# Brackets

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## What are Brackets?

The first question the student gets in this topic is “How can we define brackets”.

In evaluating an expression containing a bracketed sub-expression, brackets denote a type of grouping, the operators in the sub-expression take precedence over those surrounding it. Additionally, for the different brackets, there are many uses and definitions.

### Types of Brackets

The frequently used bracket types are:

• Parentheses ( )

• Square brackets [ ]

• Curly brackets { }

• Angle brackets ⟨ ⟩

### Parentheses

Among the four different types of brackets used, parentheses are the most commonly used bracket type.

In mathematical problems, the primary use of parentheses is to group numbers together. Use the order of operations to solve the problem when we see multiple numbers and operations in parentheses.

For three key purposes, parentheses are used in mathematics:

• To divide numbers for clarification.

• To signify multiplication.

• To group numbers together.

To separate numbers for clarification, parentheses may be used. For instance, if we have an additional problem with a negative number, to distinguish the two signs, parentheses will be used. In order to distinguish a number from its exponents, parentheses may also be used. Typically, this occurs if we lift a negative number to control.

### What is the Use of Brackets?

• Brackets especially Parentheses () are used in elementary algebra to define the order of operations. In accordance with the BODMAS rule, words within the bracket are evaluated first.

Ex: 5 * (2 + 4) is 30, (5 * 3) + 2 is 30.

• Brackets are often used in mathematical expressions in general to signify grouping where appropriate to prevent ambiguities and increase clarity.

• In the Cartesian system of coordinates, brackets are used to designate point coordinates.

Ex: (4,8) denotes the points on the x-y coordinate system with x-coordinate being 4 and y-coordinate being 8.

• The arguments for a function are always surrounded by parentheses.

Ex: f(x), g(x).

• For denoting an open end of an interval, a bracket may be used.

Ex: [0,8) denotes a half-closed interval that includes all real numbers, except 8 from 0 to 8.

• Wide parentheses around two numbers denote a binomial coefficient, one above the other.

• As in (a,b,c), parentheses around a set of two or more numbers denote an n-tuple of numbers which are connected in a particular way.

• A matrix is indicated by broad brackets around an array of numbers.

• To denote the largest common divisor, parentheses are used.

### BODMAS Rule

Brackets find their main application in BODMAS or PEMDAS rule where the sequence of operations to be performed when an expression is resolved. BODMAS or PEMDAS stands for:

B - Brackets, P- Parentheses

O - Order, E- Exponents

D - Division

M - Multiplication

S - Subtraction

The BODMAS rule explains the sequence of operations to be done until an expression is resolved. According to the BODMAS law, if there are brackets ((), {}, []) in an expression, we first have to overcome or simplify the bracket followed by the order, then divide, multiply, add and subtract from left to right. In the wrong order, solving the issue would result in a wrong answer.

### Basic Problems on Brackets and its Application:

1) Solve (2 + 4) - (6 - 3)

Ans: Two parentheses are involved in the given expression. We can solve both of them separately by BODMAS rule and then combine their results.

(2 + 4) = 6……….(1)

(6 - 3) = 3………..(2)

Now subtracting (1) with (2), we get

(2 + 4) - (6 - 3) = 6 - 3 = 3

2) Solve (3 + (5 * 4)) - ((4 * 6) - 10)

Ans: Four parentheses are involved in the given expression. We will solve it by using the BODMAS rule to find the answer.

First parentheses is (5 * 4) = 20……………………………..(1)

Second parentheses is (3 + (5*4))=(3 + 20) =23………(2)

Third parentheses is (4 * 6) = 24……………………………(3)

Fourth parentheses is ((4 * 6) - 10) = (24 - 10) = 14……(4)

Now subtract (2) and (4) we get

(3 + (5 * 4)) - ((4 * 6) - 10) = 23 -14 = 9.