TheInfoList

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 ${\displaystyle (\varepsilon \eta )_{X}=\varepsilon _{Cod\,\eta _{X}}\eta _{X}}$, used in the definition of composition of two natural transformations, the parentheses around ${\displaystyle \varepsilon \eta }$ serve to indicate that the indexing by ${\displaystyle X}$ is applied to the composition ${\displaystyle \varepsilon \eta }$, and not just its last component ${\displaystyle \eta }$.

The arguments to a function are frequently surrounded by brackets: ${\displaystyle f(x)}$. When there is little chance of ambiguity, it is common to omit the parentheses around the argument altogether (e.g., ${\displaystyle \sin x}$).