Why 1 + 2 × 3 equals seven, not nine
Order of operations ranks arithmetic so expressions stay unambiguous. Multiplication outranks addition—hence 1 + 2 × 3 is 7, not 9—a convention baked into modern algebra. Parentheses override the ranking; calculators usually resolve equal-precedence ops left to right.
Precedence lets notation stay brief without constant brackets. Inside parentheses one still obeys the hierarchy, working inward-out if nests stack. Functional or Polish notation can encode order in the writing itself, making the infix convention unnecessary. Exponents, roots with vinculum bars, and function parentheses add further grouping habits shared across math, science, and most programming languages.
Replace division by multiplying reciprocals and associative laws free factor order inside terms—another reason textbooks drill PEMDAS-style mnemonics.
The rules are social contracts among calculators, not laws of nature; break them with parentheses whenever clarity demands.
Some cases have no settled answer. Mathematicians read −3² as −(3²), giving −9, yet Excel gives the unary minus priority over exponents. No universal rule governs mixing ÷ with ×, and multiplication written by juxtaposition often binds tighter, which is why the meme 8 ÷ 2(2 + 2) splits readers between 1 and 16; the Physical Review advises writing (a/b)/c or a/(b/c) rather than a/b/c. Stacked exponents are normally worked from the top down, but with a caret, MATLAB and Excel group a^b^c from the left while Google Search and Wolfram Alpha group it from the right. Nested brackets are cleared from the innermost outward, with square or curly braces available to aid reading. American and French pupils learn PEMDAS, German ones learn dot operations before line operations, and critics say such acronyms breed errors like doing every addition before any subtraction.
Source: Order of operations
