In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses ( ), square brackets [ ], curly brackets { }, and angle brackets < >. In the typical use, a mathematical expression is enclosed between an "opening bracket" and a matching "closing bracket". Generally such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed subexpression, the operators in the subexpression take precedence over those surrounding it. Additionally, there are several specific uses and meanings for the various brackets.
Historically, other notations, such as the vinculum, were similarly used for grouping; in presentday use, these notations have all specific meanings.
In the Z formal specification language, curly braces denote a set and angle brackets denote a sequence.
A variety of different symbols are used to represent angle brackets. In email and other ASCII text it is common to use the lessthan (<) and greaterthan (>) signs to represent angle brackets. Unicode has three pairs of dedicated characters: U+2329 (〈) and U+232A (〉) (left/rightpointing angle bracket), U+27E8 (⟨) and U+27E9 (⟩) (mathematical left/right angle bracket), and U+3008 (〈) and U+3009 (〉) (left/right angle bracket). In LaTeX the markup is \langle and \rangle: .
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 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, to provide a visual distinction. This usage is not technically correct, but is quite common.
Also in mathematical expressions in general, parentheses are used to indicate grouping (that is, which parts belong together) when necessary to avoid ambiguities, or for the sake of clarity. For example, in the formula (εη)_{X} = ε_{X}η_{X}, used in the definition of composition of two natural transformations, the parentheses around εη serve to indicate that the indexing by X is applied to the composition εη, and not just its last component η.
The arguments to a function are frequently surrounded by brackets: f(x). It is common to omit the parentheses around the argument when there is little chance of ambiguity, thus: sinx.
In the cartesian coordinate system brackets are used to specify the coordinates of a point: (2,3) denotes the point with xcoordinate 2 and ycoordinate 3.
The inner product of two vectors is commonly written as , but the notation (a, b) is also used.
Both parentheses, ( ), and square brackets, [ ], can also be used to denote an interval. The notation [a, c) is used to indicate an interval from a to c that is inclusive of a but exclusive of c. That is, [5, 12) would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In Europe, the notation [5,12[ is also used for this.
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.
Curly brackets { } are used to identify the elements of a set: {a,b,c} denotes a set of three elements.
Angle brackets are used in group theory to write group presentations, and to denote the subgroup generated by a collection of elements.
An explicitly given matrix is commonly written between large round or square brackets:
The notation
stands for the nth derivative of function f, applied to argument x. So, for example, if f(x) = exp(λx), f^{(n)}(x) = λ^{n}exp(λx). This is to be contrasted with f^{n}(x) = f(f(...(f(x))...)), the nfold application of f to argument x.
The notation (x)_{n} is used to denote the falling factorial, an nth degree polynomial defined by
Confusingly, the same notation may be encountered as representing the rising factorial, also called "Pochhammer symbol". Another notation for the same is x^{(n)}. It can be defined by
In quantum mechanics, angle brackets are also used as part of Dirac's formalism, braket notation, to note vectors from the dual spaces of the Bra <A and the Ket B>.
In statistical mechanics, angle brackets denote ensemble or time average.
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^{−1}h^{−1}gh. In ring theory, the commutator [a,b] is defined as ab − ba. Furthermore, in ring theory, braces are used to denote the anticommutator where {a,b} is defined as ab + ba.
The Lie bracket of a Lie algebra is a binary operation denoted by [·, ·] : × → . 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 JacobiLie bracket.
Square brackets, as in [π] = 3, are sometimes used to denote the floor function, which rounds a real number down to the next integer. However the floor and ceiling functions are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing, as in or .
Curly brackets, as in {π} < ^{1}/_{7}, may denote the fractional part of a real number.
The notation (a, b) is sometimes used to denote the highest common factor of a and b. This may be extended to three or more arguments.
Unicode name  MATHEMATICAL LEFT ANGLE BRACKET 

Codepoint  U+27E8 
⟨
Punctuation


