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
:
.
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
, used in the definition of composition of two
natural transformations, 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 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 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
, but the notation (''a'', ''b'') is also used.
Intervals
Both parentheses, ( ), and square brackets,
can also be used to denote an
interval. The notation