Inaccurate, this has nothing to do with the mnemonic PEMDAS, this has to do with the actual order of operations it tries to instill. That order of operations is not ambiguous, there is a correct way to solve simple equations like the one above, and there is one and only one correct answer to it. That answer is 16.
No, 2( does not bind more tightly than ÷. 2( is simply 2×(…, and ÷ and × occur at the same level of priority. After resolving the addition in the parentheses, the remaining operations are resolved left to right.
No, the fact that a good many people are incorrect about how math works does not entail that math is an open question. It’s not, math has actual rules to its equations and an unambiguous right answer. In this case, that answer is 16.
2(2+2) is identical to 2×(2+2), and follows the order of operations in an absolutely prescribed way. If you want that whole thing in the denominator of the division operator, you need to add an additional set of parentheses and make it 8÷(2×(2+2)). Without those, you read the operations left to right, and you get 16.
Implicit multiplication takes priority over division. You can’t write an implicit multiplication across a fraction, if you wrote the fraction out that way across 2 lines you would have 8(2+2) / 2 instead.
Variables work differently in standard practice due to natural reading. There is no special order of operations inserted, when variables are being used like that, you’re resolving the variable as if it had an implicit parentheses. This is common practice due to readability, nothing more, and it certainly doesn’t extend to actual digits being calculated with a clear and correct order of operations.
How can you apply “natural reading” to one situation but not another? You couldn’t split 2(2+2) across the fraction any more than you could split 2x across the fraction.
PEMDAS is actually a fairly new explanation, and both algebra and implicit multiplication predate it. Thus a strict interpretation of PEMDAS isn’t always appropriate - it’s a way of explaining things to school kids before they learn algebra and implicit multiplication.
Alternatively, perhaps a simpler way of looking at it is that 2(2+2) is all part of resolving the brackets (parentheses), so must come first. This would at least allow the new rule to hold true for all notations that came before it.
8 / 2(2+2) = 8 / 2(4) = 8 / 8 = 1
If you wrote the full fraction out the way you’re describing it, with numerator above the denominator, you would end up with 8(2+2) / 2. You can’t split an implicit multiplication across the fraction.
You apply natural reading to 2x because that looks and reads like a single number, and so you take it as a whole. This is convention only, and is implicitly reading it as (2x).
The same is not the case for 2(2+2). There is no variable in that, and it is accurately and correctly understood as 2×(2+2).
There is no order of operations which states that removal of the multiplication sign occurs prior to multiplication and division, and nothing outside the parentheses has any bearing on resolving the parentheses order of operations.
The answer to this is 16. Reading this and getting any other answer is misunderstanding it.
The correct answer is 16. Multiplication and Division happen at the same level of priority, and are evaluated left-to-right.
No it’s ambiguous, you claiming there is one right answer is actually wrong.
It is not ambiguous at all, there absolutely is one right answer, and it is 16.
You’re taking something you learned when you were like 9 years old and assuming it’s correct in every situation forever.
Unfortunately this isn’t the case and you’re incorrect.
Inaccurate, this has nothing to do with the mnemonic PEMDAS, this has to do with the actual order of operations it tries to instill. That order of operations is not ambiguous, there is a correct way to solve simple equations like the one above, and there is one and only one correct answer to it. That answer is 16.
And in the “actual” order of operations, if we want to pretend one exists,
2(binds more tightly than÷if you’re going via prescriptivism, then you’re wrong, because there are plenty of authoritative sources following the left hand model
if you’re going via descriptivism, then you’re wrong, because this thread exists
No, 2( does not bind more tightly than ÷. 2( is simply 2×(…, and ÷ and × occur at the same level of priority. After resolving the addition in the parentheses, the remaining operations are resolved left to right.
No, the fact that a good many people are incorrect about how math works does not entail that math is an open question. It’s not, math has actual rules to its equations and an unambiguous right answer. In this case, that answer is 16.
No it isn’t. 2(a+b)=(2a+2b) The Distributive Law
But there actually is only 1 right answer, and unfortunately for the person you’re replying to it’s 1.
PEMDAS be damned?
PEMDAS should be read as Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. There are four levels of priority, not six.
PEMDAS is a relatively recent learning tool made to help school kids. It doesn’t consider implicit multiplication.
If you wrote out the fraction, you couldn’t write 2(2+2) with one on the numerator and one on the denominator.
2(2+2) is identical to 2×(2+2), and follows the order of operations in an absolutely prescribed way. If you want that whole thing in the denominator of the division operator, you need to add an additional set of parentheses and make it 8÷(2×(2+2)). Without those, you read the operations left to right, and you get 16.
Replace (2+2) with x:
8 / 2x = 4 / x = 4 / (2+2) = 1
Implicit multiplication takes priority over division. You can’t write an implicit multiplication across a fraction, if you wrote the fraction out that way across 2 lines you would have 8(2+2) / 2 instead.
Variables work differently in standard practice due to natural reading. There is no special order of operations inserted, when variables are being used like that, you’re resolving the variable as if it had an implicit parentheses. This is common practice due to readability, nothing more, and it certainly doesn’t extend to actual digits being calculated with a clear and correct order of operations.
How can you apply “natural reading” to one situation but not another? You couldn’t split 2(2+2) across the fraction any more than you could split 2x across the fraction.
PEMDAS is actually a fairly new explanation, and both algebra and implicit multiplication predate it. Thus a strict interpretation of PEMDAS isn’t always appropriate - it’s a way of explaining things to school kids before they learn algebra and implicit multiplication.
Alternatively, perhaps a simpler way of looking at it is that 2(2+2) is all part of resolving the brackets (parentheses), so must come first. This would at least allow the new rule to hold true for all notations that came before it.
8 / 2(2+2) = 8 / 2(4) = 8 / 8 = 1
If you wrote the full fraction out the way you’re describing it, with numerator above the denominator, you would end up with 8(2+2) / 2. You can’t split an implicit multiplication across the fraction.
You apply natural reading to 2x because that looks and reads like a single number, and so you take it as a whole. This is convention only, and is implicitly reading it as (2x).
The same is not the case for 2(2+2). There is no variable in that, and it is accurately and correctly understood as 2×(2+2).
There is no order of operations which states that removal of the multiplication sign occurs prior to multiplication and division, and nothing outside the parentheses has any bearing on resolving the parentheses order of operations.
The answer to this is 16. Reading this and getting any other answer is misunderstanding it.