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.
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:
Non-mathematical angled brackets include: There are additional dingbats with increased line thickness,[5] a lot of angle quotation marks and deprecated characters.
This notation is extended to cover more general algebra involving variables: for example (x + y) × (x − y).
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.
The arguments to a function are frequently surrounded by brackets:
With some standard function when there is little chance of ambiguity, it is common to omit the parentheses around the argument altogether (e.g.,
Both parentheses, ( ), and square brackets, [ ], can also be used to denote an interval.
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
For example, {a,b,c} 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: The notation stands for the n-th derivative of function f, applied to argument x.
is used to denote the falling factorial, an n-th degree polynomial defined by Alternatively, the same notation may be encountered as representing the rising factorial, also called "Pochhammer symbol".
It can be defined by 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
In statistical mechanics, angle brackets denote ensemble or time average.
Square brackets are used to contain the variable(s) in polynomial rings.
is the ring of polynomials with real number coefficients and variable
This subring consists of all the elements that can be obtained, starting from the elements of A and b, by repeated addition and multiplication; equivalently, it is the smallest subring of B that contains A and b.
is the subring of Q consisting of all rational numbers whose denominator is a power of 2.
In ring theory, the commutator [a,b] is defined as ab − ba.
Furthermore, braces may be used to denote the anticommutator: {a,b} is defined as ab + ba.
However, Square brackets, as in [π] = 3, 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 ]π[ = 4.
Braces, as in {π} < 1/7, may denote the fractional part of a real number.