n=x(x-1)(x-2)\cdots(x-n+1)=\frac. Alternatively, the same notation may be encountered as representing the ''rising factorial'', also called " Pochhammer symbol". Another notation for the same is x^. It can be defined by :x^=x(x+1)(x+2)\cdots(x+n-1)=\frac.


Quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, angle brackets are also used as part of
Dirac Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
's formalism,
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
, to denote vectors from the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
s of the bra \left\langle A\ and the ket \left, B\right\rangle. In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, angle brackets denote ensemble or time average.


Polynomial rings

Square brackets are used to contain the variable(s) in
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
s. For example, \mathbb /math> is the ring of polynomials with
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
coefficients and variable x.


Subring generated by an element or collection of elements

If is a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
of a ring , and is an element of , then denotes the subring of generated by and . This subring consists of all the elements that can be obtained, starting from the elements of and , by repeated addition and multiplication; equivalently, it is the smallest subring of that contains and . For example, \mathbf
sqrt In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
/math> is the smallest subring of containing all the integers and \sqrt; it consists of all numbers of the form m+n\sqrt, where and are arbitrary integers. Another example: \mathbf /2/math> is the subring of consisting of all rational numbers whose denominator is a power of . More generally, if is a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
of a ring , and b_1,\ldots,b_n \in B, then A _1,\ldots,b_n/math> denotes the subring of generated by and b_1,\ldots,b_n \in B. Even more generally, if is a subset of , then is the subring of generated by and .


Lie bracket and commutator

In
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and ring theory, square brackets are used to denote the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
. In group theory, the commutator /nowiki>''g'',''h''/nowiki> is commonly defined as ''g''−1''h''−1''gh''. In ring theory, the commutator /nowiki>''a'',''b''/nowiki> is defined as ''ab'' − ''ba''. Furthermore, braces may be used to denote the anticommutator: is defined as ''ab'' + ''ba''. The Lie bracket of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
is a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
denoted by cdot,\cdot\mathfrak\times\mathfrak\to\mathfrak. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. There are many different forms of Lie bracket, in particular the Lie derivative and the Jacobi–Lie bracket.


Floor/ceiling functions and fractional part

Square brackets, as in , are sometimes used to denote the floor function, which rounds a real number down to the next integer. Respectively, some authors use outwards pointing square brackets to denote the ceiling function, as in . However, the floor and ceiling functions are usually typeset with left and right square brackets where only the lower (for floor function) or upper (for ceiling function) horizontal bars are displayed, as in or . Braces, as in {{math, 1={π} < 1/7, may denote the
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
of a real number.


See also

*
Binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
* Bracket polynomial * Bra-ket notation *
Delimiter A delimiter is a sequence of one or more characters for specifying the boundary between separate, independent regions in plain text, mathematical expressions or other data streams. An example of a delimiter is the comma character, which acts a ...
*
Dyck language In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of square brackets and The set of Dyck words forms the Dyck language. Dyck words and language are named after the mathematici ...
*
Frölicher–Nijenhuis bracket In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection ...
*
Iverson bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. ...
*
Nijenhuis–Richardson bracket In mathematics, the algebraic bracket or Nijenhuis–Richardson bracket is a graded Lie algebra structure on the space of alternating multilinear forms of a vector space to itself, introduced by A. Nijenhuis and R. W. Richardson, Jr (1966, 196 ...
, also known as ''algebraic bracket''. * Pochhammer symbol *
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
*
Schouten–Nijenhuis bracket In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different ...


Notes

Arithmetic Mathematical notation