What are Brackets?
You would have come across symbols such as “( ), { } or [ ]” in math textbooks. These symbols are called brackets. Brackets are unique symbols for grouping different expressions or numbers or letters (used as algebraic variables) together. When any particular expression is placed in brackets, its importance takes precedence over others.
What Are the Different Types of Brackets?
There are three kinds of brackets used for mathematical expressions:
- Parentheses or round brackets represented as: ‘( )’
- Curly or brace brackets represented as: ‘{ }’
- Square or box brackets represented as: ‘[ ]’
What Are Parentheses Brackets?
Parentheses brackets or round brackets are the most common types of brackets. They are mostly used to group together the terms of various expressions and equations. When round brackets are used for numbers placed beside each other, it denotes multiplication as an operation in a specific operation.
For example, \(\left( 4 \right)\left( 6 \right)=\text{ }4 \times 6 = 24\)
Parentheses are used to express negative integers in mathematical expressions or equations. For example, 7 + (-2) = 5
Parentheses are useful in distinguishing numbers from their exponents.
What Are Curly Brackets?
Curly brackets, ‘{ }’, are quite similar to round brackets, and are used for grouping different terms in mathematical expressions. But along with that, curly brackets are used for expressing math sets or nested expressions.
What are Square Brackets?
Square brackets, ‘[ ]’, are useful in differentiating between sub-expressions of a complex mathematical expression. For example, [50 – (6 – 2) + (3 x 6)].
What Is the Order of Operations for Brackets?
While evaluating a mathematical expression involving different brackets, the order of operations of brackets is followed. The general order of operations of brackets states, ‘[ { ( ) } ]’, as the order. Here, the values in the innermost brackets (round brackets) are evaluated first, followed by the second brackets (curly brackets), and finally the outer brackets (square brackets). The expressions are solved from left to right, depending on the operator. The use of the PEMDAS rule will help in solving such mathematical problems.
Solved Examples
Example 1: Find the value of the expression, (6 + 2) – (5 – 2)
Solution:
The expressions are given in round brackets.
So, solve the expressions in the brackets individually.
So, (6 + 2) – (5 – 2)
(8) – (3) [Solving the values in the brackets]
(8) – (3) = 5 [Subtract]
Hence, the value of the expression, (6 + 2) – (5 – 2), is 5.
Example 2: Find the value of the expression, {(8 − 3) × 5} ÷ 5.
Solution:
The given expression is, {(8 − 3) × 5} ÷ 5
First, we will solve the parentheses.
{(8 − 3) × 5} ÷ 5
= {(5) × 5} ÷ 5
= {5 × 5} ÷ 5
Now, we will solve for the curly brackets,
{5 × 5} ÷ 5
={25} ÷ 5
= 25 ÷ 5
= 5
Hence, the value of the expression, {(8 − 3) × 5} ÷ 5 is 5.
Example 3: Find the value of [{(4 + 27) × 16} −(20 ÷ 5)]
Solution:
In order to solve this expression, we will use the PEMDAS rule.
First, we will solve the values in ‘( )’ brackets,
[{(4 + 27) × 16} − (20 ÷ 5)]
= [{(31) × 16} −(4)]
Now, we will solve the values in the curly ‘{ }’ brackets,
[{31 × 16} −4]
=[{496} −4]
=[496 −4]
Finally, we will solve for the values in the square brackets,
[496 – 4]
= 492
So, the value of the expression, [{(4 + 27) × 16} −(20 ÷ 5)], is 492.
Brackets are useful for mathematical expressions, denoting the coordinates of a point on a map, or describing the variable of a function.
Brackets are important for mathematical equations in which the use of brackets differentiates the components or terms, and arranges them based on the ‘order of operations’.
Angle brackets represented as ‘< >’ are usually used to express a list of numbers or a sequence of numbers.
The full form of PEMDAS is,
P – Parentheses
E – Exponents
M – Multiplication
D – Division
A – Addition
S – Subtraction