We Need to Talk About PEMDAS

I have a fair deal of experience with maths, and as a result it hurts me on a deep level when I see viral maths questions which people always get wrong. Today I would like to take a look at the worst offender of this and why the answer is NOT 1.

[sladelucas]

I wanted to make a follow up comment just to say that even though I am not replying to comments, I do read them all, and the conclusion seems to be that yeah, I may be a little bit wrong. I am not going to remove the video because (aside from it being my most viewed video by a landslide) I don't want to just cover up mistakes and act like they don't exist. I am still proud of this video in many ways and I hope that it can have some worth still. And it was fun being able to do something completely different to the content I normally do, so I don't want to have to get rid of that.

[jensschroder]

According to the rules taught in German schools: 6/2(1+2) = 1
parentheses, exponents, IMPLIED MULTIPLIKATION, multiplication, division, addition, subtraction
6/2(1+2)= 6/ 2(3) = 6/ 2*3 = 6/6 = 1

I calculated it with a calculator prescribed by the German school authorities.
TI calculators are banned in German schools because they calculate according to PEMDAS
CASIO pocket calculators are required in German schools because they calculate according to PEJMDAS: 6/2(1+2)=1

[TengouX]

Honestly, if mathematicians can't even agree on the order of operations depending on where they're from, you know, the most foundational cornerstone of algebra, then how can we call math an exact science?

This is not a math problem, this is a people problem. That doesn't mean I'm critical of specific people or anything. People across the world have learned mathematics differently, and they will have an inevitable bias for the method they've been taught to use for decades. But we need to reduce ambiguity in writing.

[LezChap]

I still think it's 1, and that's the hill I'll die on. 2(1+2) = 2(1) + 2(2), which simplifies down to 6. We can then substitute 6 for 2(1+2) in the original equation, and learn that the answer is 1...

Or we can all agree the original equation is ambiguously and poorly written.

[dmatter]

That is not how PEMDAS is defined. M and D are equal in operation and are completed from left to right as they appear in the equation. The same is true for A and S.

A x B ÷ C : you would multiply A and B then divide that result by C.
A ÷ B x C : you would divide A and B then multiply that result by C.

PEMDAS could be written PEDMSA and it would not change how it functions. BODMAS follows the same rules.

[Bobis]

2(2+1) is considered a juxtaposition within higher level mathematics and would be done ahead of any regular multiplication or division any calculator should take juxtaposition into account unless its designed for the NA market my TI-81 would recognize the 6/2(2+1) as 6/[2(2+1)] which would output 1 as the answer for the answer to be 9 it would need to be written as 9 = 6/2x(2+1)

[MuffinsAPlenty]

In light of some of the discussion in this comment section, I want to point out an interesting source, but I need to explain the disagreement first.

There are some people here who claim that use of parentheses makes things no longer juxtaposed. So, for example, 1/ab would be equivalent to 1/(ab), but 1/a(b) would be equivalent to (1/a)b. The reasoning is that a(b) is no longer juxtaposition, so it cannot get priority as multiplication by juxtaposition.

Anyway, here's the reference:

Book: _Concrete Mathematics: A Foundation for Computer Science_ by Ronald Graham, Donald Knuth, and Oren Patashnik

On page 64, exercise 23 (of chapter 2), they claim that 1/k(k+1) can be replaced by the partial fraction decomposition 1/k − 1/(k+1). This is, of course, only true if 1/k(k+1) is interpreted the same as 1/(k(k+1)).

This isn't an accident that Graham, Knuth, and Patashnik made in their book, since we see this happen multiple times. I'll just give one more example:

On page 394 on the line above equation (8.24), they have (n-2)!/n! = 1/n(n-1). Knowing what factorial means, it is clear that 1/n(n-1) is meant to be interpreted as 1/(n(n-1)), otherwise this wouldn't be a true statement.

[knottheory]

Former US Math Teacher - Brackets are used with parentheses here, but brackets take precedence (in the sense you have to PEMDAS everything inside the bracket first before you do anything else).

I find a lot of these viral math problems either have improper notation to begin with, or people forget that multiply/divide are in fact the same operation, as are add/subtract (and technically all four are addition, all arithmetical operations are addition).

I used to teach students to rewrite the terms with the missing signs and parentheses.

Now when I went to school, you did PEMDAS, but you went left to right as the video describes. That seems to have gone out the window.

But now that I don't do math problems all day, I can get it wrong sometimes.

[NeilWick]

The problem is this (as stated in the Wikipedia article for Order of Operations): "In some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n." Some textbooks use that rule and some do not. This means that the problem is ambiguous and is not a well formed problem, so the answer is also ambiguous. As far as I'm concerned, multiplication and division are both done at the same time from left to right, so PEMDAS and BODMAS ought to lead to the same solution.

[tjd]

If you read any textbooks in physics, engineering, and many other sciences, if you have "ab/cd" written in one line of text, it is assumed to mean (a*b)/(c*d). If strictly following PEMDAS from left to right, however, this is a * b / c * d, that likely gives you a much different answer. In other words, those in the sciences and higher math use PEJMDAS in practice, where the "J" is juxtaposition multiplication or implied multiplication, that is given a priority than regular multiplication or division in the order of operations. Under PEJMDAS (from left to right), then 6÷2(1+2) = 6÷2(3) = 6÷6 = 1, whereas strict PEMDAS gives you 9. All these stupid viral problems do is use purposely and poorly written equations to pit those who quit math not long after basic algebra using PEMDAS against those who have advanced to a PEJMDAS background in advanced math, physics, engineering, or other sciences. To make matters even more annoying, typing this exact equation into a variety of calculator brands will also give you either 9 or 1 depending on whether they are programmed in PEMDAS or PEJMDAS, respectively, as educators and/or scientists demand one or the other depending on what level of math is being learned. In other words, check the calculator manual (or test it) before you assume how it will calculate. Perhaps calculators should have a PEMDAS or PEJMDAS toggled setting, such as using degrees or radians for angles.

[capchemist]

Any time I look at these exemplar math problems, it's a situation that could be easily remedied by adding a set of parentheses. I always understood PEMDAS/BODMAS to help support students in thinking critically in how to manage their algebra, and less for arithmetical pedantry.

[wyboo]

i just hate these viral "math problems." first, pemdas is just a convention as to how to evaluate expressions, and NOT math. in many, many papers, you'll see something like ab/cd to mean (ab)/(cd) while pemdas would imply ab/cd=(ab/c)d. the important thing is if the expression is clear in context, not pemdas. and these viral 'math problems' are so far removed from actual math; people have arithmetic and math confused way too much. math is so much more general, beautiful, and serious than arguing over whose interpretation of what a certain expression means is correct

[jammycakes]

I was never taught about order of precedence in maths classes in school. I learned about it from learning computer programming.

In just about every programming language that I've ever used, multiplication and division have the same precedence, and so too do addition and subtraction. They are applied by reading from left to right. So 12/2*3 gives 18 as does 3*12/2, 12*3/2 and 3/2*12.

Multiplication and division take precedence over addition and subtraction though. So 6/2+1 gives 4, not 2.

[rlbarry]

I disagree.
2(3) is still in parentheses order.
First inside parentheses then outside parentheses when no other sign is expressed. Your way would have been written 2X(1+2).

[caecandy]

I've been so annoyed by this viral thing. The answer is that it's both. Some places (academic journals or other math-heavy places) consider "implied multiplication" due to brackets to be higher priority. Others don't. The actual answer is that it's a poorly written problem.

[BillLeonard-cs]

Wrong, wrong, wrong. Clearing the brackets is not accomplished by simply performing the operation within the brackets. By so doing you have not cleared the brackets only rearranged the value within. Clearing the brackets occurs once you perform the operation indicated by the brackets. Changing the brackets to some other notation isn’t clearing the brackets.

[Of_UnCommon_Sense]

Say we have (2x1+2x2)=(6) and decide that is a lot to write. So we remove the common factor.
We then write it as 2(1+2)=(6)=6
Now we see we can write it even shorter as...
2(3)=(2x3)=(6)=6

No matter how we write it it = 6. !!!
Writing the common factor outside the brackets changes nothing. It is still 6 and it is sill one term. Although we write it outside the parenthesis, it does not mean 2 times the brackets. It means we multiply every element INSIDE the brackets by 2.
This is the misconception fueling this controversy. The 2 does not really exist outside the brackets, but wholly inside.

ALL MULTIPLICATIONS OCCUR WITHIN THE BRACKETS

So now we decide to divide 6 by this term. So it is 6 / 6 = 1 no matter how we write it.
6/(2x1+2x2)=1
6/2(1+2)=1
6/2(3)=1
6/(2x3)=1
6/(2+4)=1

[Mesa_Mike]

The problem is not really that people don't understand the "order of operations" (it's really precedence of operations, BTW) so much as that hardly anyone recognizes the fact that the expression is ambiguous, and thinks that there is a definitive and objectively correct evaluation of it.
If you never got past 5th grade math, then you'll appeal to PEMDAS (or BIDMAS, or whatever) along with it's "go left-to-right" rule and blindly "apply" it as if it were deterministic algorithm for evaluating all math expressions whatsoever, without considering nuances of notation.
If you've gotten to the stage of math where you routinely use letters or other symbols as stand-ins for unknown or otherwise unspecified numerical values, then you will realize that implied multiplications are very often treated as an operation with a higher precedence.

In reality, this is a poorly written expression that needs to be sent back for clarification as to it's intended meaning.

[dgkcpa]

Neither PEMDAS, nor BODMAS, nor the order of operations allow for the rewriting of 6 ÷ 2(1+2) as 6 ÷ 2 x (1+2), a much different expression.

[johng.]

this is the problem when you follow the "Rules" that are for elementary and pre-elementary children. oh and it isn't the order of operations that is wrong, no matter which way you calculate this, you are following the order of operations, the problem is the incorrect formatting of the question. you see when you go from a multi-level depiction of the division to an inline one, to preserve the grouping you need to add brackets, so this should be either (6/2)(1+2) or 6/(2(1+2))
but those educated correctly would know that multiplication by juxtaposition doesn't terminate the grouping for the division
6/2*(1+2) = 9
but
6/2(1+2) = 1
for example if we want to divide 6 by 5 + 1 it is written as 6/(5+1) and not 6/5+1 the explicit operation terminates the division grouping. likewise 6/2(3) = 1 but 6/2*3 = 9.