What Is the Order of Operations?

What Is the Order of Operations?

The order of operations is a convention, not a mathematical law, that standardizes how we interpret expressions containing multiple operations. It prevents ambiguity. When you see 3 + 5 × 2, the order tells you to multiply before adding: 5 × 2 = 10, then 3 + 10 = 13. Without it, someone might add first (3 + 5 = 8, then 8 × 2 = 16), producing a different result for the same expression.

The standard hierarchy is summarized by PEMDAS in the US (BODMAS/BIDMAS in much of the Commonwealth):

• Parentheses (also brackets or grouping symbols), highest priority
• Exponents (powers and roots)
• Multiplication and Division, equal priority, solved left to right
• Addition and Subtraction, equal priority, solved left to right

That last point is where PEMDAS is often misinterpreted. The acronym suggests M comes before D, and A before S. In practice, multiplication and division are performed in the same step, working from left to right. Same for addition and subtraction.

A common example that catches people: 8 ÷ 4 × 2. Often misread as 8 ÷ (4 × 2) = 1, the correct application is left-to-right: 8 ÷ 4 = 2, then 2 × 2 = 4. The rule is unambiguous.

The best way to internalize the order is to work through increasingly complex examples.

Example 1: Simple expression
3 + 6 × 2
Step 1: Multiply 6 × 2 = 12
Step 2: Add 3 + 12 = 15

Example 2: Parentheses first
(3 + 6) × 2
Step 1: Parentheses 3 + 6 = 9
Step 2: Multiply 9 × 2 = 18
Note: The parentheses change the result completely.

Example 3: Nested parentheses and exponents
2 × (3 + 4²) ÷ 5
Step 1: Innermost, exponent: 4² = 16
Step 2: Inside parentheses: 3 + 16 = 19
Step 3: Now expression reads 2 × 19 ÷ 5
Step 4: Left to right: 2 × 19 = 38, then 38 ÷ 5 = 7.6

Example 4: Fraction bar as grouping
(4 + 8) / (2 + 1)
A fraction bar acts as an implicit grouping symbol. Evaluate numerator and denominator separately, then divide.
Step 1: Numerator 4 + 8 = 12
Step 2: Denominator 2 + 1 = 3
Step 3: 12 ÷ 3 = 4

Example 5: The infamous 8 ÷ 2(2 + 2)
This expression went viral on social media because it can be interpreted two ways, depending on whether the implicit multiplication 2(2+2) is treated as having higher priority than explicit division. The modern convention (used by professional mathematicians and calculators like the TI-84 Plus) treats implicit multiplication the same as explicit, so left-to-right applies: 8 ÷ 2 × 4 = 4 × 4 = 16. However, some older textbooks treat implied multiplication as binding tighter. The safest rule: when ambiguity arises, add an explicit multiplication sign or extra parentheses.

Despite the simplicity of PEMDAS, several scenarios cause consistent confusion.

Misunderstanding 1: PEMDAS as a strict sequence
People often think M always comes before D and A before S. The correct rule is equal priority, left to right. This distinction matters in expressions like 12 ÷ 3 × 2. Left-to-right gives 8. Reversing the multiplication and division would give 2.

Misunderstanding 2: The viral 8 ÷ 2(2 + 2) problem
As discussed above, this expression splits mathematicians. The correct modern answer using standard convention is 16, but many people (and some older textbooks) treat the implicit multiplication as grouping, yielding 1. The lesson: ambiguous notation is the real problem. Use parentheses to clarify intent whenever there could be doubt.

Misunderstanding 3: Negatives and exponents
Expressions like -3² are often misread as (-3)² = 9. By order of operations, the exponent is applied before the negation: -3² = -(3²) = -9. To square a negative number, parentheses are required: (-3)².

Misunderstanding 4: Nested grouping symbols
In expressions with parentheses inside brackets inside braces, for example 2 + [3 × (4 – 1)], work from the innermost outward. Evaluate (4 – 1) = 3, then [3 × 3] = 9, then 2 + 9 = 11.

Where the math breaks down in practice
Spreadsheets and programming languages follow the same hierarchy, but with small variations. In Excel, the exponent operator ^ is right-associative (2^3^2 = 2^(3^2) = 512), which differs from the left-to-right convention many expect. When in doubt, test a small expression in your tool to confirm its behavior.

In an era of calculators, spreadsheets, and AI, the order of operations might seem like a relic. It isn't. The convention is baked into every tool that processes numeric expressions.

Calculators and spreadsheets
Scientific calculators and spreadsheet programs like Excel and Google Sheets implement the order of operations automatically. If a formula contains = 10 + 5 * 2, the software multiplies before adding. Users who do not understand the hierarchy will get unexpected results. A 2023 survey by the National Council of Teachers of Mathematics found that roughly 60% of college students entering STEM courses incorrectly simplified expressions involving multiple operations.

Programming languages
Every general-purpose programming language, Python, JavaScript, C++, Java, and others, follows the same hierarchy with minor exceptions. Misunderstanding the order is a primary source of bugs in financial modeling code, scientific simulations, and even simple scripts. Learning PEMDAS translates directly to better programming.

Real-world estimation
Outside formal math, the order of operations governs how people estimate costs, split bills, and compare offers. A car salesman might say "take 20% off the MSRP, then add $500 in fees." Misinterpreting that as "add $500, then take 20% off" changes the outcome. Accurate estimation requires understanding what operations to apply and in which order.

Expert Tips

• Always start from the innermost parentheses and work outward, never skip levels.
• For expressions with multiplication and division, work left to right, not by the order of letters in PEMDAS.
• When typing into a calculator or spreadsheet, use explicit parentheses for any grouping you intend, don't rely on implicit multiplication conventions.
• If you need to square a negative number, always write it as (-3)^2, never -3^2.
• Fraction bars are grouping symbols: treat the entire numerator and denominator as if they are in parentheses.
• When in doubt, test a simple version of your expression in a calculator or spreadsheet to verify the tool's behavior.

Mistakes to Avoid

• Ignoring left-to-right for same-priority operations. Multiplication and division have equal weight; addition and subtraction do too. Always go left to right within those groups.
• Treating PEMDAS as four sequential steps. It's four levels of priority, not four mandatory rounds. M and D happen in one pass.
• Assuming implicit multiplication binds tighter. In modern convention, 2(3) means the same as 2 × 3, no extra priority. Use explicit parentheses if you want grouping.
• Forgetting that exponentiation is right-associative in many contexts. Expression 2^3^2 in Excel gives 2^9 = 512, not 8^2 = 64. Check your tool.

Pros and Cons

Bottom Line

The order of operations is a foundational convention that anyone who works with numbers, students, programmers, spreadsheet users, or anyone making everyday calculations, should know cold. PEMDAS is a useful memory aid, but understanding the equal-priority left-to-right rule for M/D and A/S is what separates reliable solvers from those who produce wrong answers in real-world settings. Learn the hierarchy, practice with varied examples, and always add parentheses to remove any trace of ambiguity.