In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets [ ], braces { } and angle brackets ⟨ ⟩, are frequently used in mathematical notation. 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. Sometimes, for the clarity of reading, different kinds of brackets are used to express the same meaning of precedence in a single expression with deep nesting of sub-expressions. 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. 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×(3 + 4) is 14, is 2 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—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 , 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 . Functions The arguments to a function are frequently surrounded by brackets: . When there is little chance of ambiguity, it is common to omit