Mathematics · Arithmetic

The grammar of an expression

If you read left to right like a sentence — 2 + 3 = 5, then 5 × 4 = 20 — you get 20, and it's wrong. The agreed answer is 14, because multiplication is done before addition. This isn't a quirk; it's a convention every mathematician, calculator, and programming language shares, so that one written expression has exactly one meaning. Without it, 2 + 3 × 4 would be ambiguous, and code that disagreed on it would compute different things.

The rule is a fixed pecking order. Do what's inside parentheses first, then exponents, then multiplication and division (left to right), and finally addition and subtraction (left to right). The picture is simply: the 3 × 4 binds tighter than the +, so it resolves first.

Parentheses are the override

If you actually want the addition first, you say so with parentheses: (2 + 3) × 4 = 20. They're the one tool that lets you bend the order to your intent, and the habit of adding them — even when not strictly required — is how you write a formula nobody can misread. When in doubt, bracket it.

The other half: number sense

Knowing the rules isn't enough — you also want to know roughly the answer before you grind it out, so a slip stands out. That's estimation: round the messy numbers to friendly ones and do the easy version in your head. Asked for 312 × 49, think "300 × 50 = 15,000," and now any real answer near 6,000 or 60,000 is obviously a typo. This single habit — carrying a ballpark in your head — catches more mistakes than careful arithmetic does.

Work one, then finish one

Worked: evaluate 2 + 3 × 4². Exponent first: 4² = 16. Then multiply: 3 × 16 = 48. Then add: 2 + 48 = 50. Order, every time, top of the list down.

Your turn: evaluate (8 − 2) × 3 + 4², and first estimate it to the nearest ten. (Answer: parentheses 6, exponent 16, so 6×3 + 16 = 18 + 16 = 34.)

Why this earns a place in your toolkit

Every formula you'll meet in this course — a loss function, the inside of a neuron, an attention score — is one expression that has to be read in exactly this order, and code runs on precisely these rules. When you write w * x + b in a model, the language multiplies then adds, with no mercy for what you meant; mis-bracketing is one of the most common bugs in real ML code. And estimation is the working scientist's reflex: when a training run reports a loss of 4,000,000 and you expected something near 1, the ballpark in your head is what tells you, instantly, that something is broken. Reading expressions correctly and sanity-checking them by eye are the two habits that keep the math from quietly lying to you.