Brackets and Order

3 + 4 × 2 could be 14 or 11, depending on which part is done first. That is a real problem.

Brackets settle it

(3 + 4) × 2 is 14. 3 + (4 × 2) is 11. The brackets say what happens first.

Without brackets

Multiplication and division happen before addition and subtraction. Brackets override that.

Writing from words

"Add 8 and 7, then triple it" is written 3 × (8 + 7). The brackets hold the "then".

Why an order had to be agreed

2 + 3 × 4 could be 20 if you work left to right, or 14 if you multiply first. Both are defensible, so mathematicians agreed on one convention so that every written expression has exactly one meaning. Without the agreement, written maths would be ambiguous.

The order, in full

Brackets first. Then exponents. Then multiplication and division, working left to right. Then addition and subtraction, left to right. Multiplication and division sit at the same level, as do addition and subtraction — the left-to-right rule breaks the tie.

Brackets let you say what you mean

If you want the addition first, write (2 + 3) × 4 = 20. Brackets are how you override the default order. When an expression is getting complicated, adding brackets that were not strictly needed makes your intention unambiguous to any reader.

The left-to-right slip

The commonest error is doing all the additions before the multiplications, or working strictly left to right throughout. Scan the whole expression before starting and mark what must be done first. The order is a reading rule, and reading carefully is most of it.