Brackets and Parentheses in Math: Definition, Method and Examples

What are brackets and parentheses?

Brackets and parentheses are symbols used in pairs to group numbers, variables, and operations together. They organize mathematical ideas and clarify the exact sequence in which to evaluate numerical expressions.

Without grouping symbols, we read expressions using standard priority rules. When we introduce a set of brackets, we force the enclosed part of the expression to be calculated before anything else.

Types of grouping symbols

Mathematics uses several different shapes of grouping symbols to keep expressions readable. While they look different, their primary function in arithmetic is the same.

Parentheses are the most common grouping symbols. When an expression requires multiple layers of grouping, we use square brackets and curly braces to make the different pairs easier to see.

• Parentheses: Written as $()$. These are also called round brackets. They are the standard choice for grouping numbers and operations.
• Square Brackets: Written as $[]$. These are typically used as an outer layer of grouping when parentheses are already used inside.
• Curly Braces: Written as $\{\}$. These act as a third layer of grouping in complex expressions. They are also universally used to define a set of numbers, such as $\{2,4,6,8\}$.
• Angle Brackets: Written as $⟨⟩$. These are sometimes used in advanced mathematics and physics. However, they are rarely used for standard grouping because they can be easily confused with the less-than and greater-than inequality signs.

How brackets change an expression

Adding brackets to an expression often changes its final value because it changes the operation sequence.

Consider the expression $5+3\times 4$. If we evaluate it without any grouping symbols, standard multiplication takes priority over addition. The result is $5+12=17$.

If we add parentheses around the addition, we get $(5+3)\times 4$. The grouping symbols force us to add first. The new result becomes $8\times 4=32$.

Nested grouping symbols

When an expression contains brackets inside other brackets, the grouping symbols are nested. Always evaluate the innermost set of brackets first, then work outward.

Always evaluate the innermost pair of grouping symbols first.

For example, in the expression $[10+(2\times 3)]\times 5$, the round parentheses are completely inside the square brackets. We evaluate the multiplication inside the parentheses first to get $[10+6]\times 5$. Then we complete the operation inside the square brackets to get $16\times 5$, which equals $80$.

Brackets in the order of operations

Brackets form the first step in the formal order of operations. This mathematical priority system ensures everyone calculates the same answer for a given expression.

Many learners memorize this system using acronyms like PEMDAS, where the "P" stands for Parentheses. In other regions, the accepted acronym is BODMAS, where the "B" stands for Brackets. Both terms mean exactly the same thing: resolve any operations enclosed in grouping symbols before calculating exponents, multiplication, division, addition, or subtraction.

Worked examples

The following examples demonstrate how to correctly identify and evaluate expressions containing grouping symbols.

Example 1: Evaluating a single pair of parentheses

Question: Evaluate $(12-5)\times 3$.

Method:

1. Identify the operations inside the parentheses.
2. Perform the subtraction first because it is grouped.
3. Multiply the result by the outside number.

Answer: $(12-5)\times 3=7\times 3=21$.

Check: Without parentheses, the multiplication would happen first, yielding $12-15=-3$. The parentheses correctly changed the operation order to yield $21$.

Example 2: Evaluating multiple pairs of parentheses

Question: Evaluate $(4+6)÷(7-2)$.

Method:

1. Evaluate the first set of parentheses independently.
2. Evaluate the second set of parentheses independently.
3. Perform the division using the two results.

Answer: $(4+6)÷(7-2)=10÷5=2$.

Check: Since the addition and subtraction are fully grouped, neither can interact with the division symbol until they are resolved. $10÷5$ correctly yields $2$.

Example 3: Evaluating nested grouping symbols

Question: Evaluate $2\times [15-(3+4)]$.

Method:

1. Locate the innermost grouping symbols, which are the round parentheses.
2. Add the numbers inside the parentheses.
3. Subtract that result from the number inside the outer square brackets.
4. Multiply the final bracketed result by the outside number.

Answer: $2\times [15-7]=2\times [8]=16$.

Check: Working from the inside out ensures we calculate $15-7$ before we attempt to multiply by $2$. The steps are valid.

Common mistakes

When working with brackets and parentheses, learners often misapply priority rules or misinterpret notation. Avoid these frequent errors to ensure accurate calculations.

Ignoring grouping priority

The most common mistake is reading an expression from left to right and ignoring the brackets entirely. Always scan the entire expression for grouping symbols before performing any arithmetic.

Failing to apply the outside operation to the entire bracketed result

If a number sits directly next to a bracket, it implies multiplication. Sometimes, learners multiply only the first term inside. However, you must either resolve the entire inside first, or properly use the distributive property of multiplication to multiply the outside term by every term inside.

Frequently asked questions

Do parentheses mean multiplication?

Parentheses primarily mean grouping. However, when a number is written immediately adjacent to an opening parenthesis, such as $4(5)$, it indicates multiplication. The expression is equivalent to $4\times 5$.

What is the difference between brackets and parentheses?

Mathematically, they serve the exact same grouping function. Parentheses $()$ are simply the standard rounded shape. Square brackets $[]$ are used visually as an outer layer when parentheses are already present inside, making the expression easier to read.

Can an expression have brackets but no parentheses?

Yes. While it is unconventional, an expression written as $[8+2]\times 3$ is mathematically valid and behaves exactly like $(8+2)\times 3$. By convention, we generally use standard parentheses for the first layer of grouping.