This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
When I became a math and science writer, I had no idea that one of the most common requests I would get would be to weigh in on order of operations problems that somehow go viral in some segment of the internet. The latest one I've seen is 8÷2(2+2).
My favorite headline for this one: “Viral math problem baffles mathematicians, physicists.” I am greatly amused at the implication that large numbers of mathematicians and physicists have abandoned their analytic number theory or quantum gravity to get to the bottom of an ambiguous arithmetic problem.
Questions like 8÷2(2+2) boil down to whether you compute the answer left to right or do the multiplication on the right side of the obelus (division symbol) first. Part of the reason this expression is ambiguous is that some people the acronym American students learn, PEMDAS (parentheses exponents multiplication division addition subtraction), as meaning multiplication comes before division when they should have equal priority. (Acronyms used in some English-speaking countries put the D before the M.) One method yields 1 as the answer and the other 16.
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I used to explain that when people sent me messages about some viral order of operations questions, but for the past few cycles (they seem to bubble up every several months like an algal bloom in a lake) I have adopted a strict policy of refusing to answer. I do believe one of the answers is more correct than the other, but I will not tell you which one.
The real answer, the one I believe any mathematician, physicist, engineer, other number-cruncher would tell you is to make sure your expressions aren’t ambiguous. There’s no extra charge for another set of parentheses. Just toss them in. If you want the answer to be 16, write (8÷2)(2+2). If you want it to be 1, write 8÷(2(2+2)). Problem solved. Some people leave school math classes believing math is a minefield studded with gotchas. It’s not supposed to be like that, and it’s a real shame so many people end up with that impression. Part of the job of anyone who writes an expression like that is to make sure it can be understood. If it’s ambiguous and someone gets the “wrong” answer, the blame belongs to the person who wrote the ambiguous question.
The bar is low, but I think the question of why these viral questions keep popping up is much more interesting than what the answers are. Is it cathartic for people who were traumatized by what they viewed as arbitrary rules in math? “See? Math is confusing and ambiguous!” Is it wanting to be right and shame people who are wrong? Maybe it’s the arithmetic equivalent of the dress or Yanny/Laurel: We can all understand the question, think we know the answer, and can’t imagine how anyone could see anything different. Is it just a relief to have passionate internet arguments about something much less important than healthcare or immigration? If you like these questions, what is it you like about them? (You can tell me on Twitter.)
Steven Strogatz has already written about this topic more elegantly for the New York Times. I’ll close by echoing his appeal to those of us who talk about math in public to help people see beyond mathematics as a subject with arbitrary rules and black and white answers to the more interesting questions that motivate mathematicians.