Order of Operations
The order of operations is a universally agreed-upon set of rules that determines the sequence in which arithmetic and algebraic operations must be performed in any mathematical expression. Without these rules, the same expression could yield multiple different answers depending on the order in which a person chose to calculate. For example, the expression 2 + 3 × 4 could equal 20 if you add first, or 14 if you multiply first — and mathematicians worldwide have agreed that 14 is correct. This convention ensures that every person, calculator, and computer arrives at the same unambiguous result. The rules are commonly remembered using acronyms. In North America, the acronym PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. In the United Kingdom and many other countries, BODMAS or BIDMAS is used: Brackets, Orders (or Indices), Division, Multiplication, Addition, and Subtraction. Despite the different letters, both acronyms encode exactly the same hierarchy. A helpful mnemonic for PEMDAS is 'Please Excuse My Dear Aunt Sally.' It is critical to understand that multiplication and division have equal priority and are performed left to right, as are addition and subtraction. Order of operations is not merely a classroom convention — it is a foundational principle embedded into every programming language, scientific calculator, and computer algebra system. Understanding it deeply allows students to correctly simplify algebraic expressions, evaluate functions, and work confidently with formulas from physics, chemistry, finance, and beyond. Mastery of this concept is essential before advancing to equation solving, polynomial algebra, and calculus.
- Parentheses (Brackets)
- Exponents (Orders / Indices)
- Multiplication and Division (Left to Right)
- Addition and Subtraction (Left to Right)
- A Critical Nuance: Implied Multiplication
Parentheses (Brackets)
Always evaluate expressions inside parentheses (or brackets) first. This includes all grouping symbols: ( ), [ ], and { }. Work from the innermost pair outward. Example: 2 × (3 + 4) → evaluate (3 + 4) = 7 first, then 2 × 7 = 14. Nested example: 2 × [(1 + 3) × 5] → (1+3)=4, then 4×5=20, then 2×20=40.
Exponents (Orders / Indices)
Next, evaluate all exponents, powers, and roots. For instance, in 3 + 2³ × 5, first compute 2³ = 8, giving 3 + 8 × 5. Square roots and other radicals count as fractional exponents (√x = x^(1/2)) and are handled at this stage. Right-to-left associativity applies for stacked exponents: 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64.
Multiplication and Division (Left to Right)
Multiplication (×, ·, or implied) and division (÷, /) share the same level of priority. When both appear, process them strictly from left to right. Example: 12 ÷ 4 × 3 = (12 ÷ 4) × 3 = 3 × 3 = 9. Do NOT do 12 ÷ (4 × 3) = 12 ÷ 12 = 1. The left-to-right rule prevents ambiguity entirely.
Addition and Subtraction (Left to Right)
Addition (+) and subtraction (−) are last in the hierarchy and also share equal priority. Process them left to right. Example: 10 − 3 + 2 = (10 − 3) + 2 = 7 + 2 = 9. Doing it right to left would give 10 − (3 + 2) = 10 − 5 = 5, which is incorrect. Subtraction is equivalent to adding a negative number: a − b = a + (−b).
A Critical Nuance: Implied Multiplication
Expressions like 2(3 + 4) use implied multiplication (juxtaposition). In standard mathematical notation, this still follows the same hierarchy: evaluate parentheses first, then multiply. However, some calculators and software differ in how they handle expressions like 6/2(1+2), which is why explicit parentheses are always recommended in ambiguous cases. Always write grouping symbols clearly to eliminate ambiguity.
The Key Equation
- b^2 — Exponent term (dimensionless): Computed first inside the radical (square root), since exponents precede multiplication and subtraction at that level.
- 4ac — Product term (dimensionless): Multiplication of three terms, evaluated after the exponent b² but before the subtraction inside the radical.
- \sqrt{b^2 - 4ac} — Discriminant radical (dimensionless): The entire expression under the radical is a grouping (like parentheses), computed before adding −b or dividing by 2a.
- 2a — Denominator (dimensionless): Evaluated last: the full numerator (−b ± √(b²−4ac)) is divided by 2a only after all numerator operations are complete.
Worked Example
Problem. Evaluate the expression: 18 ÷ (2 + 4) × 3² − 7 + √(16)
Answer. 24
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz made enormous contributions to the standardization of mathematical notation, including the use of the × symbol for multiplication and the integral sign ∫. His careful attention to notation helped lay the groundwork for consistent conventions like order of operations. He co-invented calculus (independently of Newton) and championed the importance of rigorous symbolic manipulation in mathematics. Leibniz designed one of the first mechanical calculators, the 'Stepped Reckoner,' capable of performing multiplication and division — making him deeply aware of how ambiguous notation could produce computational errors.
Related Concepts
- Algebraic Expressions — Simplifying algebraic expressions requires applying order of operations to terms with variables, coefficients, and exponents.
- Laws of Exponents — Exponents occupy the second tier of PEMDAS; understanding exponent rules is essential for correctly evaluating expressions involving powers.
- Distributive Property — The distributive property interacts directly with order of operations: a(b + c) = ab + ac uses the priority of parentheses and multiplication.
- Operations with Fractions — Fraction bars act as implicit grouping symbols (like parentheses), requiring order-of-operations thinking for numerator and denominator separately.