Order of operations: why 2 + 3 × 4 is 14, not 20

Type 2 + 3 × 4 into a basic calculator that works left to right and you may get 20. A scientific calculator gives 14. The second is correct, because mathematics has an agreed order for working out an expression.

The order

1. Brackets — anything inside parentheses first.
2. Powers — exponents such as 2^3.
3. Multiplication and division — equal rank, worked left to right.
4. Addition and subtraction — equal rank, worked left to right.

Schools teach this as BODMAS, BIDMAS or PEMDAS; the letters differ, the rules are the same.

Worked examples

• 2 + 3 × 4 = 2 + 12 = 14
• (2 + 3) × 4 = 5 × 4 = 20
• 10 ÷ 2 × 5 = 5 × 5 = 25 — left to right, not 10 ÷ 10
• 2^3^2 = 2^9 = 512 — powers are worked from the right

The last one surprises people. Stacked powers are evaluated from the top down, so 2^3^2 means 2 to the power of 9. If you meant (2^3)^2, which is 64, write the brackets.

Using the scientific calculator

Type a whole expression into the Scientific Calculator and press Enter. It follows the order above, keeps a history you can click to reuse, and has memory keys for a running total. You can type on your keyboard as well as using the keypad.

Supported functions include:

• sin, cos, tan and their inverses, plus sinh, cosh and tanh;
• log (base 10), ln (natural), log2, square and cube roots, abs and exp;
• floor, ceil, round, factorial with !, and the constants π and e.

Degrees or radians

Trigonometric functions depend on the angle unit. In degrees, sin(30) is 0.5. In radians, sin(30) is about −0.988, because 30 radians is a very different angle. If a trigonometry answer looks wrong, check the mode first; the current mode is shown next to the input.

Viral puzzles and ambiguous expressions

Puzzles such as 8 ÷ 2(2 + 2) circulate online because people get different answers — 16 or 1. The disagreement is about whether a multiplication written without a sign, as in 2(2 + 2), binds more tightly than ordinary division. Conventions differ between textbooks and calculators, so the expression is genuinely ambiguous. The fix is not to argue about the rule but to write the brackets: (8 ÷ 2) × (2 + 2) is 16, and 8 ÷ (2 × (2 + 2)) is 1. Different calculators — even the one on your phone in its basic and scientific layouts — may follow different conventions, so unambiguous input is the only reliable kind.

Errors that are actually right

• log of zero or a negative number is undefined in real numbers, so it produces an error rather than a misleading result.
• Unbalanced brackets are reported, rather than guessed at.
• Unknown names — a mistyped function — are named in the error so you can fix them.

Tips for fewer mistakes

• When in doubt, add brackets. Extra brackets never change a correct answer.
• Write the whole calculation as one expression rather than chaining partial results.
• Estimate first: if you expect about 50 and get 5,000, something is out of place.

For percentages — a percentage of a number, percentage change, adding tax — the Percentage Calculator lays out the working for you.