Pre-Algebra — Numbers & First Variables

Order of operations (PEMDAS)

ELI5: when an expression mixes operations, everyone must agree on the order or we'd get different answers. The rule is PEMDAS: Parentheses, then Exponents, then Multiply/Divide (left to right), then Add/Subtract. So 2 + 3 × 4 = 14 (multiply first), not 20.

pem = 2 + 3 \cdot 4

14

multiply before adding: 14

✓ pass

pem == 14

not 20 — × beats +

pem_{paren} = \left(2 + 3\right) \cdot 4

20

parentheses jump the queue: 20

✓ pass

pem_{paren} == 20

now the addition goes first

Real-world hook: every spreadsheet, calculator, and programming language obeys exactly this order — get it wrong and a budget or a dosage comes out wrong.

Try it yourself: evaluate 5 + 2 × 3² (exponent first, then ×, then +).

pemdas_{you} = 5 + 2 \cdot 3^{2}

23

✏️ Your turn: evaluate 5 + 2 × 3². Do 3² first (= 9), then ×2, then +5.

✓ pass

\mathrm{abs}\left(pemdas_{you} - 5 + 2 \cdot 3^{2}\right) < 10^{-9}

green when you follow PEMDAS correctly