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In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets [ ], braces { } and angle brackets ⟨ ⟩, are frequently used in mathematical notation.[1] 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.[2]

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.[3]

In elementary algebra, parentheses ( ) are used to specify the order of operations.[2] Terms inside the bracket are evaluated first; hence 2×(3 + 4) is 14, 20 ÷ (5(1 + 1)) is 2 and (2×3) + 4 is 10. This notation is extended to cover more general algebra involving variables: for example (x + y) × (xy). 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 a

In elementary algebra, parentheses ( ) are used to specify the order of operations.[2] Terms inside the bracket are evaluated first; hence 2×(3 + 4) is 14, 20 ÷ (5(1 + 1)) is 2 and (2×3) + 4 is 10. This notation is extended to cover more general algebra involving variables: for example (x + y) × (xy). 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

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 .

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 a , b {\displaystyle \langle a,b\rangle } ,[1] but the notation (a, b) is also used.

Both parentheses, ( ), and square brackets, [ ], can also be used to denote an interval.[1] 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. Here, 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 some European countries, the notation

In some European countries, the notation is also used for this, and wherever comma is used as decimal separator, semicolon might be used as a separator to avoid ambiguity (e.g., ).[7]

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.

Braces { } are used to identify the elements of a set. 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 Angle brackets are used in group theory and commutative algebra to specify group presentations, and to denote the subgroup[8] or ideal generated by a collection of elements.

An explicitly given matrix is commonly written between large round or square brackets: