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The PEMDAS Paradox

David Linkletter Share this page

It looks trivial but it keeps going viral. What answer do you get when you calculate $6\div 2(1+2)$? This question has reached every corner of social media, and has had millions of people respond with two common answers: $1$ and $9$.

You might think one half of those people are right and the other half need to check their arithmetic. But it never plays out like that; respondents on both sides defend their answers with confidence. There have been no formal mathematical publications about the problem, but a growing number of mathematicians can explain what's going on: $6\div 2(1+2)$ is not a well-defined expression.

Well-defined is an important term in maths. It essentially means that a certain input always yields the same output. All maths teachers agree that $6\div (2(1+2)) = 1$, and that $(6\div 2)(1+2) = 9$. The extra parentheses (brackets) remove the ambiguity and those expressions are well-defined. Most other viral maths problems, such as $9-3\div 1/3 + 1$ (see here), are well-defined, with one correct answer and one (or more) common erroneous answer(s). But calculating the value of the expression $6\div 2(1+2)$ is a matter of convention. Neither answer, $1$ nor $9$, is wrong; it depends on what you learned from your maths teacher.

The order in which to perform mathematical operations is given by the various mnemonics PEMDAS, BODMAS, BIDMAS and BEDMAS:

  • P (or B): first calculate the value of expressions inside any parentheses (brackets);
  • E (or O or I): next calculate any exponents (orders/indices);
  • MD (or DM): next carry out any multiplications and divisions, working from left to right;
  • AS: and finally carry out any additions and subtractions, working from left to right.

Two slightly different interpretations of PEMDAS (or BODMAS, etc) have been taught around the world, and the PEMDAS Paradox highlights their difference. Both sides are substantially popular and there is currently no standard for the convention worldwide. So you can stop that Twitter discussion and rest assured that each of you might be correctly remembering what you were taught – it's just that you were taught differently.

The two sides

Mechanically, the people on the "9" side – such as in the most popular YouTube video on this question – tend to calculate $6\div 2(1+2) = 6 \div 2 \times 3 = 3\times 3 = 9$, or perhaps they write it as $6\div 2(1+2) = 6\div 2(3) = 3(3) = 9$. People on this side tend to say that $a(b)$ can be replaced with $a\times b$ at any time. It can be reduced down to that: the teaching that "$a(b)$ is always interchangeable with $a\times b$" determines the PEMDAS Paradox's answer to be $9$.

On the "1" side, some people calculate $6\div 2(1+2) = 6\div 2(3) = 6\div 6 = 1$, while others point out the distributive property, $6\div 2(1+2) = 6\div (2+4) = 6\div 6 = 1$. The driving principle on this side is that implied multiplication via juxtaposition takes priority. This has been taught in maths classrooms around the world and is also a stated convention in some programming contexts. So here, the teaching that "$a(b)$ is always interchangeable with $(ab)$" determines the PEMDAS Paradox answer to be $1$.

Mathematically, it's inconsistent to simultaneously believe that $a(b)$ is interchangeable with $a\times b$ and also that $a(b)$ is interchangeable with $(ab)$. Because then it follows that $1 = 9$ via the arguments in the preceding paragraphs. Arriving at that contradiction is logical, simply illustrating that we can't have both answers. It also illuminates the fact that neither of those interpretations are inherent to PEMDAS. Both are subtle additional rules which decide what to do with syntax oddities such as $6\div 2(1+2)$, and so, accepting neither of them yields the formal mathematical conclusion that $6\div 2(1+2)$ is not well-defined. This is also why you can't "correct" each other in a satisfying way: your methods are logically incompatible.

So the disagreement distills down to this: Does it feel like $a(b)$ should always be interchangeable with $a\times b$? Or does it feel like $a(b)$ should always be interchangeable with $(ab)$? You can't say both.

(Image from Quora)

In practice, many mathematicians and scientists respond to the problem by saying "unclear syntax, needs more parentheses", and explain why it's ambiguous, which is essentially the correct answer. An infamous picture shows two different Casio calculators side-by-side given the input $6\div 2(1+2)$ and showing the two different answers. Though "syntax error" would arguably be the best answer a calculator should give for this problem, it's unsurprising that they try to reconcile the ambiguity, and that's ok. But for us humans, upon noting both conventions are followed by large slices of the world, we must conclude that $6\div 2(1+2)$ is currently not well-defined.

Support for both sides

It's a fact that Google, Wolfram, and many pocket calculators give the answer of 9. Calculators' answers here are of course determined by their input methods. Calculators obviously aren't the best judges for the PEMDAS Paradox. They simply reflect the current disagreement on the problem: calculator programmers are largely aware of this exact problem and already know that it's not standardised worldwide, so if maths teachers all unified on an answer, then those programmers would follow.

Consider Wolfram Alpha, the website that provides an answer engine (like a search engine, but rather than provide links to webpages, it provides answers to queries, particularly maths queries). It interprets $6\div 2(1+2)$ as $9$, interprets $6\div 2x$ as $3x$, and interprets $y=1/3x$ as the line through the origin with slope one-third. All three are consistent with each other in a programming sense, but the latter two feel odd to many observers. Typically if someone jots down $1/3x$, they mean $\frac{1}{3x}$, and if they meant to say $\frac{1}{3}x$, they would have written $x/3$.

In contrast, input $y=\sin 3x$ into Wolfram Alpha and it yields the sinusoid $y=\sin (3x)$, rather than the line through the origin with slope $\sin 3$. This example deviates from the previous examples regarding the rule "$3x$ is interchangeable with $3\times x$", in favor of better capturing the obvious intent of the input. Wolfram is just an algorithm feebly trying to figure out the meaning of its sensory inputs. Kinda like our brains. Anyway, the input of $6/x3$ gets interpreted as "six over $x$ cubed", so clearly Wolfram is not the authority on rectifying ugly syntax.

On the "1" side, a recent excellent video by Jenni Gorham, a maths tutor with a degree in Physics, explains several real-world examples supporting that interpretation. She points out numerous occasions in which scientists write $a/bc$ to mean $\frac{a}{bc}$ . Indeed, you'll find abundant examples of this in chemistry, physics and maths textbooks. Ms. Gorham and I have corresponded about the PEMDAS Paradox and she endorses formally calling the problem not well-defined, while also pointing out the need for a consensus convention for the sake of calculator programming. She argues the consensus answer should be 1 since the precedence of implied multiplication by juxtaposition has been the convention in most of the world in these formal contexts.

The big picture

It should be pointed out that conventions don't need to be unified. If two of my students argued over whether the least natural number is 0 or 1, I wouldn’t call either of them wrong, nor would I take issue with the lack of worldwide consensus on the matter. Wolfram knows the convention is split between two answers, and life goes on. If everyone who cares simply learns that the PEMDAS Paradox also has two popular answers (and thus itself is not a well-defined maths question), then that should be satisfactory.

Hopefully, after reading this article, it's satisfying to understand how a problem that looks so basic has uniquely lingered. In real life you should use more parentheses and avoid ambiguity. And hopefully it’s not too troubling that maths teachers worldwide appear to be split on this convention, as that’s not very rare and not really problematic, except maybe to calculator programmers.

For readers not fully satisfied with the depth of this article, perhaps my previous much longer paper won't disappoint. It goes further into detail justifying the formalities of the logical consistency of the two methods, as well as the problem's history and my experience with it.


About the author

David Linkletter

David Linkletter

David Linkletter is a graduate student working on a PhD in Pure Mathematics at the University of Nevada, Las Vegas, in the USA. His research is in set theory - large cardinals. He also teaches undergraduate classes at UNLV; his favourite class to teach is Discrete Maths.

Comments

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I have stopped teaching my students BEDMAS (or it's equivalents) as it is misleading in so many ways.
I now use GEMA.
One: I don't like the idea of a large MAS in someones BED!
Two: GEMA is such a lovely name.
Three: the DM (or MD) and the AS (or SA) misleads so many students.

G=Grouping
E= Exponents
M=Multiplication(and division is just inverse multiplication)
A= Addition (and subtraction is just inverse addition)

It is time we put PEDMAS, BIDMAS, BOMDAS, etc to BED and woke up with GEMA

I agree with GEMA, because it is much less misleading given that there are only 4 steps in the order of operations, but stick to the math and please do not associate a good idea with woke ideology!

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I would argue that there is no answer because it is not written in any standardized form of mathematics. There is a reason why no math teacher on earth would accept this as an equation, it is ambiguous. Unlike languages that can over time due to the common use of a term or word by common people, see the Oxford dictionary's inclusion of slang like ain't, mathematical notation can only change by the agreement of scholars. Those that shout PEMAS as the answer need to learn math beyond the 4th grade. When math is used for a purpose, such as engineering, it needs to be clear and unambiguous. Proper notation allows for the identification of units of measure. We don't use "slang" math notation for a reason if it isn't agreed upon than it can become useless.

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The way I see it, if there is no 'multiply' sign between the 2 and the (1+2), it acts just like if you were to have the expression 6/2x. Naturally, if x is replaced by (1+2), or (3), 2x=6, so the answer would be 1, as 6/6=1.

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In my view, this entire article is unnecessary. Unless I'm wrong, you always solve within parenthesis first, then reset, so to speak, and go about the problem from the beginning, which in this case would place the mathematician back at the division portion. From there, you have a simple problem resembling 6/2*3=9. And you would solve it left to right to come up with the obvious answer that any online calculator I've tried comes up with; 9. Was this posted on April 1st as some elaborate joke? This is obviously 9, unless there are multiple versions of the rule of the order of operations.

Unfortunately, I see it as being unnecessary in the opposite direction. Unless we have given up on the distributive method, the only way to resolve the parentheses is to expand the multiplier of 2 across the contents of the parentheses. Putting a "6/" in front of 2(2+1) does not mean that the first 2 no longer gets distributed.

Where it says “solve within parentheses first”
6/2(1+2) solve inside parentheses first
6/2(3) simplify parentheses last
6/6
Answer is 1
It doesn’t say “solve inside parentheses.”
It states “first”.

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I have spent 4 months arguing, (especially with Americans) over this matter, because they were taught differently, and in my opinion, wrongly.

At the crux of this matter are two principles that are ignored by many:
1. That multiplication is commutative, always.
2. That division is not commutative, and is only partly distributive.

Therefore, a rational person must ask themselves: when I find a math problem that involves both division and multiplication, how do I approach it, in order to retain the full power of the principles above?

Do I enter then shut the door, or do I shut the door then enter?

Do I reproduce then die, or do I die then reproduce?

That level of logic is what is missing.

Multiplication allows you (by commutation), to "walk in or out" as you wish. It allows you to "reproduce" early or late, for as long as you are alive. Once you divide, commutativity is lost (like death - reproduction is impossible afterwards).

Division, on the other hand, shuts the door; or is like the event of death. Once the door is shut, you cannot enter or leave. Once one is dead, they cannot reproduce!

How does this apply here?

Suppose we have 6 ÷ 2(2+1).
We cannot start with division, simply because it BLOCKS multiplication from being commutative.

You see, clearly, 2*(2+1) = (2+1)*2 and that MUST always remain valid and feasible. We cannot prefer or apply any method that denies this principle. This should be self explanatory.

Anyone who suggests anything that denies the commutativity of multiplication opposes math itself!

So, 6 ÷ 2(2+1) can only be resolved by safeguarding the commutativity of multiplication, before we "shut the door."

That is why 6 ÷ 2(3) = 6 ÷ 6 = 1, and not 9.

We cannot ever decide to die first, then start wondering whether reproduction is possible!

Multiplication is commutative. a * b = b * a. The problem is how you define your variables. You assume that a = 2 and b = 3, independently of the division. The reality is, a = 6 / 2 = 3 because of PEMDAS rules. Then you have 3 * 3 (a * b) = 3 * 3 (b * a) = 9.

You can't just ignore the division and do commutation on the multiplication by itself. There's no such thing as "shutting the door", whatever that means.

I remember getting problems wrong all the time because I dident simplify in my shown work. When you solve for what is in the parentheses 6/2(1+2) you get 6/2(3) and then you have solved for what's in the parentheses already. That's done, you did you P in PEMDAS. And so 2(3) just becomes 2×3 at that point cause the parentheses have been solved. This simplifying of equations was to help NOT get this kind of confusion among students. So in reality 6/2(3) is = to 6/2×3... by insisting the parentheses must stay after the inside has been solved and then insisting that you have to do that multiplication is completely bonkers to me. The 2 in front of the parentheses is not part of the set (1+2). There's a whole thing called Set Theory that a guy named Georg Cantor came up with almost 150 years ago that I believe explains this principle.

Just change the rule to PEJMDAS and you can do left-to-right and *still* handle the Juxtaposition in the normally accepted (but not demanded) way (i.e. takes precedence over explicit multiply-divide)
Presumably PEJMDAS is then well-formed and avoids the annoying ambiguity, but i haven't looked for other corner cases.

For the life of me, I do not understand what happened to the distributive method of resolving parentheses. While I find the article at the top of this page to be very informative, and it does mention the distributive method, it fails to answer why so many people seem to have forgotten about it! I have had multiple people tell me it is only used when you have variables. This makes no sense as the equation comes out differently with variables versus constants. Whether it has constants or variables, the parentheses need to be expanded first, and this is only done with one method: (2*2)+(2*1)=6, then you can divide this resultant into 6.

Refreshing point. That is why old text books would evaluate the items on either side of the divide ahead of the divide - i'll be they were protecting the principle. I think I recall so, but it has been many decades but I think the principles were introduced just ahead of these kinds of examples. One of the videos on this says this was an out of vogue view. You make a great point
that it cannot be out of vogue, its essential.

Like the other guy said, multiplication is commutative, however, in the problem the 2 is actually 1/2 due to the ./. [fraction] sign
so it's 6*(1/2)(2+1)

It's really about what operation is being performed here, then perform the correct order. 2(2+1) has the multiplication operation being used. The problem is [logically] solved as 9 using this simplicity.

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Hi David. I'm sure you reached a satisfactory conclusion and certainly don't intend to discuss this particular topic any further, but use it to move on. I do hope you read this as I note your research is in set theory and the title of your article here includes the word "paradox".

There's also a famous paradox in set theory known as the Russell paradox (and its equally famous "barber" version) and it's my contention that this too is really an order of operations of operations problem just like the one here, and could just as easily be resolved with brackets, conventions like PEMDAS, or some other order marker.

In fact this paradox has little to do specifically with sets at all. It works for just about any other subject-verb- object formation, for example: man-shave-man, set-belong to-set, dog-eat-dog, including those occasions when that object is the reflexive pronoun him-/her-/itself. I suggest that the confusion arises not with the nature of the grammatical subject but with the tense of that verb. It arises from failure to recognise that each successive appearance of the verb in the argument that sets out the paradox is logically, if not grammatically, a different tense and refers to a different time. The paradox disappears when we make that difference explicit, when we differentiate with respect to logical time. The barber shaves today all and only those people who didn't shave themselves yesterday, so if he didn't shave himself yesterday, he does today. The Russell set includes all and only those sets which didn't previously include themselves. So if it didn't include itself before it does now, but if it did before then it doesn't now.

In math when faced with the contradictory result arising from failure to put explicitly different relative times on the division, multiplication, and addition operations in an expression like 6 ÷ 2(1 + 2) = 4 we eventually find a way of nailing them down, so too with the operations of set inclusion and exclusion, or shaving and not shaving, or whatever.

There is a solid reason why some math problems take centuries to resolve: it is because simplistic people run away from the challenge.

***
Take a simple example: the area of a right triangle = ½bh or ½*b*h. Suppose it has basic side lengths 3, 4 & 5.

Irrespective of which is the base or height,
Area = ½*3*4 = 6.

So, given the Area (6) and one side (base), what is the other length (height)?

Area ÷ ½ * b = h.
Area ÷ ½ * h = b.

I am arguing that ½*b is one term, even without parentheses.

So that:
6 ÷ ½ * 3 = 4.
6 ÷ ½ * 4 = 3.

Professors, even from Harvard and Cambridge (UK) want to insist here that division "ranks equally with multiplication." It does not!

So, their 6 ÷ ½ * 3 = 36.
And their 6 ÷ ½ * 4 = 48.
"BECAUSE Google and WolframAlpha also said so!"

Men and women of the world, that is completely stupid. You know the area of a right triangle is settled math.

Why do you ignore your own brain, and trust robots? Computers are not perfect. They are less than 70 years old in combined developmental age. The human brain has at least 200 million years of arithmetic progress. Within this timeline, it has made these confused computers. Trust your head more.

The problem is that, given the final expression 6 ÷ ½ * 3 we don't really know where it came from. OF COURSE if we know is the result of calculating the area of a triangle we could maybe infer that ½ * 3 is half the length of a side or a height and then the only sensible thing to do is group that. But triangle areas are not the only subject we could arrive this expression from.

For example, let's take another super-basic mathematical calculation, the cross-multiplication, where you know that a couple of ratios are equal: a ÷ b = c ÷ d, and you must calculate one of the topmost variables, either a or c (direct cross-multiplication). For example:

a ÷ b * d = c
c ÷ d * b = a

Now we should conclude that a ÷ b and c ÷ d are one term, even without parentheses, right?

So that:
1 ÷ 2 * 6 = 3.
3 ÷ 6 * 2 = 1.

And therefore we would consider these other calculations (that use your logic of grouping the multiplication) nonsensical:

1 ÷ 2 * 6 = 1 ÷ 12.
3 ÷ 6 * 2 = 3 ÷ 12 = 1 ÷ 6.

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1 (meaning that juxtaposition has higher precedence) is what makes sense

it makes life easier and it is more intuitive

people only say 9 because they are using the oversimplification they learned at kindergarten

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Math syntax is only a syntax error when there are undefined symbols, or when the expression has no numbers in it. So 6÷2(1+2)=9 with no syntax error.
(3×(3)+3)=12
(3+(3)×3)=12
[3×(3]+3)=18
[3+(3]×3)=12
(3×[3)+3]=12
(3+[3)×3]=18
[3×[3]+3]=12
[3+[3]×3]=12
Also, parentheses have precedence over brackets.

Who's syntax check? Here's an interesting case where
=6/2(1+2)
is a syntax error:
LibreOffice Calc found an error in the formula entered.
Do you want to accept the correction proposed below?
=6/2*(1+2)

Accepting the proposal yields 9. But it refused to assume that was the intended meaning.
This is a great choice because otherwise you would have to have a setting as to
which convention would be used to resolve the ambiguity of whether the 2( was distributed or
just multiplied. It is perhaps interesting to note that did not offer=6/(2*(1+2)) instead
suggesting simply multiply might be the usual meaning.

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Late to this post but it's a pretty good article so I wanted to discuss it further.

Isn't there a further contradiction with the whole a(b) = (ab) argument in the choice of a?

If you have 6/2(1+2), then what determines that a = 2 and not 6/2?

I was going to comment this exact thing. Calling this an issue with an acronym for order of operations and suggesting a paradox doesn't really make sense when the real reason for the ambiguity of the expression is because of dissension on whether the term a in a(b) in this case is 2 or 6/2.

If this had been written using a vinculum instead of a solidus (or worse, the obelus) the ambiguity would be resolved:

6
-- (1 + 2) = 9
2

versus

      6
--------- = 1
2(1 + 2)

Saying that there is no recognized standard also doesn't seem accurate. What about ISO 80000-1 and ISO 80000-2? ISO 80000-1:2009 section 7.1.3 which discusses printing rules for mathematical expressions indicates:

 a
--- = a / (bc), NOT a/b · c
bc

ISO 80000-2:2009(E) also states that only the vinculum and solidus are acceptable as general division operators. A colon should only be used for ratios and the obelus should. not be used at all.

There's also the Physical Review Style and Notation Guide by the American Physical Society, but if I were to choose a standard, I'd go with an internationally recognized standard.

My question is, since there are in fact published international standards regarding presenting mathematical expressions (and in part their interpretation as noted above), why isn't the academic community either aware of or using those standards when teaching, particularly in getting rid of using the obelus altogether?

We are in agreement with the answer of 1, but I will take it one step further. Regardless what indicator is used to show division, the equation is still clear. I know of only one method to resolve parentheses and that is with the distributive method. Regardless how the equation is written, you will always end up with (2*2)+(2*1)=6.

"If you have 6/2(1+2), then what determines that a = 2 and not 6/2?"

If a = 6/2 then the expression would be presented as (6/2)(1+2).

Is it not usual when a constant or variable is defined as a term or expression of numbers that the numbers be enclosed in parentheses?

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Using a real life problem to reach the solution, imagine you have 6 sweets and 2 groups of 1 girl and 2 boys to share them around.

The kids all get one sweet each, so the answer is 1. If the answer were 9, you'd be creating 3o sweets out of thin air!

Permalink In reply to by Lucky13pierre (not verified)

Using a real life problem to reach the solution, imagine you have 6 index cards and one group of 1 girl and 2 boys. Your instructions are to first get rid of half of the cards, then give one card to each child, then tell each child to tear their card into 3 pieces.

You now have 9 pieces, so the answer is 9.

And who is to say which "real life problem" is the correct one? Nobody. We can always invent a word problem to fit whatever arithmetic steps we want.

(and, yes, it's been 8 months, so probably nobody will ever read what I wrote...)

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David, in your exceptional article, you mentioned that the people in the 1 category will sometimes point to the distributive method for how to get an answer of 1. If the distributive method is not used in this type of equation, where is it used? Be gentle with me! I am an engineer, not a mathematician!

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I do wish the international body of mathematicians could sort this one out. In algebra the implicit multiplier is always understood to take precedence. In other 'random' mathematical expressions whoever gave the mathematical world the permission to arbitrarily express 2*(1+2) as 2(1+2)? In order to follow logical consistency we spend years teaching kids algebraic mathematics where 2X is always 2X (inseparable and implicit takes precedence). If you wish to express 6 / 2 * (1+2) you simply do not have any permission to lazily and arbitrarily express it as 6 / 2(1+2) for this very reason. I know that even the maths professors are contorting themselves over this, but I think it should be sorted out. And logically it should simply be nailed down that implicit multiplication takes precedence because otherwise all of algebra is wrong. ab no longer means ab. I thought it was. I was taught Brackets Of Division (2 of something takes precedence over division and multiplication). But it turns out even that is not agreed on as more seem to hold to BODMAS meaning 'Brackets Orders Division...'

The International Organization for Standardization (ISO) has a document ISO 80000-2 whose topic is "Mathematical signs and symbols to be used in the natural sciences and technology".

It says (in a 2020 version), regarding the two standard symbols for multiplication, "Either symbol may be omitted if no misunderstanding is possible". It does not follow up by saying that when the symbol has been omitted the multiplication automatically rises to a higher level of precedence.

Consider four expressions:
4+(5-2)
4-(5-2)
4*(5-2)
4/(5-2)
The notational convention is that in one of those four cases the operator adjacent to the opening bracket can be omitted. That case is multiplication.

The idea that omitting the operator automatically gives it higher precedence than division is ludicrous. Omitting the operator can not, and should not, change its meaning.

Of course, it is true that algebraic notations such as xy or 2x are considered grouped. But the similarity of, for example 2(3), to such algebraic expressions superficial, and is not a basis for considering 2(3) grouped.

That's my perspective, anyway.

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This is a complete use or words but not use simple Arithmethic for resolve.
1. Arithmetic try to solve many terms into 1.
2. When we dont see any agrupation signal we know there's an order to solve any problem.
3. Calculators can't make maths, just only results.
4. Your single example corrspond to the form c:b(a+b). Simple algebraic definition. ¿Where is the rule that show me Wolframs is superior to math for follow your suggested formula?
5.This nosense publication correspond to try to insert deconstructional Gramsci thinking where it is impossible to do. And follow same steps in the science of maths, bases starts in arithmethics.

2 Results:
2.1. Arithmetican: 1

Lol

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I have followed a lot of the comments regarding the subject expression and the controversy created. Many agree that the expression is ambiguous but some settle for either PEMDAS and the order of operations from left to right while others use juxtaposition to resolve the group first. A lot of the confusion results from the use of the obelus (:-) for division with some noting that it is a syntax error and one should use either the slash (/) or more preferably the vinculum (---) instead. However, I have seen a different way of solving the expression that uses the uncontroversial rule for how to solve the division of fractions. i.e, multiply the first fraction by the reciprocal of the second fraction. To wit: the expression 6:- 2(1+2) uses the obelus to indicate division. But all can agree that any value is equal to itself if divided by 1, so if we write the expression as 6 over 1 :- 2(1+2) over 1 then it becomes 6 over 1 x 1 over 2(1+2)[the reciprocal]. From here it does not matter which operation is chosen first, i.e to either resolve the denominator first: 6 over 6 or to divide the group factor (in this case 2) into the numerator it becomes 3 over 3, and both methods result in 1 as the answer. Any comment on this?

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When you bring something like x/yz into the discussion, you're talking apples and oranges.

In dealing with algebra, an expression like yz is known as a monomial. A monomial is defined as:

"an expression that has a single term, with variables and a coefficient. For example, 2xy is a monomial since it is a single term, has two variables, and one coefficient. Monomials are the building blocks of polynomials and are called 'terms' when they are a part of larger polynomials. In other words, each term in a polynomial is a monomial."

A monomial is a convention used in algebra that mathematicians understand to be a single unit. So when you run into something like x/yz, you know to interpret it as x divided by yz, not x divided by y and then multiplied by z.

Simply put, 2(1+2) is not a monomial, and shouldn't be treated the same way. 2(1+2) is exactly the same as 2 x (1+2) and should be treated as such.

Order of Operations is clear and is intended to bring unambiguity to math. Those of you talking about juxtaposition multiplication having a higher priority are just adding confusion to the mix.

Sin 2(3) is interpreted as Sin (2 * 3) = Sin 6 not as (Sin 2) * 3 so I respectfully disagree. Logically if ab is algebraically interpreted as (ab), {that is as the single number which is the product of a and b}, then a(-b) has to be interpreted as (a(-b)) = (-ab) which means that a(b) is as much of a monomial as ab is and therefore 2(3) is a monomial (a number is a monomial: the monomial 2X**0 = 2(1) = 2).

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There is no paradox. The expression is a well formed formula and it has the value 9 when evaluated using PEMDAS.

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The confusion is caused by order of operations, in this case, whether division or multiplication should be done first. The question, in fact, is:
6 / 2 * 3 as brackets always go first.
Instead of determining the order of operations, let's change the division into mutiplication:
6 * 1/2 * 3
Now, we only have multiplication, but let's make it even easier than that. Let's change all numbers into fractions:
6/1 * 1/2 * 3/1
We multiply fractions by multiplying the numerators:
6 * 1 * 3 = 18
and by multiplying denominators:
1 * 2 * 1 = 2
As a result, we have a fraction:
18 / 2
and the answer is 9.

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As one who majored in MAE with a minor in math and computer programming, I did find it more intuitive to treat a/bc as a/(bc). I did not until recently swap camps when I realized that in several problems where multiple operators were present, the commutative and associative properties of multiplication would go away. Not only that, if we look at another formerly ambiguous expression (for those who forget that association and commuting do not work with subtraction):

a - b + c = (a - b) + c
a - b + c ≠ a - (b + c)

Likewise:
-x² = -(x²)
-x² ≠ (-x)²

So for if nothing more than the sake of consistency:

a/bc = (a/b)(c) [where distribution still works]
a/bc ≠ a/(bc)

We all agree that the two following statements are true:

1. a/b/c = a/(bc)
2. a/b × c = (ac)/b

If a/bc = a/(bc), then a/bc = a/b/c

Which seems nonsensical to me.

But if a/bc = (ac)/b, then a/bc = a/b × c

Which to me makes more logical sense since the two sides of the equation both deal with multiplication (one implied and one expressed)

Just my $0.02

You said...
"If a/bc = a/(bc), then a/bc = a/b/c
Which seems nonsensical to me."

In fact that is completely correct - just put some numbers in...
1÷2÷3=(1/2)÷3=1/6=1/(2*3)
In the same way that distributing a minus over a bracketed term changes any + to a -, distributing a ÷ over any bracketed term changes a x to a ÷.
This is a well-known property in Maths which is covered when teaching multiplying and dividing fractions.
Someone once asked me "what is a/b/c/d?". Easy, a/bcd

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An obelus was added to the mathematical language in the 1600's to write simple ratios. The orginal symbol for this looked like this ÷ . Eventually overtime especially with typewriters the symbol became looking like this : .
A viniculum was added shortly after an obelus and to write an equation that had an obelus into a equation using a vinculum you would write
6
6÷2×(3+3) into. _____×(3+3)
2
But this can also be written as 6:2×(3+3)

A solidus / was added to the mathematical language in the 1700's in order to write complex fractions that would normally require the use of a vinculum in the "inline fraction form". The use of a vinculum in math can easily require 3 times more paper use. The length of the equation is equal to the longest of either the denominator or numerator respectively, and even if one of the 2 is only 1 character, and the other is 32 characters long. You are wasting 31 extra characters on a vinculum and also the opossing side of the fraction. For this fact a solidus is not only a needed symbol in mathematics but also a cost saving method.
6/2(3+3) written with a vinculum would look like

6
__________
2(3+3)

Scientific calculator in the programming can switch between if they recognize a solidus or an obelus. Very expensive calculators can utilize both symbols in the same equation. Many physics professors will have 2 identical calculators with a label on each that states obelus, or solidus; or use the corresponding symbol on the label. This is done to help teach students that do not understand the difference.

If you have an equation such as ab÷cd and have a calculator set up to only solve for solidus than you need to convert that into a(b/c)d for the calculator to solve that equation correctly.

If you have an equation such as ab/cd and have a calculator only capable of solving for obelus(ratio) equation than it will need to be converted into (ab)÷(cd)

In the 1900's a printing press thought a solidus looked better than an obelus. So instead of placing the key for an obelus down as they where suppose to do, they changed the symbol and printed that mathematical text wrong. Still to this day there are many examples of this still found in high-school math text books.
For example when they have an excel spread sheet of mathematical notation a vinculum is referenced as being a horizontal or a vertical line and the horizontal symbol is removed from the symbols column. With ÷ and : symbols they are in there own column together and they both simply reference division symbols and not there intended references of being simple ratio symbols. In a column far away from obelus symbols generally next to a vinculum it has a solidus / symbol. Again only referencing it as a division symbol, instead of writing in-line fraction bar. If they could actually be used interchangeably than the orginal pre-printing company edits would have had all 3 symbols on the same excel column.

During the 1900's many public education school teachers wouldn't even have an associates degree. As a result many of the books sold to schools where never peered reviewed after printing.

A good scientific calculator not only should be able to switch between if it reads a solidus and obelus in the equation by going into setting. But also when it is set to obelus; to show an obelus on the screen. When it is set to do math in terms of a solidus than it shows a solidus on screen.

All these problems exist solely do to publishers such as wolfram misprinting mathematical textbooks. They have a web page dedicated to the fact that all peer reviewed mathematical textbooks used by people with phd. and peer reviewed by people with Phd's have the solidus and obelus meaning different things.

For this reason wolfram is not used in any college that teaches physics, programming, engineering or any other degrees that require advanced mathematics. As they would be peer reviewed and thrown out do to the level of misprinted textbooks they produce.

Instead of wolfram correcting there misprinted textbooks and printing a retraction and an apology to the mathematical community and public educational facilities, they ask others to join in with them for there misprints on there web page that dedicated to the fact that all peer reviewed advanced math text books would have 6÷2×(3+3)=18 and 6/2(3+3)=1÷2
A solidus is a necessary symbol in mathematics, it allows for easier reading of complex fractions, it saves on paper, ink.
https://mathworld.wolfram.com/Solidus.html

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Ok we have that problem and 2 "plausible" answers so make a little trick.
Change the 1 to be a X.
So instead of 6/2(1+2)= you make that 2 equations:
6/2(X+2)=9 and 6/2(X+2)=1. I never reach the obvious X=1 with one of them 🤷‍♂️

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Educators developed an acronym to help students remember how to apply the order of operations. Unfortunately it has a flaw, it does not consider multiplication by juxtaposition. How this came to be is unknown, it could have been deliberate or simply an oversight.

When calculators came on the scene that did apply multiplication by juxtaposition then educators had a problem. In North America manufacturers were asked to drop multiplication by juxtaposition in calculators since it created a conflict with textbooks which applied PEMDAS strictly. A few manufacturers did comply (Casio comes to mind) which resolved the conflict in NA and assured of sales of calculators in that market. This then created another conflict in other nations where multiplication by juxtaposition was taught. Several manufacturers have subsequently switched back to applying multiplication by juxtaposition. And then we have online calculators which also disagree with each other.

This is a case of educators trying to make life easier for students and then breaking mathematics using arithmetic rules.

In my mind Arithmetic is a mechanical calculation to arrive at a single evaluated number, or the 'answer'. Mathematics goes way beyond Arithmetic and is more involved in solving problems through the application of logic.

Another reason North America caught the PEMDAS bug is that America loves multiple choice tests. It allows educators to grade students using automation and is focused on students getting the right 'answer'. I was always graded in school mostly on my method and I would only lose a small amount of my final grade if the answer I arrived at was "wrong' due to a calculation mistake. Showing your method was a requirement to get a passing grade. In the US that has largely been lost in grade schools. I believe this is why Americans that have been taught PEMDAS and have been graded on their ability to get the right answer argue for the strict application of PEMDAS rather than look at the problem mathematically. Adding parentheses to "encapsulate" multiplication by juxtaposition would make mathematical formulas harder to read and write (by hand) and certainly make solving mathematical problems burdensome. Mathematicians are less interested in the minutia of evaluation and focus on the problem at hand.

In my mind logic should prevail insead of being a slave to arithmetic rules that may have limitations.