Why does PEMDAS place multiplication before addition—historical and mathematical reasoning?
Executive summary
The convention that multiplication is performed before addition—memorably encoded in PEMDAS—grew out of the needs of algebraic notation in the 1600s and was later formalized in school textbooks as part of standard pedagogy around the 19th–20th centuries [1] [2]. Mathematically, the distributive relationship between multiplication and addition makes treating multiplication as a higher-level operation sensible and notationally compact; historically, the mnemonic devices and explicit “order of operations” language are relatively modern [1] [3].
1. Why the rule feels natural: distributivity and notation
Mathematicians treat multiplication as a way to combine repeated sums, and the distributive law (a(b + c) = ab + ac) makes multiplication structurally above addition in algebraic manipulations, so granting multiplication higher precedence lets concise expressions stand for unambiguous expanded sums without parentheses [1] [3].
2. The 17th-century origin story: algebraic symbols create the need
As algebraic notation matured in the 1600s, authors began to adopt conventions so that expressions could be written briefly without excessive words or grouping; historians trace the “multiplication-before-addition” convention back to work such as van Schooten’s 1646 edition of Vieta and to the general needs of early symbolic algebra [1] [4].
3. Formalization came much later: textbooks and mnemonics
While the implicit hierarchy existed among practitioners, the explicit phrase “order of operations” and pedagogical acronyms such as PEMDAS/BEDMAS were solidified only with the rise of standardized textbooks in the late 19th and early 20th centuries, as curricula sought uniform rules for learners [1] [2].
4. Practical textbook conventions—and some persistent disagreements
Even after textbooks codified the order, disagreement lingered about details such as whether multiplication strictly precedes division or whether they share equal rank and are evaluated left-to-right; notable commentators and historians like Florian Cajori recorded such debates as late as the 1920s [5] [1].
5. Pedagogy vs. mathematicians’ informal consensus
Many historians and educators note that mathematicians long relied on informal agreement rather than rigid rules—using parentheses to resolve ambiguity—while educators turned those informal conventions into mnemonic rules to teach students consistently; some sources emphasize that the mnemonic was more a teaching convenience than a deep mathematical decree [2] [6].
6. Edge cases and modern confusion: implicit multiplication and notation traps
Ambiguities remain in expressions with mixed notations—such as a/2b or implicit juxtaposition 2x versus explicit 2·x—where textbooks and software sometimes disagree about precedence; authorities recommend avoiding such notations or using parentheses to make intent explicit [2] [7] [5].
7. The mathematical rationale summarized
At root, the precedence of multiplication over addition is defensible because multiplication compresses repeated addition and because distributivity allows omission of parentheses when multiplication is understood to bind more tightly; historically this made symbolic algebra practical, and pedagogical formalization later followed to standardize learning [3] [1] [2].
Conclusion: convention born of algebraic sense and cemented by schooling
The placement of multiplication before addition in PEMDAS is not an arbitrary classroom quirk but a convention that arose from the algebraic structure of operations in the 1600s and was later packaged into explicit rules and mnemonics for education as textbooks standardized curricula—while remaining subject to technical caveats and occasional disagreements even into the 20th century [4] [1] [5].