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Order of Operations (PEMDAS)

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The standard order for evaluating expressions: parentheses, exponents, multiplication/division, addition/subtraction.

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Order of operations

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  1. 1Order of operations
    A collection of conventions about which arithmetic operations to perform first in order to evaluate a given mathematical expression
  2. 2Precedence in order of operations
    The rank of an operation; an operation with higher precedence is performed before operations with lower precedence
  3. 3Four levels of the standard order of operations
    1) Parentheses, 2) Exponentiation, 3) Multiplication and division, 4) Addition and subtraction
  4. 4PEMDAS
    A mnemonic acronym standing for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction, common in the United States
  5. 5What phrase is often used to remember PEMDAS?
    "Please Excuse My Dear Aunt Sally"
  6. 6Value of 1 + 2 × 3
    7 (multiplication is performed before addition: 1 + (2 × 3) = 7, not (1 + 2) × 3 = 9)
  7. 7Value of 3 + 5²
    28 (exponentiation before addition: 3 + 25 = 28)
  8. 8Value of 3 × 5²
    75 (exponentiation before multiplication: 3 × 25 = 75)
  9. 9Why do order of operations conventions exist?
    To avoid notational ambiguity while allowing notation to remain brief
  10. 10What can be used to override precedence conventions?
    Parentheses ( ), or other brackets like [ ] or { } to avoid confusion with nested parentheses
  11. 11Value of (2 + 3) × 4
    20 (parentheses force addition to precede multiplication)
  12. 12Infix notation
    The usual mathematical notation where operators appear between operands (e.g., 2 + 3)
  13. 13Standard direction for evaluating operations of the same precedence
    From left to right
  14. 14Radical symbol
    The square root symbol √, traditionally extended by a bar (vinculum) over the radicand to avoid needing parentheses
  15. 15How is the expression sin 3x typically interpreted?
    As sin(3x), because monomials are conventionally included within function arguments without parentheses
  16. 16How is -3² typically interpreted in written mathematics?
    As -(3²) = -9 (exponentiation before negation)
  17. 17Implied multiplication
    Multiplication denoted by juxtaposition (e.g., 2x instead of 2·x or 2×x), also known as multiplication by juxtaposition
  18. 18How is 1/2n often interpreted in academic literature?
    As 1/(2·n) rather than (1/2)·n, giving implied multiplication higher precedence than division
  19. 19Ambiguity in expressions like "8 ÷ 2(2 + 2)"
    It can be interpreted as 8 ÷ [2·(2 + 2)] = 1 or as (8 ÷ 2)·(2 + 2) = 16, depending on whether implied multiplication has higher precedence
  20. 20How is a^b^c typically evaluated in written mathematics?
    As a^(b^c), working from top down (right to left)
  21. 21German mnemonic for order of operations
    "Punktrechnung vor Strichrechnung" (dot operations before line operations), referring to · (multiplication), : (division), + (addition), − (subtraction)
  22. 22What common mistake do students make with PEMDAS?
    Interpreting it as "addition first, subtraction afterward" instead of "addition and subtraction have equal precedence, evaluated left to right"
  23. 23Operator associativity
    The order within any single precedence level, usually left to right (left associative) but sometimes right to left (right associative)
  24. 24What criticism has been made of mnemonic acronyms like PEMDAS?
    They don't develop conceptual understanding, don't address purpose or flexibility, and students routinely make mistakes; they also crowd out substantive mathematical content

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