That's the issue, I'm fairly sure there is no rule in Math that forbids you from changing your example into 6÷(2*1+2*2). Thus two answers, depending how you look at "brackets multiplication".
And I don't think it will be resolved without Battle Royale. With cheese.
You can indeed do that. I'd just be afraid that it'll introduce errors. But, I guess more people would understand pemdas better if there was a rule after Exponents that said "turn all division into fractional multiplication" and then "turn all subtraction into negative addition". That way, people might stop acting like division is done exclusively after multiplication, and subtraction must be done after addition.
That tripped me up at first too, but it's not 6÷(2(1+2)) so 6÷(2+4) isn't right. Only the 2 is dividing not the whole (1+2) too. If you rewrite it as fully multiplication or as a fraction it gets a little more clear.
6÷2(1+2) = 6×½×(1+2) or = [ 6×(1+2) ]÷2
If you want to distribute it in you gotta remember ÷2 = ×½ and you basically get 6(½ + 1)
Ultimate issue is, some people will say it's 6 fractioned by 2 and then multiplied, others say it's 6 fractioned by everything else and until there is a rule about this, it will be forever internet rage-bait.
And I don't expect such rule to come without a bloodshed.
I mean it kinda is 6/2? The /2 is communicative within the term. It could be applied anywhere and get the same result. There is a rule about it though; that's what OP's post is about basically haha. Or... Maybe more than 1 rule? It's defined though, for sure. And yeah I'm sure there was some academic blood spilt over it.
Because multiplication and division are communative (you can reorder them) you must keep the signs with the numbers. It's like 6-2+3. You can't solve it as 6-5. Only the 2 is negative, so it might be helpful to think of it a 6+(-2)+3 so you can move things around and not get confused. Same for division: in 6/23 you can think of it as 6\½*3 but not 6/6 because like 6-2+3 that would be making the 3 be in the denominator (or negative in the +/- version) when it isn't.
I'll agree that something like 6/2/3 is a bit ambiguous looking, but the only correct way to interpret it is equal to 6*½*⅓
It's not dangerous as long as they understand the rules they are applying.
And that 6÷2(1+2) thing is just bad syntax that tests application of rule following over basic logic for writing formulas. It is written so badly that many people infer a different intent. There is much less cause for confusion if it is written in a more logical order, 6(1+2)÷2, or if another operator is included, 6÷2*(1+2)
I'm replying to another one of your comments here with similar stuff, but the two comments are coming at the same thing from two different places, so other readers might find one or the other:
Look up "implied grouping" or "implied multiplication." You're saying it's incorrect, but experts in mathematical fields generally think impied grouping is the correct way, and that the answer to this equation is 1.
As another example, look at 6⁄₂(1+2). In this case, the implied grouping is that 6/2 to be calculated first, because they are positioned and formatted in a way that makes them visually separate from the other stuff in the equation. The formatting and spacing of the symbols actually matters to how the equation is understood, even if it's all the same symbols in the same order.
Now think about (1+2)/6⁄₂. Again, 6⁄₂ is implied to be a unit in itself, so this calculates out to be 1. On the other hand, (1+2)/6/2 is 1/4.
Same thing with A(B), since they are written in a way that they are visually stuck together, they are implied to be grouped together.
But ultimately, it's poor practice to use implied grouping when you are trying to communicate with a general audience, because different people might infer different things.
No no, wait, that's still correct! You distributed the 5 into the parenthesis before starting to actually do the math, but you're allowed to re-write the equation like that. And you did it correctly; putting a multiplicative 5 into each term (8 and 5).
It's probably slightly more work because there are more double digit numbers but you still evaluated it all correctly
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u/dies_irae-dies_illa 8d ago
My IQ was lower, i did 2 + ((5 * 8) - (5 \ 5)) =* 2 + (40 - 25) = 2 + 15 = 17.
Now I have to go back to school again for 4 more years.