[…] the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!
As youngsters, math students are drilled in a particular
convention for the “order of operations,” which dictates the order thus:
parentheses, exponents, multiplication and division (to be treated
on equal footing, with ties broken by working from left to right), and
addition and subtraction (likewise of equal priority, with ties similarly
broken). Strict adherence to this elementary PEMDAS convention, I argued,
leads to only one answer: 16.
Nonetheless, many readers (including my editor), equally adherent to what
they regarded as the standard order of operations, strenuously insisted
the right answer was 1. What was going on? After reading through the
many comments on the article, I realized most of these respondents were
using a different (and more sophisticated) convention than the elementary
PEMDAS convention I had described in the article.
In this more sophisticated convention, which is often used in
algebra, implicit multiplication is given higher priority than explicit
multiplication or explicit division, in which those operations are written
explicitly with symbols like x * / or ÷. Under this more sophisticated
convention, the implicit multiplication in 2(2 + 2) is given higher
priority than the explicit division in 8÷2(2 + 2). In other words,
2(2+2) should be evaluated first. Doing so yields 8÷2(2 + 2) = 8÷8 = 1.
By the same rule, many commenters argued that the expression 8 ÷ 2(4)
was not synonymous with 8÷2x4, because the parentheses demanded immediate
resolution, thus giving 8÷8 = 1 again.
This convention is very reasonable, and I agree that the answer is 1
if we adhere to it. But it is not universally adopted.
Everyone in this threading referencing PEMDAS and still thinking the answer is 1 are completely ignoring the part of the convention is left to right. Only way to get 1 is to violate left to right on multiplication and division.
The problem is that BIDMAS and its variants are lies-to-children. Real mathematicians don’t use BIDMAS. Multiplication by juxtaposition is extremely common, and always takes priority over division.
Nobody in their right minds would saw 1/2x is the same as (1/2)x. It’s 1/(2x).
That’s how you get 1. By following conventions used by mathematicians at any level higher than primary school education.
There are plenty of contexts where writing in a single flat line is necessary, so it’s still useful to address the issue.
Just using more brackets is always a solution, but it can become messy and hard to read if you take it to the extreme (there’s a Minute Physics video where he does this and it unintentionally shows you just how bad it is), so it eventually becomes a matter of agreeing on convention and using brackets judicially where there’s actual ambiguity.
The problem is that you’re thinking of BIDMAS as a set of hard rules, rather than the set of rough guidelines created in the early 20th century by one random teacher for the purposes of teaching 10-year-olds how to do the level of maths that 10-year-olds do.
This video and this one point to some examples of style guides in academia as well as practical examples in the published works of mathematicians and physicists, which are pretty consistent.
If you want to come up with a hard rule, doing BIJMDAS, adding in “multiplication indicated by juxtaposition” with the J, is a much better way to do it than what you learnt when you were 10. But even that’s still best to think of as a handy guideline rather than a hard and fast rule.
No fr I had no idea that those acronyms weren’t the whole picture, I just assumed some mathematicians a long time ago decided how that stuff should be written out and that BEDMAS/PEMDAS/whatever contained all the rules in it. Thank you for the info, Idk why this isn’t more widely taught, ig because those acronyms are what all the questions are already written for? It doesn’t seem that hard to just teach it as BEADMAS, where the A refers to numbers adjacent to variables
Only way to get 1 is to violate left to right on multiplication and division
Actually the only way to get 16 is to ignore one of more rules of Maths - sometimes it’s Terms, sometimes it’s The Distributive Law, but always something. If you follow all the rules of Maths you get 1.
[…] the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!
Yeah nah. Actual Maths textbooks and proofs - did you not notice the complete lack of references to textbooks in the blog? It’s funny that he mentions Cajori though, given Cajori has a direct reference to Terms #MathsIsNeverAmbiguous
I think I’m gonna trust someone from Harvard over your as-seen-on-TV looking ass account, but thanks for the entertainment you’ve provided by trying to argue with some of the actual mathematicians in here
If you wrote out a proper fraction, numerator above denominator, you would have to write 8(2+2) to get 16. 2(2+2) only works if both are in the denominator.
If you wrote 8/2 vertically and then put the parenthesis (2x2) next to that, it’s a multiply. If you wrote 8/2 diagonally and put (2x2) next to that, it could be a divide, but I would specify an operator in that case because it’s ambiguous.
https://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html
Everyone in this threading referencing PEMDAS and still thinking the answer is 1 are completely ignoring the part of the convention is left to right. Only way to get 1 is to violate left to right on multiplication and division.
The problem is that BIDMAS and its variants are lies-to-children. Real mathematicians don’t use BIDMAS. Multiplication by juxtaposition is extremely common, and always takes priority over division.
Nobody in their right minds would saw 1/2x is the same as (1/2)x. It’s 1/(2x).
That’s how you get 1. By following conventions used by mathematicians at any level higher than primary school education.
The problem is the /. Usually you’d use a fraction bar, which groups it and makes it unambiguous
There are plenty of contexts where writing in a single flat line is necessary, so it’s still useful to address the issue.
Just using more brackets is always a solution, but it can become messy and hard to read if you take it to the extreme (there’s a Minute Physics video where he does this and it unintentionally shows you just how bad it is), so it eventually becomes a matter of agreeing on convention and using brackets judicially where there’s actual ambiguity.
Division (operator) and fraction bar (grouping symbol) aren’t the same. It already is unambiguous.
I only minored in math but I definitely see (1/2)x. That’s how I always entered polynomials into my ti-83 plus
Edit for clarity: I’d enter “1/2x”, as in 0.5x. Casio people have to use parenthesis or explicit multiplication though
Source? I have a hard ass time believing that nobody in their right mind would do pemdas the way you’re supposed to do it xD
The problem is that you’re thinking of BIDMAS as a set of hard rules, rather than the set of rough guidelines created in the early 20th century by one random teacher for the purposes of teaching 10-year-olds how to do the level of maths that 10-year-olds do.
This video and this one point to some examples of style guides in academia as well as practical examples in the published works of mathematicians and physicists, which are pretty consistent.
If you want to come up with a hard rule, doing BIJMDAS, adding in “multiplication indicated by juxtaposition” with the J, is a much better way to do it than what you learnt when you were 10. But even that’s still best to think of as a handy guideline rather than a hard and fast rule.
That’s cool, but still wrong :3
No fr I had no idea that those acronyms weren’t the whole picture, I just assumed some mathematicians a long time ago decided how that stuff should be written out and that BEDMAS/PEMDAS/whatever contained all the rules in it. Thank you for the info, Idk why this isn’t more widely taught, ig because those acronyms are what all the questions are already written for? It doesn’t seem that hard to just teach it as BEADMAS, where the A refers to numbers adjacent to variables
Terms already are taught!
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Cajori (1928) for starters, plus any old Year 7 Maths textbook, any era (we know from Lennes letter that textbooks were already doing this in 1917).
Actually the only way to get 16 is to ignore one of more rules of Maths - sometimes it’s Terms, sometimes it’s The Distributive Law, but always something. If you follow all the rules of Maths you get 1.
That’s cool, but still wrong :3
Exactly, explicit multiplication is part of the parenthesis so it comes first in order of operations
Yeah nah. Actual Maths textbooks and proofs - did you not notice the complete lack of references to textbooks in the blog? It’s funny that he mentions Cajori though, given Cajori has a direct reference to Terms #MathsIsNeverAmbiguous
I think I’m gonna trust someone from Harvard over your as-seen-on-TV looking ass account, but thanks for the entertainment you’ve provided by trying to argue with some of the actual mathematicians in here
So you’re going with the appeal to authority argument - ok, got it.
But if you’re gonna do that then make sure you check out Cajori’s credentials, since that’s, you know, who we both quoted.
You mean the dude who claimed to be, and was quoting wikipedia? BWAHAHAHAHA
deleted by creator
If you wrote out a proper fraction, numerator above denominator, you would have to write 8(2+2) to get 16. 2(2+2) only works if both are in the denominator.
If you wrote 8/2 vertically and then put the parenthesis (2x2) next to that, it’s a multiply. If you wrote 8/2 diagonally and put (2x2) next to that, it could be a divide, but I would specify an operator in that case because it’s ambiguous.
For the record I think the answer should be 16.