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
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 ...
( ),
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; 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 mathem ...
. 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
Christopher Clavius, SJ (25 March 1538 – 6 February 1612) was a Jesuit German mathematician, head of mathematicians at the Collegio Romano, and astronomer who was a member of the Vatican commission that accepted the proposed calendar inve ...
, 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 ...
.
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 ...
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 standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
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
:
.
Non-mathematical angled brackets include:
* and , used in East-Asian text quotation
* and , which are
dingbat
In typography, a dingbat (sometimes more formally known as a printer's ornament or printer's character) is an ornament, specifically, a glyph used in typesetting, often employed to create box frames, (similar to box-drawing characters) or as ...
s
There are additional dingbats with increased line thickness, and some angle quotation marks and deprecated characters.
Algebra
In
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
, 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 examp ...
.
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 ...
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 expression
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, f ...
s 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
, used in the definition of composition of two
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s, the parentheses around
serve to indicate that the indexing by ''
'' is applied to the composition
, and not just its last component
.
Functions
The arguments to a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
are frequently surrounded by brackets:
. When there is little chance of ambiguity, it is common to omit the parentheses around the argument altogether (e.g.,
).
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 ...
, 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 pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of two vectors is commonly written as
, but the notation (''a'', ''b'') is also used.
Intervals
Both parentheses, ( ), and square brackets,
can also be used to denote an
interval. The notation