Order of operations: PEMDAS, BODMAS and the rules explained
The order of operations is the convention for working out an expression: brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. So 3 + 4 × 2 = 11, not 14. PEMDAS in the United States and BODMAS in the United Kingdom are two names for the same rule.
Without an agreed order, 3 + 4 × 2 could be 14 or 11. The order of operations settles it, and it exists so that algebra can be written without a forest of brackets. This guide gives the rule, compares PEMDAS, BODMAS and BIDMAS, explains why multiplication and division share a step, how stacked powers are read and why 8 ÷ 2(2+2) divides the internet, with worked examples and a free calculator to check them.
Key ideas
- Brackets, then powers, then × and ÷ together from left to right, then + and − together from left to right.
- PEMDAS, BODMAS, BIDMAS and BEDMAS are one rule under different names; M before D in PEMDAS does not mean multiplication goes first.
- Stacked powers are read from the top down, so 2^3^2 = 2^9 = 512, and −3² = −9, because the power comes before the minus.
- 8 ÷ 2(2+2) is 16 when implied multiplication is treated like ×, and 1 when it binds tighter; brackets remove the doubt.
What is the order of operations?
The order of operations is a convention that says which part of an expression to work out first: brackets, then powers, then multiplication and division, then addition and subtraction. Operations of the same rank are done from left to right. With it, 3 + 4 × 2 has a single value, 11, because the multiplication comes before the addition.
The convention keeps notation short. A polynomial such as 17x⁸ + 2x⁷ + 3x + 4 is understood as (17(x⁸)) + (2(x⁷)) + (3x) + 4: powers first, then multiplications, then additions. If every operation were simply done from left to right, all those brackets would have to be written out. The rule that multiplication comes before addition has been part of algebraic notation since the 1600s.
Brackets override everything: (3 + 4) × 2 = 14. Inside brackets the same order applies, and nested brackets are worked from the innermost outwards: 7 + (6 × 5² + 3) = 7 + (6 × 25 + 3) = 7 + 153 = 160.
PEMDAS vs BODMAS vs BIDMAS: what is the difference?
Only the names differ. PEMDAS, common in the United States, stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, and is often remembered as “Please Excuse My Dear Aunt Sally”. BODMAS, used in the United Kingdom and other Commonwealth countries, stands for Brackets, Orders, Division and Multiplication, Addition and Subtraction. BIDMAS says Indices for powers, and BEDMAS is common in Canada and New Zealand.
All of them hide the same trap: the letters suggest a strict sequence that does not exist. Multiplication and division form one step, and so do addition and subtraction. PEMDAS puts M before D and BODMAS puts D before M, yet both mean the same thing. Reading the acronym literally, with every addition before any subtraction, gets 10 − 4 + 3 wrong: worked from left to right it is 9, not 3.
Why do multiplication and division go from left to right?
Because they have the same rank, they are worked in the order they appear. 20 − 10 ÷ 5 × 3 is 20 − 2 × 3 = 20 − 6 = 14; doing the multiplication first would give the wrong 20 − 10 ÷ 15. Chains of division are the sharpest test: 100 ÷ 10 ÷ 2 = 10 ÷ 2 = 5, not 100 ÷ 5 = 20.
Powers run the other way. A stack of exponents is read from the top down, so 2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64. Software is not unanimous here: some spreadsheet programs read 2^3^2 from left to right and return 64. A leading minus also comes after the power: −3² = −(3²) = −9, while (−3)² = 9, although some spreadsheet programs again give 9 for −3².
The famous 8 ÷ 2(2+2) shows where the convention runs out. The brackets give 8 ÷ 2(4). If the implied multiplication 2(4) is treated like any other ×, left to right gives 8 ÷ 2 × 4 = 16. If a number written right against a bracket counts as one unit, as is common in academic writing, it becomes 8 ÷ 8 = 1; the submission instructions of the journal Physical Review, for example, give multiplication precedence over division.
Neither reading is a blunder: the expression is ambiguous, and calculators disagree as well, with one widely used school model answering 1 for 6 ÷ 2(1+2) and another answering 9. The mathematician Hung-Hsi Wu calls puzzles of this kind contrived. The honest fix is to write what you mean: (8 ÷ 2)(2 + 2) = 16 or 8 ÷ (2(2 + 2)) = 1.
How do calculators apply the order of operations?
A simple calculator that works through the keys as you press them can ignore the rule: on a basic chain calculator, 1 + 2 × 3 gives 9, because it adds before it multiplies, while a scientific calculator gives 7. Before trusting a long calculation, test the calculator with 1 + 2 × 3.
Worked examples to try by hand and then check: 2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50. (2 + 3) × 4² = 5 × 16 = 80. 18 ÷ 3 × 2 = 6 × 2 = 12. 10 − 4 − 3 = 6 − 3 = 3, not 10 − 1 = 9.
The free scientific calculator at VITON13 SCHOOL follows the convention described here and shows the result as you type, so you can watch a bracket change the answer. Brackets come first, then functions such as sin or √ and signs written after a number, such as ! and %; powers are read from right to left, so 2^3^2 gives 512 and −2² gives −4. Then × and ÷ go from left to right, then + and −. A number written next to a bracket or π is multiplied like ×, so 8 ÷ 2(2+2) gives 16; type 8 ÷ (2(2+2)) for the other reading. The last 30 calculations stay in your browser, and a result equal to a simple fraction is shown as a fraction too.
In everyday language
The order of operations is like getting dressed: socks go on before shoes, whatever order you list them in. Brackets are the bag you open first, powers come next, then × and ÷, and + and − are the last layer, done from left to right.
Try it yourself
Open the scientific calculator and type 20 − 10 ÷ 5 × 3, then 8 ÷ 2(2+2), then 8 ÷ (2(2+2)).
Expected result
14 for the first, 16 for the second, because implied multiplication is treated like ×, and 1 for the third, where the brackets group 2(2+2) first.
Check your answer: Work out 6 + 2 × (5 − 1)² ÷ 4 by hand.
14. Brackets: 5 − 1 = 4. Power: 4² = 16. Then × and ÷ from left to right: 2 × 16 = 32 and 32 ÷ 4 = 8. Last, 6 + 8 = 14.
Questions
Do you always do multiplication before division?
No. Multiplication and division share one step and are worked from left to right, whichever comes first. 12 ÷ 3 × 2 is 8, not 2.
Is PEMDAS the same as BODMAS?
Yes. Both give brackets, then powers (exponents or orders), then multiplication and division together, then addition and subtraction together. Only the words and the order of the letters differ.
Learn the rule as four steps rather than six letters, and check your working with the free VITON13 SCHOOL calculator: type the expression, compare its answer with yours and add brackets wherever a reader might hesitate.
Sources and further reading
- Order of operations — Wikipedia ↗Sources checked:
- Precedence — Wolfram MathWorld ↗Sources checked:
- H. Wu, University of California, Berkeley: “Order of operations” and other oddities in school mathematics ↗Sources checked: