Back to School: Order of Operations and the Infamous “48÷2(9+3)” Meme

On the surface, this may seem like a math problem, but it’s not;  it’s a language and culture problem!

Culture

What is the proper way to greet somebody?

It’s easy to say a handshake, a smile or a “hello”, but this is based solely on our own cultural convention.  In North America, a handshake is quite acceptable, but in Japan it’s a bow, in some European countries, a kiss, and in dog society, it’s butt sniffing.

In the same way, my initial answer to this question was “288”, because I grew up on BEDMAS.  I was pretty sure that this was the answer, but upon hearing about the huge debate on the matter, a quick Google search revealed to me other cultures that I had yet to come across.  Before then, I had never heard of PEMDAS or BODMAS, the “division before multiplication” interpretation of BEDMAS or the implied multiplication rule.

We may argue about the conventionality of some conventions (e.g. the implied multiplication rule is not that common) or the logical integrity of others (e.g. multiplication is the same thing as division, so “multiplication before division” is a bit silly), but it is important to realize that we are arguing over nothing more than conventions.

Concepts Versus Notation

Whether we say the answer is 2 or 288, we are saying nothing about any mathematical concepts.  It’s important to distinguish between mathematical concepts and mathematical notation.  Mathematical notation is a specialized language that we use to convey mathematical concepts.  Like all languages, it arises out of convention, and conventions diverge in different locales into subcultures (like the BEDMAS split between the equal precedence interpretation and the division before multiplication interpretation).  The Internet has brought together these sublocales and what we are seeing in this meme is more of a cultural clash than a mathematical debate.

(Which is why I hate the emphasis on order of operations so much in school, and its portrayal as mathematical truth.  The teacher will say, “The answer to 5×4+3 is 23.”  The teacher generally does not say, “Notice that 5x(4+3)=60 and (5×4)+3 = 23, so by convention we do multiplication first.”  Of course, teaching conventions produces good workers, and gets practical things done.)

Psychology

There’s a psychological trick in the way the question is posed.

Does  48 ÷ 2(9 + 3) equal 2?

or

Does 48 ÷ 2(9 + 3) equal 288?

We look at it and say, “How can it be both things?”  It has to be one or the other.  When in fact, we actually do not have enough information because we have not yet specified which convention we are using.  (For example, what does “take” mean?  It could mean “grab something”, or it could mean “bamboo”, depending on if we interpret the letters to be in English or in romanized Japanese.)

If we used a less ambiguous notation, the answer would be clear:

48 ÷ (2(9 + 3)) = 2

(48 ÷ 2)(9 + 3) = 288

But this doesn’t mean we’ve circumvented convention, it’s just that most of our conventions happen to give braces (or parentheses) highest precedence.  We are still operating on conventions.  If it were conventional to represent the concept of multiplication with the symbol “=====” and the concept of “3” with the symbol “)”, then we would write 8×3 as “8=====)”.

Time Well Wasted?

Does it mean that the whole meme is a waste of time?   Well, that depeneds on what we do with it, what we learn from it, what perspectives have we glimpsed into, and which ones we choose to peer from.

How quickly are we to take our own cultural conventions and think of them as Truth, like I did.  But once we realize that there are other cultures and different perspectives out there, we can use this meme as a lesson for the other times we act culturocentrically, like in culturally based morality, gender roles, or political organization.  Because when we take our cultural conventions to be Truth in those contexts, it’s not just an interesting argument.  People die and suffer because of culturocentrism.  But if we open ourselves up to other possibilities, then not only will we have better mathematical notation, we will have a more peaceful, compassionate world.

In math, as in life, it’s not necessarily the solution that is the crown jewel, but the stuff we learn along the way.