What do double parentheses mean in math?

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

What do double parentheses mean in math?
, the part of the expression within the parentheses,
What do double parentheses mean in math?
, 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,

What do double parentheses mean in math?
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

What do double parentheses mean in math?
, which means that values of the function
What do double parentheses mean in math?
are dependent upon the values of
What do double parentheses mean in math?
.

4. Large parentheses around two numbers, one above the other, denotes a binomial coefficient

What do double parentheses mean in math?
.

5. Parentheses around a set of two or more numbers, as in

What do double parentheses mean in math?
, denote an
What do double parentheses mean in math?
-tuple of numbers that are linked in some special way.

6. Large parentheses around an array of numbers, e.g.,

What do double parentheses mean in math?
indicate a matrix. (However, in this work, the symbol
What do double parentheses mean in math?
is used instead.)

7. Parentheses may also be used to denote the greatest common divisor, e.g.,

What do double parentheses mean in math?
.

8. Parentheses are used to denote a congruence, as in

What do double parentheses mean in math?
.