Order of Operations (PEMDAS)
Why 2 + 3 × 4 is 14 and not 20, and the order every calculator agrees on.
The explanation
If everyone evaluated expressions in their own order, the same expression would have different answers. So mathematics fixed one order and everybody uses it.
Parentheses first. Then exponents. Then multiplication and division. Then addition and subtraction.
The part people miss: multiplication and division are the *same* rank, done left to right as you meet them. Same for addition and subtraction. In 12 ÷ 3 × 2 you do 12 ÷ 3 first because it comes first, giving 4 × 2 = 8. Not 12 ÷ 6.
The convention has two tiers of information. The first is precedence: grouping symbols, then exponentiation, then multiplicative operations, then additive operations. The second is associativity, which resolves ties within a tier — and this is the part the mnemonic PEMDAS hides.
Multiplication and division share one precedence level and are left-associative, so a ÷ b × c means (a ÷ b) × c. Addition and subtraction likewise. Exponentiation is the exception: it is right-associative, so 2^3^2 means 2^(3²) = 2⁹ = 512, not (2³)² = 64.
Grouping symbols also include implicit ones: a fraction bar groups its entire numerator and denominator, and a radical groups everything under it. (3+5)/(2·2) needs no written parentheses when set as a fraction, which is a common source of transcription errors when rewriting work on one line.
Worked example
Evaluate 4 + 2 × (9 − 5)² ÷ 8.
- Parentheses: 9 − 5 = 4, giving 4 + 2 × 4² ÷ 8.
- Exponent: 4² = 16, giving 4 + 2 × 16 ÷ 8.
- × and ÷ left to right: 2 × 16 = 32, then 32 ÷ 8 = 4.
- Addition: 4 + 4.
Answer: 8
Common mistakes
- Doing all multiplication before any division. They tie, so go left to right.
- Treating a fraction bar as if it only groups the first term of the numerator.