Slate has an interesting article, by Tara Haelle, discussing a math problem that recently received some attention on Facebook. The problem is to evaluate this expression:
Obviously, the challenge here is not the arithmetic itself. It is to figure out the order in which to do the operations. I suspect most people would naturally do
the parentheses first, leading to this:
but what now? We could argue that we should first multiply the two by the three, leading to this:
which is obviously equal to 1. Alternatively, we could break up the expression this way:
which is equal to 9. The first one strikes me, personally, as more natural, but as it happens, the accepted convention says the second one is correct.
You surely learned the acronym PEMDAS back in elementary school. (Perhaps you also learned the mnemonic, “Please excuse my dear aunt Sally.”) The acronym stands for: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. That is, it tells you the order in which to carry out the operations in an expression like this. Even with the acronym, however, there is still some interpreting to be done. Doing the parentheses and the exponents first rarely causes confusion, but it needs to be understood that multiplication and division are done at the same time, working from left to right.
As I said, this is purely a convention. It’s not that the first interpretation is unreasonable, it’s just that, as it happens, that’s not the convention that’s been agreed upon. Haelle provides a very good discussion of these conventions, as well as some of the various notations that have been used for division over the years.
Frankly, though, PEMDAS is an abomination. I periodically teach courses intended for future elementary school mathematics teachers, and we inevitably spend a lot of time teaching about PEMDAS. The powers that be have decided that interpreting ambiguous algebraic expressions is a terribly important skill.
As a mathematician I find this all stupid and offensive. You see, no mathematician would ever write something as ridiculous as the expression with which we started. In fact, mathematicians never use the symbol for division. I just now had to look up how to make LaTeX produce such a thing. We always use the fraction notation. For example, if we meant our first interpretation we could have written:
whereas the second interpretation could have been written
which rather takes the fun out of the problem, since either one is completely unambiguous. Especially in research-level math, we routinely write very complicated algebraic expressions with tons of elaborately arranged symbols. It’s hard enough to decipher these expressions without adding the burden of an ambiguous order of operations. We’ve actually gotten pretty good at writing things clearly. It’s amazing what you can do with the careful use of parentheses and brackets, along with some sensibly chosen notation.
Yet we take school children at the peak of their curiosity, and teach them that math is all about memorizing arbitrary conventions for interpreting unnecessarily ambiguous algebraic expressions. And then we wonder why people hate math!
On a related note, I’m not a big fan of using a diagonal line for fractions. It can be useful for simple fractions, like 1/2 or 2/3. Formatting things in the more usual “vertical” manner can mess up the line spacing. But what are we to make of something like this: 1/2+3? Is that the fraction 1/5, or are we adding the whole number 3 to the fraction 1/2? Ugh!
One more thing. Mathematicians routinely refer to the top and bottom of a fraction. We say numerator and denominator too, but top and bottom is just fine. The reason it’s fine is that if you talk about the top and bottom of a fraction everyone knows what you mean, and clarity is what we care about. But if any schoolkid uses such language, he gets told that he is wrong to do so, and that he has to learn the jawbreakers if he wants to do math. Ridiculous!