What is the difference between PEMDAS and BODMAS in order of operations?
Executive summary
PEMDAS and BODMAS are pedagogical acronyms that encode the same conventional “order of operations” used to disambiguate arithmetic and algebraic expressions: evaluate grouping symbols first, then powers/roots, then multiplication and division (as a tied pair, left-to-right), and finally addition and subtraction (as a tied pair, left-to-right) [1] [2] [3]. The difference is linguistic and regional—PEMDAS is common in the United States while BODMAS (or BIDMAS/BOMDAS) appears in the United Kingdom and Commonwealth countries—but not a difference in mathematical priority when the rules are applied correctly [1] [4] [5].
1. What the letters mean and why they exist
PEMDAS stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction; BODMAS stands for Brackets, Orders, Division/Multiplication, Addition/Subtraction (with “orders” meaning exponents and roots) and the acronyms were created as classroom mnemonics to standardize how students interpret written expressions [1] [4] [6].
2. The core mathematical rule beneath the acronyms
Mathematically, the key point is that multiplication and division share the same precedence and are performed in the order they appear from left to right, and the same holds for addition and subtraction—so PEMDAS’s ordering of M before D and BODMAS’s D before M are not intended to make one operation universally prior to the other but to remind learners of grouped precedence with left-to-right evaluation for equal-priority operators [2] [7] [8].
3. Where confusion comes from and how it shows up in practice
Confusion arises when learners treat the letter order as a strict ranking (e.g., “do multiplication always before division”), which produces contradictory results on expressions where division and multiplication intermix; educators and calculators instead enforce “do whichever comes first reading left-to-right,” and misinterpretations are a frequent source of classroom mistakes and online debates [7] [8].
4. Historical and regional context that created the acronyms
The formalization of order-of-operations mnemonics emerged as standardized textbooks spread in the late 19th and early 20th centuries, and differences in vocabulary—parentheses vs. brackets, exponents vs. orders—reflect regional English usage rather than alternate mathematical systems [1]. Different acronyms (BIDMAS, BEDMAS, GEMDAS, etc.) exist but point to the same operational hierarchy [9] [5].
5. Practical guidance for applying the rules correctly
To avoid ambiguity, always simplify expressions inside grouping symbols first, then evaluate exponents, then process multiplication and division left-to-right as they appear, and finally addition and subtraction left-to-right; when in doubt, add parentheses to make intended grouping explicit or use a calculator or programming language whose parsing rules are documented [3] [10] [8].
6. Sources, alternative views and what reporting doesn’t settle
Authoritative summaries across educational sites and reference works consistently state that PEMDAS and BODMAS encode the same convention and that the perceived difference arises from mnemonic ordering and language [1] [4] [5], though historical scholarship notes earlier disagreements about the precedence of multiplication vs division prior to the 20th century—an academic nuance that explains why mnemonic clarity was needed [1]. The provided sources do not supply primary curriculum policy texts from specific countries; therefore this account relies on educational summaries and math-education commentary rather than national standards documents [1] [4].