I keep chanting BIDMAS like it’s a spell, but the middle bit is melting my brain. If D stands for Division and M for Multiplication, does that mean division “wins”? Or are they actually the same level and I should march left-to-right after brackets and indices? I get tangled on things like 12 ÷ 3 × 2 versus 12 ÷ (3 × 2). When there aren’t any extra brackets, how am I supposed to read it so I don’t invent a new number system by accident?
Bonus burr in my sock: expressions like 8/2(2+2). My brain happily does the brackets, then stares at the slash and the sneaky multiplication holding hands, and suddenly the whole thing feels like a logic seesaw. I’ve heard some people treat “implicit multiplication” (like 2(…)) as tighter than division, while others say multiplication and division are equals that go left-to-right. How do you decide what’s intended, and what’s a safe way to rewrite these so they’re unambiguous? Any tips or rules of thumb to keep my notebook from turning into a scribbly crime scene would be amazing.
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4 Responses
A good way to read BIDMAS is: do Brackets, then Indices, and after that treat Division and Multiplication as equals processed left-to-right (same for Addition and Subtraction). So division doesn’t “win”-it’s just on the same level as multiplication, and we march left-to-right because division breaks the associativity that multiplication alone would have. That’s why 12 ÷ 3 × 2 means (12 ÷ 3) × 2 = 4 × 2 = 8, whereas 12 ÷ (3 × 2) is a different instruction and equals 12 ÷ 6 = 2. The tricky case 8/2(2+2) becomes clear if we stick to those rules: do the brackets first, then go left-to-right for × and ÷. Worked example: 8/2(2+2) → 8/2×4 → 4×4 → 16. Some people like to give “implicit multiplication” (2(…)) extra priority, but that’s not a standard algebra rule and different tools disagree, so it’s risky. Safer habit: never mix a single slash with implicit multiplication unless you add parentheses. If you intend the whole product to be in the denominator, write 8/(2(2+2)); if not, write (8/2)(2+2). Clear grouping beats any chant every time.
I like to think of BIDMAS as traffic rules on a straight road: after Brackets and Indices, multiplication and division share the same lane, so you just drive left-to-right. Neither “wins” by default. That’s because multiplication and division are inverse operations, and division isn’t associative, so we need a reading order to keep things sane. Example: 12 ÷ 3 × 2 is read left-to-right as (12 ÷ 3) × 2 = 4 × 2 = 8. But if the author wants the 3 × 2 to stick together, they have to say so: 12 ÷ (3 × 2) = 12 ÷ 6 = 2. Same story later with addition and subtraction: equal rank, go left-to-right.
Now for the famous banana peel: 8/2(2+2). The bracket gives 2(2+2) = 2·4, but the slash sitting next to an “implicit” multiplication makes the whole thing typographically ambiguous. Some people (and some calculators) read it left-to-right as (8/2)·(2+2) = 4·4 = 16. Others read the slash as a long fraction bar with the whole 2(2+2) in the denominator: 8/[2(2+2)] = 8/8 = 1. There isn’t a universal math law that makes implicit multiplication bind tighter than division, so don’t rely on that. When you want to be unambiguous, rewrite it: either (8/2)·(2+2) if you mean 16, or 8/(2(2+2)) if you mean 1. My rule of thumb when using a single slash is to wrap the entire denominator (or numerator) in brackets, or use a proper horizontal fraction bar; and if two things are meant to be multiplied, I show the × or a dot so the reader doesn’t have to guess.
So the safe, sanity-saving checklist is: after brackets and powers, do a left-to-right sweep for × and ÷; then a left-to-right sweep for + and −. If a slash meets a neighboring bracketed product, add parentheses to show exactly what’s in the numerator or denominator. That tiny bit of punctuation keeps your notebook from turning into a scribbly crime scene-and saves you from inventing a brand-new number system by accident!
Good news: × and ÷ (and + and −) are equals-after brackets and indices, go strictly left-to-right; implicit multiplication isn’t tighter, so if something looks squint-worthy (like 8/2(2+2)), add parentheses or a fraction bar to say exactly what you mean. Example: 12 ÷ 3 × 2 = (12 ÷ 3) × 2 = 8, but 12 ÷ (3 × 2) = 2; by the usual convention 8/2(2+2) = (8/2)×(2+2) = 16, though you can write 8/[2(2+2)] if that’s the intent (Khan Academy explainer: https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/a/order-of-operations).
Great question-BIDMAS can sound like a magic spell that backfires in the middle! The key is that Multiplication and Division are equals: they share the same priority and you do them left to right (same for Addition and Subtraction). So 12 ÷ 3 × 2 is read left-to-right: first 12 ÷ 3 = 4, then 4 × 2 = 8; that’s different from 12 ÷ (3 × 2) = 12 ÷ 6 = 2 because those brackets force the grouping. About 8/2(2+2): under the most common convention (used in many textbooks, calculators, and programming languages), the slash behaves like division with the same level as multiplication, so it’s 8 ÷ 2 × (2+2) = 4 × 4 = 16. Some people prefer to treat the 2(2+2) as a glued-together denominator, getting 8/(2(2+2)) = 8/8 = 1-but that’s a different grouping, and the plain slash doesn’t guarantee it. My rule of thumb: implicit multiplication (like 2(…)) is not “stronger” than division; if you want it to be the denominator, show it. When a slash meets a product or parentheses, add brackets so nobody has to guess: write (8/2)(2+2) if you mean 16, or 8/(2(2+2)) if you mean 1. If you can, use a horizontal fraction bar or extra parentheses-future you (and your teacher) will thank you.