DIE PEMDAS DIE - The Problem with PEMDAS - Wednesday's Why Episode 8

The order of operations is not wrong, and neither is PEMDAS. What is wrong is our understanding of the relationship between addition and subtraction, as well as the relationship between multiplication and division. It causes all sorts of drama! This video tackles those issues in a hope that you’ll be better prepared to handle points of mathematical confusion in your future. Stay tuned until the end of the video when I show you two different ways of answering this problem correctly.

Thank you again for your support.

[davemiller]

There is a woman who is a mathematician who has a video explaining that PEMDAS is only for elementary school children. And it is wrong. She also explains that the answer to this problem is in fact 1. This is due to juxtaposition, also called implied multiplication and it has precedence over other multiplication/division. This is how mathematicians, engineers, and physicists do it. They don't use PEMDAS.
She cites papers and websites showing that this is in fact true.
So if you are in elementary school, the answer is 9, but if you are a professional mathematician, engineer, or physicist, the answer is 1.

Watch the video here:

[bendono]

I fully agree that addition and subtraction and multiplication and division are the same thing, respectively. And applying PEMDAS (or whatever you want to call it) does result in 9. However, the bigger problem here is that outside the classroom a great many people (both mathematicians and physicists included ) intend for 6÷2(2+1) to really mean 6÷(2(2+1)). More generally, often the form a÷b(...) is intended to mean a÷(b(...)). Probably a better solution is to try to avoid the issue and just use additional parentheses to be clear, even if some people may argue that it is redundant.

[edl]

However, my calc book treats f(x) = 1/2x as 1/(2x) not (1/2)x

[drooplug]

But what about implied multiplication?

[brucesmith]

the 2(1+2) is like a function. If I wrote this like a computer function, the 2(3) would be performed first. It is valuable to have such ability to override the order of operation.
Almost Orwellian, to say otherwise. The answer is 1.

2*(1+2) is not same as 2(1+2). in 2(1+2) the ( )s serve as a function. In 2*(1+2), the ( )s serve to group operations.

To solve this use Keneth Iversons APL.

[tkjyxrb]

It is true that subtraction and division are shorthand notations for a combination of the group binary and unary operations, a⋅b⁻¹. However, order of operation doesn't care about relations between the operations and the underlying field structure, it's pure syntax.
The real problem with PEMDAS is that nobody tells students it's just an oversimplified and incomplete (what's the Knuth's up-arrow operator priority?) set of rules existing only to be memorized at elementary school, not some "great truth".
BTW, do those mnemonic rules specify that exponentiation is right-associative?

[brianegendorf]

The biggest problem I have with PEMDAS is that too many people interpret 2(3) as 2x3...and that's not really correct. its still 2x(3). You still have to resolve the parens. Which then equals (6), which you can THEN consider a single unit, and then naturally get rid of the parentheses so that it becomes 6.

[marcjimenez]

Agreed untill
6 ÷ 2(2 + 1) = 9
There is a term in math called juxtaposition in math in this problem its multiplication by juxtaposition also known as implied multiplication and this is given a higher priority than explicit multiplication
there is also division by juxtaposition called fractions and this is why we understand
3 ÷ 1/2 as 3 ÷ 0.5 = 6
and not as
3 ÷ 1 ÷ 2 = 1.5
this understanding of juxtaposition is also why we can not say that the
6 ÷ 2 should be put in a fraction format with out putting the (2 + 1) with the 2 in the denominator
if i use the inverse opporation to get rid of the ( ) which would be
(2 + 1) = 3 = 3/1
in fraction format so something like
6 ÷ 2(3) = 6 ÷ 2 3/1
Although this could be considered a mute point when considering the distribution property of ( ) which is technically supposed to be done first and this makes
6 ÷ 2(2 + 1) =
6 ÷ ( 2(2) + 1(2) ) =
6 ÷ (4 + 2) =
6 ÷ 6 = 1
This also brings into consideration that the multiplication happens inside the ( ) so it can not be said that
6 ÷ 2(3) = 6 ÷ 2 × 3
but rather that
6 ÷ 2(3) = 6 ÷ (2 × 3)
Even though it is inproper to do whats inside the ( ) first it is a trick tought in schools to help make problems easier

[ghislainmaury]

I agree with you until you try to calculate 6/2(1+2)

In the definition of PEMDAS/BODMAS or any other name you give them, the usage is only with EXPLICIT MULTIPLICATION

But here we have an implicit multiplication !!!
And for that, a different rule/convention may apply : the 'unseparrated letters' notation, or the 'grouping intention' of the implicit multiplication.
Or in that historical document from 1917 : www.jstor.org/stable/2972726 (the author talked about 'intended product' and then 'product' for that case)

So the fact is that : A ÷ B * C = (A÷B)*C, ok everybody agree to that
BUT : A ÷ BC = A ÷ (B*C) => grouping intention of the implicit multiplication, or 'unseparrated letters' notation, or 'indicated product'

Some maths teachers says to check/add implied/invisible parentheses before doing any calcul
In the historical document it is said that 'If an indicated product follow the division sign, the whole product is the denominator"

With that:
6/2(1+2) = 6/(2*(1+2)) = 6/(2*3) = 6/6 = 1
But you, you wrote : 6/2(1+2) = 6/2*(1+2) !!! by our usage of implicit multiplication, you just passed (1+2) from the denominator to the numerator !!!
After that, of course : 6/2*(1+2) = 3*3 = 9

But as that is in conflict with your 'literral' interpretation of BODMAS/PEMDAS, one could say that such expression is ambiguous, as say the professor in his document.
Or by following the recommendation for that type of case, parentheses should always be used when delineating compound denominators.

Remark : when you replace the divide by 2, by multiplying by 1/2, the problem is that you forget that in fact 2(1+2) was on the denominator, not only 2 !!!

Think about it by writing : 1/2x
For you is it equal to 1/(2*x) or x/2 ?
Then replace x by 1+2 : 1/(2*(1+2)) or (1+2)/2 ?
Then multiply by 6 : 6/(2*(1+2)) or 6(1+2)/2 ?
So finally : 6/2(1+2) is equal to 6/(2*(1+2)) or 6(1+2)/2 ?

Think about that other expression:
4
___
2
___
2
Can you give an answer to that ? of course not, the 2 fractions line have the same length, and no external sign is present to indicate the 'main' fraction.
But if one rewrite it 4/2/2 and then want to follow PEMDAS/BODMAS (without using his brain before), he will come to the result of (4/2)/2 = 2/2 = 1
WRONG : nobody can say if the value is (4/2)/2 = 1 or 4/(2/2) = 4, in fact both result are wrong
So because somebody write in line a expression, without checking first if it is wrong or not, a wrong expression is transformed in a 'good' expression, but that do not say that the result is good !!!

[RealMesaMike]

Nice step in the right direction, but does nothing to resolve what the viral expression 6÷2(1+2] actually means.

It's a notation issue that PEMDAS doesn't really address. The expression is ambiguous because there is disagreement as to what it means.

[Mostlyharmless]

I think the real problem isn’t so much order of operations, but failure to teach math as a language. If the intended meaning of 6 div 2(2+1) = 9 then it should be written in a way so that the result of the simplification of the math phrase is 9. If more than one answer is possible, then your question is malformed. Thus, 1 is correct, there is a clear, logical progression to that answer, 9 is also correct, for the same reason.
It is only under the tyrany of order of operations where this gets exasperated. There is no reason you can’t do equally ranked operations at the same time, and the distribution property demands that you be allowed to distribute the 2.
By rewriting the problem as 6*(1/2) allows this distribution to happen.
I think there should be more emphasis in expressing math ideas clearly, more so than demanding that a rote adherence to convention be attended to without any understanding as to why we adhere to it.

[daveisnothere]

I was taught by one math teacher to think of division like a fraction, so the 6÷2(2+1) is technically 6 over 2(2+1). With that in mind even with PEMDAS you resolve the denominator and get 6, thus 6÷6=1. This video touches on that when he states that 15÷5 is the same as 15× 1 over 5. Resolve as 15×1 divide by 5=3

[scollyb]

The ambiguity in the answer is not about order of operations. It is about implied operations. There is no explicit multiplication operator in the original question. There is an implied one by juxtaposition

2(3)

The ambiguity is about what that implies. Is it 2x3 or (2×3)

You answer at the end assumes 2x3. Professional mathematicians and physicists tend to assume (2x3) because thats the most common assumption in style guides in journals.

But it's just different conventions. The question itself is deliberately ambiguous

[JoseMartinez-zrxp]

Brother you are awesome, it helps when you understand the concept a little instead of just following rules. You are excellent at breaking stuff down and your pace is on point!

[robertgonzalez]

No.

2(1+2) is NOT the same as 2*(1+2)

[jimmichelotti]

Thank you… you have resolved it, at least for me anyway. I struggled with how to help my grade school kids in math when it came to PEMDAS. I was quick to teach them that addition and subtraction are the same thing. Stop using the subtraction symbol as an operation, and instead use it as the “sign of a number”. Like you’ve explained, 5 – 3 is really just 5 + (-3). And as you’ve also explained, multiplication and division are the same. Dismiss with the division symbol (÷) and teach kids to multiply the dividend (the first number in a division problem) by the “reciprocal of the divisor” (divisor is the second number in a division problem). I’ve never liked the division symbol anyway… it always confused things for me. Perhaps we can/should still use the division symbol to teach really young kids how to perform the “operation” of division… but then be rid of it and move on. One last item that helped me in math was the concept of a “term”. Terms stand on their own within an equation, like words do in a sentence. And the division symbol (to me) will always be an “operation which is performed on terms”… and terms stand alone. Therefore, whatever is to the left of a division symbol, and whatever is to the right, are terms and should not be broken up. But I’ve likely just confused this whole topic even more. Thanks, again, for your insight!

[petepalmere]

By the way, and adding to my previous comment, solving for division by multiplying by the reciprocal would change the expression to 6 * 1/2(1+2) essentially changing the GCF of the group to a fraction...which changes the value of the factored expression. PEMDAS sees the 2 outside the group as the divisor/denominator when, in fact, it is the whole of the subexpression 2(1+2). The bottom line is you can't perform mathematical operations involving "quantities" until you first resolve the quantities!

[ColinMill]

I'm an old guy who started my career as a physicist, meandered into engineering and wrote a great deal of software (much of it numerical modelling and control applications) and I don't recall being fed PEMDAS or anything like it at school or subsequently. However, it seems to me that, especially where it really matters - e.g. engineering - to write an equation in such a way that it is possible for another human or a compiler or spreadsheet to misinterpret it is little short of criminal. It is not very difficult to use parentheses to make your equations completely specific so why not just use them and be done with it?

[jakemccoy]

PEDMAS (not PEMDAS) gives same the answers as Microsoft Excel and calculators. PEDMAS does not need any special rules.

[skillsrobles]

The problem with that very first equation is how its written. They should have added a + between the number and parenthesis that way their would be no confusion.