**Parentheses are fundamental symbols in mathematics**, integral to various branches like algebra, calculus, and number theory. They establish a specific order of operations or precedence rules, and clarify complex mathematical expressions.

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## Introduction to Parentheses in Mathematics

Consider the expression:

** 2 + 3 * 5** **= ?**

According to the order of operations (BIDMAS/BODMAS), **multiplication is performed before addition**, leading to an answer of 17.

However, if we introduce parentheses, as in:

(2 + 3) * 5 = ?

In this case, the addition operation takes precedence, and the result is 25.

## Parentheses in Functions and Expressions

In functions and complex mathematical expressions, parentheses are vital. They help identify dependent and independent variables, distinguish function names from multiplication, and express modifications in functions.

Consider the function:

f(x) = x^2

Here, x is the independent variable, and f(x) represents the dependent value for any x. In an expression like f(a+b), parentheses indicate that we should add ‘a’ and ‘b’ first before applying the function.

## Parentheses in Intervals and Matrices

In intervals, parentheses indicate an open end, excluding the end number of an interval. For example, in the interval (2, 5), numbers greater than 2 and less than 5 are included, but not 2 and 5.

Parentheses also define matrices, a rectangular array of numbers, symbols, or expressions. Consider a 2×2 matrix, (a, b; c, d), where each row and column is enclosed within parentheses, indicating the boundaries of the matrix.

## Congruence, GCD, and Parentheses

Parentheses define the modulo in congruence relations. For example, in the expression a ≡ b (mod m), parentheses denote the modulus.

When finding the Greatest Common Divisor (GCD), parentheses group terms in the Euclidean algorithm. Suppose we need to find the GCD of 48 and 18. We write GCD(48, 18) = 6, with parentheses grouping the two numbers.

Parentheses are invaluable in mathematics. Their wide-ranging applications contribute to the simplicity and clarity of mathematical processes. They streamline mathematical expressions, ensuring operations are conducted in the correct order. Mastering their use is fundamental in excelling in mathematical concepts from basic arithmetic to advanced number theory and calculus.

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