Why does 2 + 3 × 4 equal 14 and not 20? Because mathematicians agreed on a specific order for evaluating expressions — and once you know the rules, every expression has exactly one correct answer.
Mathematics is a language, and like any language it needs grammar rules to be unambiguous. The order of operations is the grammar of arithmetic — it tells us which operations to perform first when an expression contains multiple operations.
Without an agreed order, the same expression could mean different things to different people. The expression 2 + 3 × 4 could be read as "(2 + 3) × 4 = 20" or as "2 + (3 × 4) = 14" depending on which operation you do first. Mathematicians worldwide agreed that multiplication comes before addition, so the answer is always 14.
The order of operations ensures every valid mathematical expression has exactly one correct answer. It's a convention — an agreement — not a law of nature. But once agreed upon, everyone can communicate mathematical ideas without ambiguity.
The acronym PEMDAS (used in the US) or BODMAS (used in the UK and elsewhere) describes the order:
Important: Multiplication and division are the same level — do them left to right. Same for addition and subtraction. These are pairs of equal priority, not four separate levels.
BODMAS stands for Brackets, Orders (exponents), Division, Multiplication, Addition, Subtraction. It's the same rule — the naming difference is regional. The operations at each level still have equal priority and are evaluated left to right.
1010 − 318151010 × 5 − 35047644 × 6242733 + 11718The order of operations isn't arbitrary. It reflects the way mathematical operations relate to each other in terms of "strength" or "binding power."
Multiplication is repeated addition: 3 × 4 means 4 + 4 + 4. When you see 2 + 3 × 4, the multiplication "binds" the 3 and 4 together more tightly than the addition binds 2 and 3. It makes sense to resolve that tight binding first, giving 2 + 12 = 14.
Similarly, exponentiation is repeated multiplication: 2³ = 2 × 2 × 2. Exponents bind more tightly than multiplication, so they go first.
Parentheses override everything because they explicitly tell you: "handle this group as a unit first." They're the escape hatch that lets you write any expression in any order you want, as long as you use parentheses to clarify.
Spreadsheets follow PEMDAS exactly. In Excel, =2+3*4 returns 14, not 20. If you want 20, write =(2+3)*4. Understanding order of operations helps you write formulas correctly in Excel, Google Sheets, and every programming language.
The most common error: seeing 8 + 2 × 5 and doing 8 + 2 = 10 first because it appears first from left to right. Multiplication always beats addition regardless of position. Left-to-right only applies within the same priority level.
PEMDAS lists Multiplication before Division, but they have equal priority. 12 ÷ 4 × 3 is not 12 ÷ 12 = 1. You do it left to right: (12 ÷ 4) × 3 = 3 × 3 = 9. Same for addition and subtraction — they're equal priority.
In 2 × 3², the exponent applies to 3 only, not to 2 × 3. 2 × 3² = 2 × 9 = 18. Only if there were parentheses — (2 × 3)² — would the exponent apply to the whole product: 6² = 36.
Once you master order of operations, see PEMDAS in action inside the quadratic formula.
Ready to go deeper? These are the best free resources for exploring this topic further.