… no more airplanes on treadmills then 
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Speaking as a former math teacher (and full disclaimer, I have not read the entire thread here):
The precedence of implicit multiplication in PEMDAS is not explicitly defined. Some interpretations take it to be included as part of resolving parentheses. Some of them take it as equivalent to explicit multiplication. There are good arguments for both. But neither case is explicitly addressed in PEMDAS as traditionally taught.
You can find symbolic calculators (even two different models from the same brand) which interpret them differently.
I personally tend to think that treating implicit multiplication as part of the parentheses makes a lot of sense. We omit the multiplication symbol in these cases to indicate, specifically, that the multiplier applies to the parentheses. Therefore, there’s good sense in dealing with those prior to explicit multiplication. But again, this isn’t specifically addressed in the PEMDAS model.
It’s very important to remember that PEMDAS is not a mathematical truth; it’s a convention. Since the convention isn’t clear in this case, it makes a lot of sense to err on the side of explicitness. Deliberately writing an equation this way has only one purpose: to try to trick up people on the internet.
Pragmatically, if a working mathematician wrote a formula like this, they’d probably have their license revoked. (Actually they would probably have a nasty note from Reviewer #2 and the evaluation “accepted with minor revisions.”)
(Edit to add an image and another to remove unnecessary negativity.)
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Or we can think of a symbol in front of parentheses as a function, and then by convention understand 2(x) to mean the function that doubles everything. That’s a fairly natural way to map the numbers into functions on whatever space because it’s an operator whose eigenvalue is always 2!
And as it happens, that works exactly like you said anyway, so no problems except for people who insist on PEMDAS instead…which I think probably goes with them being more into programming than mathematics where concepts like eigenvalues come up, judging from responses here.
And yes, I know that’s a factorial, I’m taking advantage of the fact that it doesn’t matter in this case. 
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My wife’s name actually does have three 'A’s in it.
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You’re right that it’s not ambiguous at all, because the divisor here is the entire expression 2(2+2) so it’s 8 ÷ 8 which is 1.
… maybe we should argue how to compute 2 ^ 2 !
Yes, that’s just what I was saying, that the PEMDAS rule as usually taught doesn’t settle the question of how to evaluate an expression like 8/2*4
Nice find! I love examples like this that demonstrate why we shouldn’t always trust our calculators. Another good one is that many calculators will happily evaluate 0^0, where it should be undefined, because the limit of x^y as x and y approach zero differs depending on the direction you approach from.
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… Pluto will always be a planet in my heart 
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Only one man knows the true value of X.
Am I right, Elon?
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Thank you.
Fuckin’ A.
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