8÷2(2+2)

8÷2(2+2)

By Bobby Neal Winters

It is a rare day when a bit of mathematics is in the news, and it is an even rarer one when it is a bit that experts and laypersons can talk about with equal confidence, equal accuracy, and be equally wrong.

Today is one of those days.

This is because the 8÷2(2+2) meme has been going around the Internet.  It is a corker to be sure. Is it 16? Is it 1? Is it a communist conspiracy?

Well, as with all such things we should treat it as a teachable moment and in several directions.  The first thing if ought to say is that if you type it into a computer program interpretive interface, like the IPython console, for instance, it will give you 16 every time.

This is not a definitive answer, however, because dealing with computer input and output is HARD and allowances have been made.  It has been a special case since computers came into existence and there have been multiple methods of dealing with it.  One of these was quite good and it was called Reverse Polish Notation (RPN), and if you ever owned an old-fashioned HP programmable calculator in the eighties, then you’ve used it.  As you may have guessed, there is something called Polish Notation (PN) as well that has been used in dealing with computer input. It would take us too far afield to discuss either of these, but there are nice articles on Wikipedia that discuss both.

Whenever I get to teach Elementary Statistics, I use an Excel spreadsheet for some applications and I’ve learned that dealing with multi step calculations can be confusing, so I teach coping mechanisms as I go along.  We approach them with care because everything has to be typed in in a linear fashion and we can’t use all of the methods that mathematicians standing in front of a blackboard or writing in longhand on a piece of paper have at their disposal.

This brings me to something that has bothered me since the first time I saw this meme: The used of the symbol ÷ for division.  I haven’t used it in years.  You cannot find it on a computer keyboard.  To type this into a computer, you write 8/2(2+2).  When I see this, the ambiguity pops right out at me.  I would teach students to write either (8/2)(2+2) or 8/(2(2+2)) so that the problem goes away.  That ÷ symbol was put in there on purpose to make a point.

My dad was a truck driver and truck drivers have a saying about the right of way in driving: He was right, but he dead right.  As a mathematician, I need to keep that sentiment alive. If you follow the order of operations (parenthesis, exponentiation, multiplication division, addition, subtraction) you will get 1.  However, you can’t count on computers do that; you can’t count on people walking around on the street to do that; therefore, you need to order your mathematical communications in such a way as to be as unambiguous as possible.

So I’ve answered the first two questions, 16 if you are a computer and 1 if you are a mathematician, what about the third, is it a communist conspiracy?

Maybe not communist, but the use of the ÷ symbol makes me believe there is something going on here that requires a bit of technical sophistication.  I had to look up how to get my computer to do that symbol.  Maybe it is easier for others, I will be open to learning that, but I had to use [control][shift]uf7 to get Google Docs to do it. Its use in this meme not only hides the ambiguity of this expression from professional mathematicians, it makes the question accessible to people who’ve had no exposure to mathematics beyond eighth grade arithmetic.  Is this good? Is this bad? I don’t know, but it is interesting.

Perhaps I am sensitive to this because there is a fairly steady stream of anti-mathematics education memes that go around in social media.  They hate the new methods of teaching subtraction; they hate the common core; they think we should stop teaching algebra all together and teach how to balance checkbooks instead (that is not an either or, by the way, we do both, but the second we don’t do in math class).

So the final lesson that should be taken away from this is that we who teach math can’t ignore this.  We must engage it in some way.

So the answer to 8÷2(2+2) is that we need to talk.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. )