The PEMDAS Paradox

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 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.

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PEMDAS is an acronym developed by educators, and is flawed

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.

Implied Mathematics and distributive property dismissal

what happens when you replace 2 with -(1+2) or 5! or 2 arc tangent 60 degrees?
Does Right to left still dominate?

PEMDAS/BIDMAS/BODMAS are incomplete nonsense

Exactly. The more sophisticated and algebraically consistent implied multiplication should be standard. Just think about what should be the logical result when one divides collections or variables with units: distance/time = 100 km / 2 hours = 50 km/hour.

Order of Operations is intended to eliminate ambiguity

The reason this problem seems 'ambiguous' to people is because they lack understanding of the Order of Operations. Division and multiplication are the same and should be performed in the same step, from left to right. Subtraction and addition are the same and should be performed at the same step.

Note that when I say 'division and multiplication are the same,' I am saying that you can rewrite all division as multiplying by the reciprocal. The division happens to anything directly after the division sign. If there are no parentheses grouping the denominator (like a fraction bar naturally does), then only the first thing after the division sign is divided. Implied multiplication shouldn't take any more precedence than 'regular' multiplication that is shown. Also, when I say 'subtraction and addition are the same,' I am saying that subtraction can be rewritten as adding the opposite.

The order of operations in this problem is as follows:
Work inside parentheses (1+2=3).
Divide 6 by 2 (3).
Multiply 3 by 3 to get 9.

Any perceived 'ambiguity' is from a substandard understanding of the Order of Operations and/or understanding of division/multiplication features. This article suggests that someone jotting down 1/3x means 1/(3x) which is flat out untrue. Jotting that down while intending it to be 1/(3x) shows you don't understand how the Order of Operations ACTUALLY works.

Ambiguity

Order of Operations is not math. It is a convention, a linguistic agreement between a writer and a reader as to how to decode a written series of symbols into mathematical concepts. Ambiguity occurs when decoders disagree on the method of decoding. If I type 10-6×2 using common western numerals and mathematical symbols, people would come up with the answer of 8. However if I were evaluating 10-6×2 as transliteration of a mahgreb arabic math expression the correct answer would be 2. This is because mahgreb Arabs do order of operations from right to left. (Follow the order of their writing) if they had intended the answer to be 8, they would have written it as 2×6-10. Doing math left to right is just a made up rule devised by Eurocentric left to right readers. It is a convention, not a rule of math. Interpreting a/bc to mean either (a/b)×c or a/(b×c) do not break "rules of math" they just define two similar but distinct decoding interpretations Using one decoding method to attempt to disprove the other is simply circular reasoning. All you need to understand is that there are calculators that use one and calculators that use other. Textbooks and journals that use both and millions of people that USE alternate interpretations to decode math expressions. None answer is universally "right" or "wrong" it is "correctly decoded" or "incorrectly decoded" according to the convention you chose to use. If your purpose is to clearly communicate your intent to your reader, write your expression with sufficient parentheses so everyone, using either convention will correctly interpret your intent.

FB PEDMAS HELL

the dogmatism around the PEMDAS paradox is amazing - I shouldn't waste time on it, but it's a kind of mindless leisure - I ultimately found the APS submission guidelines, published by the American Physical Society, which address precisely the PEMDAS paradox, while never once mentioning PEMDAS, BODMAS, nor any other MASS, fitting expression considering the near-religious fervor of many proponents of these mnemonic devices

https://journals.aps.org/authors/bracketing-mathematical-expressions-h9…

the salient point: [...] "Use enough bracketing to make the meaning clear and unambiguous. Be especially clear with fractions formed with the solidus (/). According to accepted convention, all factors appearing to the right of a solidus are to be construed as belonging in the denominator" [...]

the real answer

The real answer to these "problems" posted on social media is "I don't care. I just want to stir up controversy and argument to create lots of comments on my post earning me increased $$$ value to my account." As long as people keep arguing this question, posters keep earning more money.

PEMDAS and the Order of Operations

PEMDAS may be a good mechanism for teaching grade school students but it has serious drawbacks especially when moving on to more complex expressions. One teacher, Jason Taff at the Burroughs School in St. Louis points out some of the shortcomings of the mnemonic and has come up with a method for teaching the order of operations based on identifying terms and factors (mathematics Teacher, Vol III, No 2, Oct 2017). Let me quote from that article, "PEMDAS fares poorly because it never was an order of operations to begin with. It has always been a hierarchy of operations. PEMDA tells us what comes first or second or third, when the more relevant concept involves which operations more closely bind numbers and expressions into their nested structure..." So here is where the real foundational differences appear between the sides with those seeing implied multiplication and juxtaposition of variables as more closely binding than explicit multiplication. The one proponents see the sub-expression 2(1+2) as a product in itself with a unique value as if it was surrounded by another set of grouping symbols. Why is this? Let's examine a few areas where grouping is implied though not shown overtly. Take for example the expression 8÷2^3 (8 divided by 2 cubed). One could not simply write as a step to calculate this 8÷2*2*2 since PEMDAS (using the strict left-to-right convention) would evaluate this as 16. So, the indicated exponentiation has to be enclosed in parentheses. Also, take 6÷3! (6 divided by 3 factorial). Again, one could not simply write 6÷3*2*1 as PEMDAS would solve this incorrectly. So, again, parentheses are implied. Now, let's talk about juxtaposition and implied multiplication. My college text, written by a Ph.D. Professor, clearly states that a÷bc is a÷(bc) or a over bc but not (a÷b)*c. For one thing, bc should be seen as a product before division takes place since we are allowed to use commutation and the associative properties of multiplication (in this case variables) to move things around so that bc=cb, and substituting this in the expression would result in different PEEMDAS solutions. that is problematic. Also, what about implied multiplication especially when it comes to factored expressions? Do we now just forget about the distributive property? After all, one could make the case that 2(1+2) is just the factored form of (2+4). Or in its expanded form, (2*1+2*2) using distribution. In each case, the viral expression would resolve to 1. Yes, not every expression showing implied multiplication is a factored version of some group, but even if not, do they not have a unique value in that the multiplier outside the group is the coefficient of that group (or in some cases as factoring also is a GCF). Further, many assert that the P in PEMDAS simply means doing the operation in the grouping. So, in the sub-expression, 2(1+2) they simply add (1+2) and assert that the group is resolved. One could take issue with that especially if it is a factored expression and the GCF is essential in computing the original value of the expression. One could go on about simplifying expressions especially when variables are involved, but let's leave things here and hopefully get some enlightening comments.

A question of notation.

6÷2(1+2) is a simple, well formed, unambiguous problem of division, a÷b, where a=6 and b= 2(1+2), which simplifies to 6. 6÷6=1
Question of notation: axb=ab, and/or any other character, for example c. If we want to show the factors of c, we write (axb).
Factor out the a from (axb) and we get a(b). Thus, ab=c=(axb)=a(b).
a(b) is not interchangeable with a x b, because a(b) represents a product, (a SINGLE number) while a x b is an expression, that is, TWO numbers connected by an operator. However, if we enclose axb in (), it becomes interchangeable with a(b).
Illustration: If 6÷6=1, and 6=2x3, does 6÷6 = 6÷2x3 or 6÷(2x3)????
Easy Rule: Whenever switching from Implicit to Explicit multiplication, ALWAYS enclose the factors involved in ().
Finally, remember that different calculators work differently. Read the instructions, and adjust input accordingly. We should be able to get the same answer on any and all calculators, every time, all the time. If we don't, the problem is probably with our input, not the calculator.

6 ÷ 2(1+2)

2x = [x + x]

So the statement "2x divided by 2x" is:

[ x + x ] ÷ [ x + x ]

...which, given that x is not zero, has a quotient of 1.

Let's say that in the statement "2x divided by 2x" the value of x is 3

Now the statement is:

[ 3 + 3 ] ÷ [ 3 + 3 ]

Inside each set of brackets, it's "3 + 3" on each side of the division sign, making the statement "6 divided by 6."

A monomial is a single term with ONE VALUE -- which is the PRODUCT of the coefficient multiplied by the variable (factor). With that understanding, a monomial never needs parentheses around it -- it's as if it already has parentheses around it, in much the same way an exponent attached to a base number never needs to be encased inside a set of parentheses. In both cases, you're told how many times to multiply a quantity.

The division sign (obelus), the slash (solidus), and the fraction bar (vinculum) all mean "divided by" -- they are synonymous. Therefore, those division symbols are interchangeable. If you do an internet search for "How to divide one monomial by another monomial," you will find example after example being worked through as a top-and-bottom vertical fraction, even if the division statement was originally written horizontally with an obelus or slash.

The statement "6 ÷ 2(1+2)," the numerator "6" can be factored out as: 2(1+2) ...making the statement: 2(1+2) ÷ 2(1+2) and if we replace what's inside the parentheses with the variable "x," the statement is:
2x ÷ 2x
which can also be written as the horizontal fraction...
2x / 2x
or as the vertical fraction...
2x
__
2x
...since all of those symbols mean "divided by."

The statement "6 ÷ 2(1+2)" is dividing one monomial by another monomial -- it's "2x divided by 2x." Because a monomial is a SINGLE TERM with a SINGLE VALUE, the coefficient cannot just be "peeled off" & used in some other operation as if it had no bearing on the variable (factor).

2x is equal to: (x + x), Therefore, the "2" in "2x" is not a stand-alone number unto itself -- it's the coefficient in a monomial which tells you how many times to multiply the factor (which is actually adding the factor to itself), in much the same way that an exponent tells you how many times to multiply the base number by itself. Just as the exponent is "attached" to the base number, the coefficient of a monomial is "attached" to the variable (factor). So the monomial division statement "2x divided by 2x" is: [ x + x ] ÷ [ x + x ]

Notice that the coefficient number of 2 "disappears" when the statement is written out in its most basic form (as the indicated additions). That proves, once and for all, that using the coefficient of the monomial (the "2" in "2x") in some other operation is not valid.

Division is fractions. Fractions is division. We all know how to work through a fraction: Do all the operations indicated in the numerator, then do all the operations indicated in the denominator, and finally divide the numerator by the denominator.

PEMDAS is incorrect for "processing" division statements.

Division of Monomials

According to Study.com:
"Monomials are the product of a coefficient, and a variable or variables."

from Cue Math teaching website:
"Practice Questions on Dividing Monomials"
"Q.1. Divide. 15a^2b^3 ÷ 5b"

The correct answer is listed as: 3a^2b^2

That means that even though an obelus was used to indicate "divided by," the statement was treated as a top-and-bottom vertical fraction, with 15a^2b^3 as the numerator & 5b as the denominator -- with NO PARENTHESES anywhere in the statement.

The "2" in "2x" is not a stand-alone number -- it's the coefficient in a monomial which tells you how many times to multiply a quantity (which is actually adding the quantity to itself), in much the same way that an exponent tells you how many times to multiply the base number by itself. Just as the exponent is "attached" to the base number, the coefficient of a monomial is "attached" to the variable (factor). Thus, the monomial division statement "2x divided by 2x" is: [ x + x ] ÷ [ x + x ]. Notice that the coefficient "disappears" when the statement is written out in its most basic form (as the indicated additions of the quantity). That proves, once and for all, that "peeling off" the coefficient of the monomial (the "2" in "2x") & using it in some other operation is not valid.

In the statement "6 ÷ 2(1+2)," the numerator "6" can be factored out as: "2(1+2)," making the statement:

2(1+2) ÷ 2(1+2)

Replacing what's inside the parentheses with the variable "x," the monomial division is:

2x ÷ 2x

If x equals 3 [expressed as (1+2) ], then the statement "2x ÷ 2x" can also be written as: "6 ÷ 2(1+2)," which is 6 divided by 6 -- which has a quotient of 1.

Division is fractions. Fractions is division. Do all the operations indicated in the numerator, then do all the operations indicated in the denominator, and finally divide the numerator by the denominator. Division has to go LAST. PEMDAS is incorrect for division statements -- because they're fractions.