In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses, square brackets , braces, and angle brackets ⟨ ⟩. 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. Additionally, there are several uses and meanings for the various brackets. 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 was suggested in 1608 by Christopher Clavius and in 1629 by Albert Girard.
Symbols for representing angle brackets
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. Unicode 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 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. Terms inside the bracket are evaluated first; hence 2× is 14 and is 2 and + 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. Also in mathematical expressionsin general, parentheses are used to indicate grouping when necessary to avoid ambiguities, or for the sake of clarity. For example, in the formula X = εCod ηXηX, 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.
Functions
The arguments to a function are frequently surrounded by brackets:. It is common to omit the parentheses around the argument when there is little chance of ambiguity, thus:.
Coordinates and vectors
In the cartesian coordinate system brackets are used to specify the coordinates of a point: denotes the point with x-coordinate 2 and y-coordinate 3. The inner product of two vectors is commonly written as, but the notation is also used.
Intervals
Both parentheses,, and square brackets, , can also be used to denote an interval. The notation is used to indicate an interval from a to c that is inclusive of but exclusive of. That is, 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, but 12.0 is not included. In some European countries, the notation 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.
Sets and groups
Braces are used to identify the elements of a set: denotes a set of three elements. 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.
Matrices
An explicitly given matrix is commonly written between large round or square brackets:
Derivatives
The notation stands for the n-th derivative of function f, applied to argument x. So, for example, if, then. This is to be contrasted with, the n-fold application of f to argument x.
The notation n is used to denote the falling factorial, an n-th 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. It can be defined by
Quantum mechanics
In quantum mechanics, angle brackets are also used as part of Dirac's formalism, bra–ket notation, to note vectors from the dual spaces of the bra and the ket . 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 is commonly defined as g−1h−1gh. In ring theory, the commutator is defined as ab − ba. Furthermore, in theory, braces are used to denote the anticommutator where 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 Jacobi–Lie bracket.
Square brackets, as in, 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 only the lower or upper horizontal bars are displayed, as in or. Braces, as in, may denote the fractional part of a real number.
