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.
You’re implicitly reading 2x as one variable, but not implicitly reading 2(2+2).
The answer is 1, per standard notation. If you put an explicit multiplication, only then would it be 16. Frankly, I’ll trust my Japanese calculator’s maths over Americans who butcher language as well :p
Really though there is a bit of academic debate on the subject. Wikipedia even has a section on it.
Incidentally, I just found another juicy rabbit hole: the UK version of the acronym is BODMAS (Brackets, Order, etc) and is widely attributed to Achilles Reselfelt. However, it seems that this person doesn’t even exist! There was a recent reddit thread on it, and as a result the textbook they tried asking about it ended up removing the reference. In any case, the earliest known version of that acronym is from 1945. Suffice it to say, though, orders of operation have been around far longer than the acronym, so it doesn’t really make sense to apply a strict interpretation of the new simplified learning tool when the nuance was established long before.
What many people don’t realize is that the “rules” we teach are only an attempt at DESCRIBING what mathematicians did for a long time without explicitly stating what rules they were following. They do not PRESCRIBE what inherently must be done, a priori. In just the same way, English grammar came long after English itself, and has sometimes been taught in a way that is inconsistent with actual practice, in an attempt to make the language seem perfectly rational.
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.
You’re implicitly reading 2x as one variable, but not implicitly reading 2(2+2).
The answer is 1, per standard notation. If you put an explicit multiplication, only then would it be 16. Frankly, I’ll trust my Japanese calculator’s maths over Americans who butcher language as well :p
Really though there is a bit of academic debate on the subject. Wikipedia even has a section on it.
Incidentally, I just found another juicy rabbit hole: the UK version of the acronym is BODMAS (Brackets, Order, etc) and is widely attributed to Achilles Reselfelt. However, it seems that this person doesn’t even exist! There was a recent reddit thread on it, and as a result the textbook they tried asking about it ended up removing the reference. In any case, the earliest known version of that acronym is from 1945. Suffice it to say, though, orders of operation have been around far longer than the acronym, so it doesn’t really make sense to apply a strict interpretation of the new simplified learning tool when the nuance was established long before.
This link perhaps explains it better: