Order of Operations

Arithmetic
rule

Also known as: PEMDAS, BODMAS, operator precedence

Grade 3-5

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The agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Required for unambiguous communication in math and programming.

This concept is covered in depth in our order of operations and math properties, with worked examples, practice problems, and common mistakes.

Definition

The agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

💡 Intuition

Without rules, 2 + 3 \times 4 could mean 20 or 14. We agree to multiply first: 14.

🎯 Core Idea

Convention ensures everyone gets the same answer from the same expression.

Example

2 + 3 \times 4 = 2 + 12 = 14 not (2+3) \times 4 = 20

Formula

Parentheses \to Exponents \to Multiplication/Division (left to right) \to Addition/Subtraction (left to right)

Notation

PEMDAS (or BODMAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

🌟 Why It Matters

Required for unambiguous communication in math and programming.

💭 Hint When Stuck

Try breaking the problem into smaller steps -- compute each operation one at a time following PEMDAS order.

Formal View

\text{eval}(E) \text{ is defined recursively: parenthesized sub-expressions first, then } \wedge, \text{ then } \{\times, \div\} \text{ left-to-right, then } \{+, -\} \text{ left-to-right}

🚧 Common Stuck Point

Forgetting that \times and \div have equal precedence (go left to right).

⚠️ Common Mistakes

  • Doing operations left to right without considering precedence
  • Forgetting parentheses first

Frequently Asked Questions

What is Order of Operations in Math?

The agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Why is Order of Operations important?

Required for unambiguous communication in math and programming.

What do students usually get wrong about Order of Operations?

Forgetting that \times and \div have equal precedence (go left to right).

What should I learn before Order of Operations?

Before studying Order of Operations, you should understand: addition, subtraction, multiplication, division.

How Order of Operations Connects to Other Ideas

To understand order of operations, you should first be comfortable with addition, subtraction, multiplication and division. Once you have a solid grasp of order of operations, you can move on to expressions and equations.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Commutative, Associative, and Distributive Properties →
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Visualization

Static

Visual representation of Order of Operations