# Parentheses, Braces, and Brackets in Math

## How These Symbols Help Determine the Order of Operations

You'll come across many symbols in mathematics and arithmetic. In fact, the language of math is written in symbols, with some text inserted as needed for clarification. Three important—and related—symbols you'll see often in math are parentheses, brackets, and braces, which you'll encounter frequently in prealgebra and algebra. That's why it's so important to understand the specific uses of these symbols in higher math.

## Using Parentheses ( )

Parentheses are used to group numbers or variables, or both. When you see a math problem containing parentheses, you need to use the order of operations to solve it. For example, take the problem: 9 - 5 ÷ (8 - 3) x 2 + 6

For this problem, you must calculate the operation within the parentheses first—even if it's an operation that would normally come after the other operations in the problem. In this problem, the multiplication and division operations would normally come before subtraction (minus), however, since 8 - 3 falls within the parentheses, you'd work out this part of the problem first. Once you've taken care of the calculation that falls within the parentheses, you'd remove them. In this case (8 - 3) becomes 5, so you would solve the problem as follows:

9 - 5 ÷ (8 - 3) x 2 + 6
= 9 - 5 ÷ 5 x 2 + 6
= 9 - 1 x 2 + 6
= 9 - 2 + 6
= 7 + 6
= 13

Note that per the order of operations, you'd work what's in the parentheses first, next, calculate numbers with exponents, and then multiply and/or divide, and finally, add or subtract. Multiplication and division, as well as addition and subtraction, hold an equal place in the order of operations, so you work these from left to right.

In the problem above, after taking care of the subtraction in the parentheses, you need to first divide 5 by 5, yielding 1; then multiply 1 by 2, yielding 2; then subtract 2 from 9, yielding 7; and then add 7 and 6, yielding a final answer of 13.

## Parentheses Can Also Mean Multiplication

In the problem: 3(2 + 5), the parentheses tell you to multiply. However, you wouldn't multiply until you complete the operation inside the parentheses—2 + 5—so you would solve the problem as follows:

3(2 + 5)
= 3(7)
= 21

## Examples of Brackets [ ]

Brackets are used after the parentheses to group numbers and variables as well. Typically, you'd use the parentheses first, then brackets, followed by braces. Here is an example of a problem using brackets:

4 - 3[4 - 2(6 - 3)] ÷ 3
= 4 - 3[4 - 2(3)] ÷ 3 (Do the operation in the parentheses first; leave the parentheses.)
= 4 - 3[4 - 6] ÷ 3 (Do the operation in the brackets.)
= 4 - 3[-2] ÷ 3 (The bracket informs you to multiply the number within, which is -3 x -2.)
= 4 + 6 ÷ 3
= 4 + 2
= 6

## Examples of Braces { }

Braces are also used to group numbers and variables. This example problem uses parentheses, brackets, and braces. Parentheses inside other parentheses (or brackets and braces) are also referred to as "nested parentheses." Remember, when you have parentheses inside brackets and braces, or nested parentheses, always work from the inside out:

2{1 + [4(2 + 1) + 3]}
= 2{1 + [4(3) + 3]}
= 2{1 + [12 + 3]}
= 2{1 + }
= 2{16}
= 32

## Notes About Parentheses, Brackets, and Braces

Parentheses, brackets, and braces are sometimes referred to as "round," "square," and "curly" brackets, respectively. Braces are also used in sets, as in:

{2, 3, 6, 8, 10...}

When working with nested parentheses, the order will always be parentheses, brackets, braces, as follows:

{[( )]}