The Elephant in the Pajamas

I’m not much on Internet mems, but this one caught my eye. What is the answer to the equation 8÷2(2+2)? If you’ve not already done so, try it and see what you get. Did you come up with 16 or 1? What answer we get says a lot not only about how we learned math, but about mathematics itself.

Mathematics, based on rigorous proof of concepts, doesn’t lend itself to ambiguity. It’s only when we learn quadratic equations that we enter the world of two equally valid mathematical answers. If mathematics was ambiguous, it would make for a poor tool, indeed. But while mathematics isn’t ambiguous, language can be. Though we
don’t think of equations (called algebraic notation) as a language, they are. It’s a rigorously constructed language, to be sure, but still a language. What an equation means depends as much on an agreed upn set of rules as any sentence in any language./p>

This sort of ambiguity in language shows up in a Groucho Marx joke from Animal Crackers: “One morning I shot an elephant in my pajamas. What it was doing in my
pajamas, I’ll never know.” The gag is the unexpected, but equally valid, interpretation of the first sentence. What it means is all in how you look at it.

Simple equations, like simple sentences, aren’t much of a problem.
“Bill hit the ball” is as clearly understood as 5 + 3. It’s only when our sentences and equations become more complex that the need for an agreed upon set of rules becomes important. In language, this is called grammar. In mathematics, it’s order of operation. What is the answer to the equation 5 + 3 ˣ 2? If you said “11,” you’re correct, because 5+3 ˣ 2 = 5+6 = 11. But why should that be the case? Why not 5+3 ˣ 2 to 8 ˣ 2 to 16? Because, by the order of operation, we multiply 3 by 2 before we add the result to 5. The order of operation is such a basic thing, we’re taught it relatively early in school, and use it unconsciously. If you went to school in the United States, you likely learned the mnemonic “Please excuse my dear Aunt Sally.” That stands for parenthesis, exponents, mathematics, division, addition, subtraction,” which is how we remember how to
evaluate an equation. This is known as PEMDAS. If you went to school somewhere else in the world, you probably learned brackets, operators, division, multiplication, addition, subtraction, or BODMAS. Someone who learned BODMAS would likely evaluate 8÷2(2+2)
as 8÷2(2+2) = 8÷2(4) = 4(4) = 16, while someone who learned PEDAS would likely evaluate the same equation as 8÷2(2+2) = 8÷2(4) = 8÷8 = 1. This is because BODMAS names division before multiplication, and PEMDAS names multiplication before division.

Having learned PEMDAS, I came up with 1. The idea that it could be 16 provoked a vicersal reaction. Nor was I alone. This equation provoked considerable, sometimes heated, discussion across the Internet. So, what is the correct answer?

That led to some casual research into orders of operation, and through a surprising foggy patch of mathematics. It seems that the question of whether multiplication should always be performed before division has been going on for a long time. In our era, the concensus is that multiplication and division are equally ranked in the order of
operation, as are addition and subtraction. This means you evaluate multiplication and division as you go through the equation from left to right. 8÷2(2+2) should
be solved by first evaluating what’s in the parenthesis, giving 8÷2(4). Then, working from left to right, 8÷2(4) = 4(4). Lastly, multiplying 4 by 4 gives the answer of 16. It doesn’t
matter if you learned BODMAS or PEMDAS: Since multiplication and division are on the same level of operation, you should come up with the same result.

If that goes against the grain, as it does to all of us who thought multiplication came before division, consider this: Is 8 ÷ 2 different than 8 ˣ ½? Is 4 – 2 different from 4 + (-2)? In both cased, they’re the same.

To those of us who thought the order of operation ranked multiplication before division (or division before multiplication, for that matter), giving division equal ranking with multiplication might seem like a new thing. The surprising answer is “Maybe.” While there’s some evidence that the order of operation we know predates algebraic notation itself, the question of whether multiplication should take precedence before division rocked on into the 20^th Century. Opinions seemed to mostly hold to the same understanding we have now, but the multiplication prior to division opinion also shows up in at least one textbook, while there are some that side-step the issue by calling it a gray area. The picture we get, looking at textbooks from a century ago, is that placing multiplication and division on the same level was already winning out.

Regardless of how we reached the current understanding of the order of operation, placing multiplication and division on the same level is how it’s understood now. PEMDAS still stands for parenthesis, exponents, multiplication, division, addition, subtraction, but can perhaps be more clearly understood as parenthesis, exponents,
multiplication and division, addition and subtraction. We have to keep in mind multiplication and division are on the same level, worked as we come to them from left to right, just like addition and subtraction.

If that still seems a little confusing, it might help to think of it like this:

First Parenthesis
Then Exponents
Then Multiplication and Division
Then Addition and Subtraction

For some of us, that might take some getting used to, but that’s the way it’s done.

Here’some simple problems. Answers given below.