The PEMDAS Paradox

By 
David Linkletter

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

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.