# No, You Can’t Say What 6/2(1+2) Equals

I am made aware that a section of Twitter argues about how to evaluate an expression. There may be more than one of these going around, but the expression I’ve seen is:

$6 \div 2\left(1 + 2\right) =$

Many people feel that the challenge is knowing the order of operations. This is reasonable. That is, that to evaluate arithmetic, you evaluate terms inside parentheses first. Then terms within exponentials. Then multiplication and division. Then addition and subtraction. This is often abbreviated as PEMDAS, and made into a mnemonic like “Please Excuse My Dear Aunt Sally”.

That is fine as far as it goes. Many people likely start by adding the 1 and 2 within the parentheses, and that’s fair. Then they get:

$6 \div 2(3) =$

Putting two quantities next to one another, as the 2 and the (3) are, means to multiply them. And then comes the disagreement: does this mean take $6\div 2$ and multiply that by 3, in which case the answer is 9? Or does it mean take 6 divided by $2\cdot 3$, in which case the answer is 1?

And there is the trick. Depending on which way you choose to parse these instructions you get different answers. But you don’t get to do that, not and have arithmetic. So the answer is that this expression has no answer. The phrasing is ambiguous and can’t be resolved.

I’m aware there are people who reject this answer. They picked up along the line somewhere a rule like “do multiplication and division from left to right”. And a similar rule for addition and subtraction. This is wrong, but understandable. The left-to-right “rule” is a decent heuristic, a guide to how to attack a problem too big to do at once. The rule works because multiplication-and-division associates. The quantity a-times-b, multiplied by c, has to be the same number as the quantity a multiplied by the quantity b-times-c. The rule also works for addition-and-subtraction because addition associates too. The quantity a-plus-b, plus the quantity c, has to be the same as the quantity a plus the quantity b-plus-c.

This left-to-right “rule”, though, just helps you evaluate a meaningful expression. It would be just as valid to do all the multiplications-and-divisions from right-to-left. If you get different values working left-to-right from right-to-left, you have a meaningless expression.

But you also start to see why mathematicians tend to avoid the $\div$ symbol. We understand, for example, $a \div b$ to mean $a \cdot \frac{1}{b}$. Carry that out and then there’s no ambiguity about

$6 \cdot \frac{1}{2} \cdot 3 =$

I understand the desire to fix an ambiguity. Believe me. I’m a know-it-all; I only like ambiguities that enable logic-based jokes. (“Would you like ice cream or cake?” “Yes.”) But the rules that could remove the ambiguity in $6\div 2(1 + 2)$ also remove associativity from multiplication. Once you do that, you’re not doing arithmetic anymore. Resist the urge.

(And the mnemonic is a bit dangerous. We can say division has the same priority as multiplication, but we also say “multiplication” first. I bet you can construct an ambiguous expression which would mislead someone who learned Please Excuse Dear Miss Sally Andrews.)

And now a qualifier: computer languages will often impose doing a calculation in some order. Usually left-to-right. The microchips doing the work need to have some instructions. Spotting all possible ambiguous phrasings ahead of time is a challenge. But we accept our computers doing not-quite-actual-arithmetic. They’re able to do not-quite-actual-arithmetic much faster and more reliably than we can. This makes the compromise worthwhile. We need to remember the difference between what the computer does and the calculation we intend.

And another qualifier: it is possible to do interesting mathematics with operations that aren’t associative. But if you are it’s in your research as a person with a postgraduate degree in mathematics. It’s possible it might fit in social media, but I would be surprised. It won’t draw great public attention, anyway.

## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

## 8 thoughts on “No, You Can’t Say What 6/2(1+2) Equals”

1. There are two more sources of ambiguity in the expression 6/2(1+2) that you didn’t mention:

(1) The fraction bar is problematic in typed math. When the math can be properly typeset, the fraction bar acts as implied brackets linking everything in the numerator together, and everything in the denominator together. The social media meme does not clearly indicate the limits of the denominator.

(2) The distributive property is a fundamental principle of math that (to me at least) feels stronger than a typical multiplication exercise. The social media meme plays on this by using the number-stuck-to-parentheses appearance of the distributive property to increase the ambiguity of the denominator.

I really dislike memes of this sort. They reinforce the cultural view of math as a confusing list of rules designed to trap the unwary student.

Perhaps I should try starting a “Number Yoga” meme: You have the digits 1, 2, 2, 6, and any math operation you like. How many different values can you create? Hmm … [Off to play with graphics!]

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1. These are great points you’ve mentioned, yeah. The meme does depend on people remembering, more or less rules without remembering why the rules are there and that’s, at least, a sign of trouble with people’s understanding of mathematics.

Although I do understand why people look for and try remembering The Rules. For an expression that isn’t written as a trick, and that is just something they want to calculate, The Rules give guidance and aren’t necessarily worse than having a detailed recipe to follow when making dinner.

That sort of number yoga could be good fun. I remember reading how Paul Dirac broke such a challenge to make numbers using exactly four 2’s. I suspect that 1, 2, 2, 6 wouldn’t lend itself to that sort of exploit, though.

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1. I’ve never heard the Paul Dirac story. Please tell! (Or do you have a link?)

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1. Ah, this comes from Graham Farmelo’s biography of Paul Dirac, The Strangest Man. There was a party game going around: how many of the counting numbers can you make using the standard arithmetic operations and exactly four 2’s?

So, for example, you could write $1 = \frac{2}{2} \cdot \frac{2}{2}$. Or $3 = 2 + 2^{2 - 2}$. Or $8 = 2^{2^2} \div 2$.

Dirac looked it over and worked out a scheme that works for any counting number.

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1. Wow! That doesn’t even seem possible.
Off to check my library for that book….

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