Order of Operations: PEMDAS Rules and Common Mistakes
Ben Orlin draws on classroom teaching experience and a deep interest in making mathematics understandable. His practical knowledge connects mathematical ideas with everyday examples, helping readers explore concepts and approach learning with curiosity.
Quick answer
The order of operations is a set of rules that tells you which part of a math problem to solve first. This way, everyone gets the same answer. In the U.S., we use PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Multiplication and division share equal rank and are done left to right. Addition and subtraction work the same way.
What the Order of Operations Actually Is
The order of operations is a collection of conventions about which arithmetic operations to perform first in order to evaluate a given mathematical expression. The word convention matters: this is an agreed-upon standard, not a law of nature. It exists to avoid notational ambiguity while allowing notation to remain brief. Without it, the expression 1 + 2 × 3 could mean two different things depending on who reads it.
With the convention in place, everyone agrees: multiply first, then add. So 1 + 2 × 3 = 1 + 6 = 7, not 9. The rules apply across mathematics, science, technology, and most computer programming languages.
The PEMDAS Sequence, Level by Level
In the United States, the standard mnemonic is PEMDAS, standing for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. A common memory phrase is “Please Excuse My Dear Aunt Sally.” Here is what each level means in practice:
| Priority | Operation(s) | Direction when tied |
|---|---|---|
| 1 (highest) | Parentheses (and other grouping symbols) | Inside out |
| 2 | Exponents | Right to left for stacked powers |
| 3 | Multiplication and Division | Left to right |
| 4 (lowest) | Addition and Subtraction | Left to right |
Notice that multiplication and division share one priority level, and addition and subtraction share another. That shared status is the source of the most common student mistake, covered below.
The Biggest Misconception: M Before D, A Before S
Many students read PEMDAS and assume multiplication must always be done before division, and addition before subtraction. That is not correct. Operations at the same priority level are evaluated from left to right, not in the order the letters appear in the acronym.
For example, consider 10 ÷ 2 × 5. If you mistakenly do multiplication first, you get 10 ÷ 10 = 1. Working left to right correctly: (10 ÷ 2) × 5 = 5 × 5 = 25. The same logic applies to addition and subtraction: in the expression a − b + c, the correct reading is (a − b) + c, not a − (b + c). Those two give different answers whenever c is not zero.
The mnemonic is a memory aid, not a strict left-to-right priority list for every letter.
Worked Example (Illustrative)
Evaluate: 3 + 4 × 23 − (6 ÷ 3)
- Parentheses first: 6 ÷ 3 = 2. Expression becomes 3 + 4 × 23 − 2.
- Exponents next: 23 = 8. Expression becomes 3 + 4 × 8 − 2.
- Multiplication (left to right): 4 × 8 = 32. Expression becomes 3 + 32 − 2.
- Addition and subtraction (left to right): 3 + 32 = 35, then 35 − 2 = 33.
Verification: Substitute back mentally. The parenthetical group (6 ÷ 3) = 2 is confirmed. The exponent 23 = 8 is confirmed. 4 × 8 = 32, and 3 + 32 − 2 = 33. Each step checks out, so the answer is 33.
How Parentheses Change Everything
Parentheses let you override the default order whenever you need a different result. The rule is simple: evaluate whatever is inside the parentheses first, working from the innermost pair outward when they are nested.
Compare these two expressions:
- 2 + 3 × 4 = 2 + 12 = 14 (multiplication first, no parentheses)
- (2 + 3) × 4 = 5 × 4 = 20 (parentheses force addition first)
Square brackets [ ] and curly braces { } serve the same grouping purpose and are often used in nested expressions to make reading easier, as in [2 × (3 + 4)] − 5 = 9. For any situation where you want to test a specific mathematical claim, writing out parentheses explicitly removes all ambiguity about what you intend.
Where the Rules Genuinely Break Down
Two situations produce real ambiguity that PEMDAS alone cannot resolve.
Implied multiplication
When multiplication is shown by placing symbols next to each other (called implied or juxtaposed multiplication, as in 2n or 2π), many mathematicians and physics journals treat it as having higher priority than explicit division. So 1 / 2n is read as 1 / (2n), not (1/2) × n. This is the convention in academic literature, but it is not universal. The viral internet problem “8 ÷ 2(2 + 2)” has two defensible answers, 1 and 16, depending on whether you treat 2(2 + 2) as a single implied-multiplication unit. The safest fix is always to use explicit parentheses.
Calculator differences
Different calculators follow different orders of operations. A basic four-function calculator processes keystrokes in the order you press them, with no priority rules. Typing 1 + 2 × 3 on such a device yields 9. A scientific calculator applies standard priority and returns 7. Even among scientific calculators, implied multiplication is handled differently: a TI-82 reads 1/2x as 1/(2x), while a TI-83 and later TI models read it as (1/2)x. When you are unsure how your calculator will interpret an expression, add parentheses to make your intent explicit.
You can explore how these kinds of conventions shape other quantitative concepts in the Math topic hub.
Common Pitfalls to Watch For
- Treating M before D as absolute: Multiplication and division are equal in priority; work left to right.
- Treating A before S as absolute: Addition and subtraction are equal in priority; work left to right.
- Forgetting nested parentheses: Always start with the innermost group and work outward.
- Misreading the unary minus: In standard written math, −32 means −(32) = −9, not (−3)2 = 9. Some spreadsheet software reverses this, so check your tool.
- Trusting a simple calculator blindly: Basic calculators ignore priority rules entirely.
- Skipping steps: Doing two operations at once in your head is where sign errors and arithmetic slips happen most often.
Frequently asked questions
Does PEMDAS apply inside parentheses too?
Yes. Once you are working inside a set of parentheses, you apply the full PEMDAS sequence to whatever is inside. Parentheses tell you where to start, not that normal rules are suspended. For nested parentheses, finish the innermost group first, then move outward.
What does BODMAS mean, and is it different from PEMDAS?
BODMAS (Brackets, Order, Division/Multiplication, Addition/Subtraction) is used in the United Kingdom and Commonwealth countries. BEDMAS is common in Canada and New Zealand. All three mnemonics describe the same underlying priority sequence; the letter differences reflect regional terminology, not different math rules.
How do I handle a fraction bar in an expression?
A horizontal fraction bar acts as a grouping symbol for both the numerator and the denominator. Evaluate the top completely, evaluate the bottom completely, then divide. For example, (1 + 2) / (3 + 4) means you compute 3 and 7 separately before dividing to get 3/7.
Does the order of operations apply to algebra with variables?
Yes, the same priority sequence applies when variables are present. You simplify or evaluate expressions with variables in the same order: grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. You just cannot combine unlike terms (for example, 3x and 5 cannot be added until x has a value).
Why do viral math problems like '8 ÷ 2(2 + 2)' have two different answers?
These problems exploit a genuine gap in the standard rules: there is no universal convention for implied multiplication next to a parenthetical group. Depending on whether you treat 2(2 + 2) as a single unit or as separate multiplication, you get 1 or 16. Both interpretations have defenders. The real lesson is to write expressions with explicit parentheses to remove all doubt.
Sources
- Order of operations - Wikipedia · en.wikipedia.org
- Precedence -- from Wolfram MathWorld · mathworld.wolfram.com
- Order of operations - Wikipedia · en.wikipedia.org
Prepared from these references and checked against them. References retrieved October 11, 2026.
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