# Braces – Definition With Examples

## Meaning of Braces in Math

Braces in math are symbols that are used twice, once to open “{“ and once to close “}” an argument, expression, or equation. These are commonly referred to as curly brackets and written as { }.

Brackets, on the other hand, is a term used to define different types of connections used in mathematics. For example,

• ( ), round brackets, which we call parentheses in math.
• { }, curly brackets, which are braces in math.
• [ ], square brackets, which we also call box brackets in math.

In general, we use braces in math for two purposes:

• For grouping a large equation, in which the second-last bracket is braces or curly brackets. For example, 7[2+{3(1+1) + 1}]
• For denoting a set, such as {x, y, z,…}

## How Do We Use Braces in Math?

Braces in math are frequently used in mathematical expressions when we have two or more than two nested groups for calculations.

So, in the first nested group, we use parentheses. In the second nested group, we use braces, and in the third nested group, we use box brackets, which contain both parentheses and braces.

For example: 3[2 – {4(2 + 2) + 2}]

Here, we have three nested groups with appropriate brackets.

So, the order of solving would be:

Fun Fact: Some conventions differentiate the order of solving brackets, which is:

We will use the first convention with curly brackets in the second position throughout this article.

You need to know the BODMAS or order of operations to solve the problem.

## What Is the Order of Operations?

When we have a long equation for multiplication, division, addition, and subtraction, we solve each function in order to find the right answer. If the problem is solved without this order, then the chances of getting a wrong answer are high!

PEMDAS (BODMAS) is one of the rules that define the most common way of following the order of operations. It stands for:

1. Parentheses or Brackets
2. Orders or Exponents
3. Divide
4. Multiply
6. Subtract

While we write the order in the above form, division or multiplication and addition or subtraction hold equal importance. This means that you can either take up multiplication first or division first.

Similarly, you can take either addition first or subtraction first. The answer will be the same. So, we usually try to solve these two from left to right.

Let’s solve the above example:

4[2 + {3(1 + 1) + 2}]

= 4[2 + {3(2) + 2}]

Now, we solve the braces or curly brackets.

= 4[2 + {6 + 2}]

= 4[2 + 8]

Then, we solve the square brackets.

= 4[10]

= 40

## Solved Examples

Example 1: If you have to solve the following equation, how will you proceed?

2[1 – {2(2 + 2) + 2}]

Solution: We solve the parentheses first:

= 2[1 – {2(4) + 2}]

= 2[1 – {8 + 2}]

Now, we solve the braces:

= 2[1 – {10}]

Finally, we solve the square brackets:

= 2[–9]

= –18

Example 2: How would you solve the following equation?

4{5(4 + 2) + 1}

Solution: First, we solve the parentheses:

= 4{5(6) + 1}

Now, we need to solve the curly brackets. But within these brackets, we have to solve multiplication and addition.

So, we multiply first and then add:

= 4{30 + 1}

= 4{31}

Finally, we multiply 4 with the value inside the braces:

= 124

Example 3: What is the process you will follow to solve an equation with more than one parentheses?

20/{1(2 + 2) + (3 + 3)}

Solution: We will start by solving the equations within the parentheses:

= 20/{1(4) + (3 + 3)}

= 20/{1(4) + (6)}

Now, we have to solve the equation within the braces, but we have multiplication within the curly brackets, so we will solve that first:

= 20/{4 + (6)}

= 20/{10}

= 2/1

= 2

## Practice Problems

1

### Solve the equation containing braces in math.57/{5 + (4 x 2) + (3 + 3)}

3
4
13
4
CorrectIncorrect
After solving the ( ), we perform addition within the { }, and then divide.
2

### Which of the following examples use braces, brackets, and parentheses correctly?

60/[(2 x 2) + (3 + 3)}
60/{(2 x 2) + (3 + 3)}
60/{[2 x 2] + (3 + 3)}
(60/{[2 x 2] + (3 + 3})
CorrectIncorrect
Correct answer is: 60/{(2 x 2) + (3 + 3)}
It uses the braces, brackets, and parentheses correctly because the innermost brackets have parentheses and then braces.
3

### If we have the following expressions inside the curly brackets, which of the expressions would you solve first?10/{(4/2) + (6 x 2) – (3 + 3) + (7 – 2)}

(4/2)
(4/2) or (6x2)
Any parentheses inside the { }, (4/2), (6 x 2), (3 + 3), (7 – 2)}
None of the above
CorrectIncorrect
Correct answer is: Any parentheses inside the { }, (4/2), (6 x 2), (3 + 3), (7 – 2)}
We can solve any of the parentheses inside the curly brackets first. Once these parentheses are solved, we have to simply add and subtract, which can be done in any order.