As most of us know, the basic operations of arithmetic are performed in a specific order. This is known as the order of operations, and is usually recalled by the mnemonic PEMDAS: parentheses; exponents; multiplication and division; addition and subtraction. But even though we learn about PEMDAS in school, it’s important to understand that there are several falsehoods associated with it.
The first is rather subtle: It’s that arithmetic operations must be done in this order. This is true…except it isn’t. In particular, while the order of operations is important, the actual order is less important than the fact that we agree on what the order should be.
The order of operations is a convention: it’s an agreement on how we do things, but there’s no mathematical justification for it. We don’t need to do multiplication before addition, any more than we need to drive on the right side of the road. It’s certainly possible to drive on the left:
What’s important is not which side of the road we drive on, but that we agree which side of the road we drive on. If some of us choose one side and some of us choose the other, then that’s a problem.
In fact, PEMDAS came about because there was no general agreement on which side of the road we should drive on…I mean, because there was no standardization of the actual order. It was only in the early years of the 20th century that the idea of a universal agreement on the order of operations came about (and a good thing, because soon after we were building electronic computers, and unless there was an agreement, a computer built in Britain might produce answers different from one built in the United States).
Another falsehood told about PEMDAS is that it’s PEMDAS. It should actually be PEM/DA/S.
The MD in PEMDAS stands for multiplication and division. In PEMDAS, multiplication is listed before division, suggesting that in an expression like , you should multiply , then divide .
But in fact, multiplication and division are equiprecedent, meaning they are handled simultaneously. Again, this is a convention, like driving on the right side of the road.
This is unfortunately impossible, since one of them must be done first. But which one?
The answer is that we do them in the order they appear, from left to right. In fact, the correct way to state the order of operations is:
All arithmetic operations are to be done from left to right, UNLESS…
Thus: , since we take care of first, then multiply the result by .
There’s another problem with the way the order of operations is usually stated: PEM/DA/S is better, but PM/DA/S is better still.
That’s because exponentiation isn’t an operation.
And here’s the lie that your math teacher told you: You don’t do exponents before multiplication and division, because exponentiation isn’t an operation.
Exponentiation is a shorthand way of writing out a multiplication. Remember that means . Consider the expression: . Generations of students, having memorized PEMDAS, “do” the exponent first, and find .
But in fact, is not an operation: it’s shorthand for . In which case, .
Why does this matter? There are several reasons. First, if you try to find using PEMDAS and finding first, you’re stuck with the product . On the other hand, if you recognize that is shorthand for “Multiply four 2s together,” then you have the much simpler task , where multiplying the 5 and 2 together gave us an easier product.
More importantly, consider scientific notation: . No one in their right mind wants to calculate (even though it’s not particularly difficult). And they don’t have to: scientific notation is shorthand. In this case, is shorthand for the number we call one trillion, so is 8 trillion. Likewise, shouldn’t be treated as the product of $5$ and whatever you get when you do the E in PEMDAS: is shorthand for millionth and this number is 5 millionths.