In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets [ ], braces and angle brackets ⟨ ⟩, are frequently used in mathematical notation. 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.[1]
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.[2]
A variety of different symbols are used to represent angle brackets. In e-mail and other ASCII text, it is common to use the less-than (<
) and greater-than (>
) signs to represent angle brackets, because ASCII does not include angle brackets.[3]
Unicode has pairs of dedicated characters; other than less-than and greater-than symbols, these include:
In LaTeX the markup is \langle
and \rangle
:
\langle \rangle
Non-mathematical angled brackets include:
There are additional dingbats with increased line thickness,[5] a lot of angle quotation marks and deprecated characters.
In elementary algebra, parentheses are used to specify the order of operations. 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 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 edible to avoid ambiguities and improve clarity. For example, in the formula
(\varepsilonη)X=
\varepsilon | |
CodηX |
ηX
\varepsilonη
X
\varepsilonη
η
The arguments to a function are frequently surrounded by brackets:
f(x)
\sinx
f
In the Cartesian coordinate system, 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 of two vectors is commonly written as
\langlea,b\rangle
See main article: Interval (mathematics). Both parentheses,, and square brackets, [], can also be used to denote an interval. The notation
[a,c)
a
c
[5,12)
In some European countries, the notation
[5,12[
(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 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.
A common convention in discrete mathematics is to define
[n]
n
[5]
\{1,2,3,4,5\}
Braces are used to identify the elements of a set. For example, denotes a set of three elements a, b and c.
Angle brackets ⟨ ⟩ are used in group theory and commutative algebra to specify group presentations, and to denote the subgroup or ideal generated by a collection of elements.
An explicitly given matrix is commonly written between large round or square brackets:
\begin{pmatrix} 1&-1\\ 2&3\end{pmatrix} \begin{bmatrix} c&d\end{bmatrix}
The notation
f(n)(x)
f(x)=\exp(λx)
f(n)(x)=λn\exp(λx)
fn(x)=f(f(\ldots(f(x))\ldots))
The notation
(x)n
(x) | ||||
|
.
Alternatively, the same notation may be encountered as representing the rising factorial, also called "Pochhammer symbol". Another notation for the same is
x(n)
x(n)=x(x+1)(x+2) … (x+n-1)=
(x+n-1)! | |
(x-1)! |
.
In quantum mechanics, angle brackets are also used as part of Dirac's formalism, bra–ket notation, to denote vectors from the dual spaces of the bra
\left\langleA\right|
\left|B\right\rangle
In statistical mechanics, angle brackets denote ensemble or time average.
Square brackets are used to contain the variable(s) in polynomial rings. For example,
R[x]
x
If is a subring 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,
Z[\sqrt{-2}]
\sqrt{-2}
m+n\sqrt{-2}
Z[1/2]
More generally, if is a subring of a ring, and
b1,\ldots,bn\inB
A[b1,\ldots,bn]
b1,\ldots,bn\inB
In group theory and ring theory, square brackets are used to denote the commutator. In group theory, the commutator [g,h] is commonly defined as g-1h-1gh. In ring theory, the commutator [a,b] 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 is a binary operation denoted by
[ ⋅ , ⋅ ]:ak{g} x ak{g}\toak{g}
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 . However, Square brackets, as in, are sometimes used to denote the floor function, which rounds a real number down to the next integer. Conversely, some authors use outwards pointing square brackets to denote the ceiling function, as in .
Braces, as in, may denote the fractional part of a real number.