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a(b) is a×b. Step 2 could be rewritten as 8 / 2 × 4. Working left to right, step 3 becomes 4 × 4.
No, because implicit multiplication binds more tightly than explicit. a/b(c) becomes a/(b×(c))
Says who?
Most maths textbooks written by mathematicians.
I don't mean when they're explaining "here's how the order of operations works". I mean in the basic way that they write more advanced problems and the answers they give for them.
This video, and the prequel to it linked in the description, go into some detail showing who uses what convention and why.
Interestingly I’ve wondered if this is regional, as a fellow Aussie I learned the same as you but it seems in other places they learn the other way
FWIW I went to school in Asia, using an internationally-focused curriculum, rather than going through the Australian curriculum here in Aus.
The video I linked includes some discussion with a calculator manufacturer who apparently is under the impression that teachers in North America are asking for strict BIDMAS, so the calculator manufacturer actually switched their calculators to doing that. Until they then got blowback from the rest of the world's teachers, so they switched back to BIDMAS with juxtaposition being prioritised over division. The video also presents the case that outside of teachers—among actual maths and physics academics—prioritising juxtaposition is always preferred, even in North America.
I'm an Australian teacher who has also taught the U.K. curriculum (so I have textbooks from both countries) and, based on these comments you mention, have also Googled some U.S. textbooks, and I've yet to see any Maths textbooks that teach it "the other way". I have a very strong suspicion that it's just a lot of people in the U.S. claiming they were taught that way, but not actually being true. I had someone from Europe claim the way we (and the U.K.) teach it wasn't taught there (from memory it was Lithuania, but I'm not sure now), so I just Googled the curriculum for their country and found that indeed it is taught the same way there as here. i.e. people will just make up things in order not to admit they were wrong about something (or that their memory of it is faulty).
Huh, I'll be darned. I'm not as much of a math nerd as I thought
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Most maths textbooks This video,
Ignore the video - she completely ignored Maths textbooks (and yes, you're right, the rules are in Maths textbooks - quoted multiple times here).
Everyone with a degree in mathematics, science or engineering.
Says who?
That's exactly where the calculators in the op differ. For more examples, Casio calculators do implicit multiplication first, while ti's treat it the same as explicit multiplication and division. I think that the latter is more predictable personally, but really you just need to know your calculator.
Casio calculators do implicit multiplication first
Actually they follow the actual rules of Maths - Terms and The Distributive Law.
Right answer, wrong words. The actual rules are Terms and The Distributive Law.
a(b) is a×b
a(b) is (a×b) - you can't remove brackets unless there is only 1 term left inside.
8/(2x4)=8/8=1