One of the symbols ( or ) used to denote grouping. Parentheses have a great many specialized meanings in mathematics. A few of these are described below.
1. Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules). In an expression like , the part of the expression within the parentheses, , is evaluated first, and then this result is used in the rest of the expression. Nested parentheses work similarly, since parts of expressions within parentheses are also considered expressions. Parentheses are also used in this manner to clarify order of operations in confusing or abnormally large expressions.
2. A parenthesis can be used to denote an open end of an interval. For example, denotes the half-closed interval which includes all real numbers from 0 to 5 except 5 itself.
3. Parentheses are used to enclose the variables of a function in the form , which means that values of the function are dependent upon the values of .
4. Large parentheses around two numbers, one above the other, denotes a binomial coefficient .
5. Parentheses around a set of two or more numbers, as in , denote an -tuple of numbers that are linked in some special way.
6. Large parentheses around an array of numbers, e.g., indicate a matrix. (However, in this work, the symbol is used instead.)
7. Parentheses may also be used to denote the greatest common divisor, e.g., .
8. Parentheses are used to denote a congruence, as in .