### Curiosities

# 8 ÷ 2(2+2) = ? Where Do You Land?

Here’s a fair warning: there will be math in this story. It won’t be particularly hard, but it will be controversial. The starting point is pretty straightforward. Solve this formula: 8 ÷ 2(2+2) = ? A math formula was floating around social media along with a taunt that went something like ‘only one in a million will get this right!’ Are you one of the one-in-a-million? Or one of the other 999,999? What answer did you get for the formula?

8 ÷ 2(2+2) = ?

## Sixteen Or One?

Many people, after much key-punching, hand wringing, and paper crumpling, came up with 16.

Other people — there didn’t seem to be any reliable statistics to tell us how many people were in one group or the other — calculated, after a roughly equal amount of key-punching, hand wringing, and paper crumpling, one.

Both groups seemed equally certain and confident that they were correct and that the others were wrong.

## How To Calculate The Different Answers

If you remember the order of operations from elementary school arithmetic, you probably remember that the first step one takes when faced with a formula like this one is to deal with the brackets first*. *Both groups agreed on this. With that step aside, we’re left with:

8 ÷ 2(4) = ?

The next question is, quite literally, “now what?” What portion of the formula needs to be worked out next? The one group tackled the 8 ÷ 2 piece first, leaving them with 4(4), which obviously got them to 16.

The second group, on the other hand, tackled the 2(4) piece first, leaving them with 8 ÷ 8, which just as obviously leaves them with one.

## Who Was Right?

When you were taught the order of operations in elementary school, did your teacher use the acronym PEMDAS, that may have stood for “Please Excuse My Dear Aunt Sally?” PEMDAS tells us to conduct our calculations in the following order: Parentheses, Exponents, Division/Multiplication, then Addition/Subtraction.

Some people of a certain age may have been taught with “BEDMAS” or “BODMAS,” which uses “brackets” instead of “parentheses” and switches up the order of “multiplication/division.”

The problem with applying PEMDAS to this formula is, of course, the meddlesome “/” between MULTIPLY and DIVIDE. It’s supposed to indicate you can M or D in either order, no big diff. Except, of course, it does make a big diff in this case.

## Again… Who Was Right? Who Was Wrong?

The possible permutations are many and various. If you use PEMDAS and M before you D because “My” comes before “Dear”, you will get one for an answer. If you use PEMDAS and D before you M (because it’s alphabetical or just because the “/” means you can do whatever you want), you will get an answer of 16.(We’ll leave the BEDMAS and BODMAS versions out of this controversy; they are not helpful additions to the mix.)

Some students were taught when in pinches like this, to work from left to right. Which would make every one of those kids — assuming they did the brackets first — get an answer of 16. Were they right?

It turned out, there really wasn’t any right answer. And there wasn’t any wrong one. Neither answer was wrong. The *question was* what’s wrong.

## The Question Is Wrong

In an August 2019 *New York Times* article called “The Math Equation That Tried to Stump the Internet,” Steven Strogatz said:

No professional mathematician would ever write something so obviously ambiguous. We would insert parentheses to indicate our meaning and to signal whether the division should be carried out first, or the multiplication.

When we can’t agree on the answer, can we all agree that the problem is with the question? After all, Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University.

Who’s to argue with him? Strogatz said this question isn’t so much a question as it is a “brickbat” — something we use to beat each other over the head with. It’s an excuse to argue, disagree, and to insist on being right.

Let’s be better than that. The world doesn’t need more brickbats and excuses to beat each other about the head. It needs more peace. More love!

And less math.

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