Inspired by this post at Education Week, by David Ginsburg, and a fruitful discussion in the comments, I thought I’d make my own post about the order of operations. I feel the need to do this because of my own experiences teaching Algebra 1 and correcting the habits of Algebra 2, Precalculus, and Calculus students. The problem is often with their misunderstanding of the ubiquitous “PEMDAS” approach to order of operations (parentheses, exponentiation, multiplication, division, addition, subtraction).
The classic example given by those who have discussed this before is an expression like the following. This example shows the problem with instructing students to simply “add then subtract”:
You can easily see that blindly following PEMDAS would give you an answer of 3, not 5. In reality, addition and subtraction can be done in any order, provided that you think of the expression this way:
I have to constantly reinforce in my students the notion that the “-” goes with the 4. It owns it. That’s the right way to think of it, since subtraction (and division) are just shorthand for addition by the additive inverse (and multiplication by the multiplicative inverse). That’s right kids, subtraction is an illusion. The typical Real Analysis approach is to use the field axioms to define the properties of addition. Here they are:
- If
and 
are in the field, so is 
(closure).
(commutativity)
(associativity)- There exists an element
such that 
for all (identity)
- For each there is an element called
such that 
(inverse)
There are almost identical ones for multiplication, and then one more axiom for distribution. Just listing the field axioms for addition helps us in the discussion of order of operations. As I said above, subtraction is never really mentioned. We only later define 
to mean 
.
Similarly, division is an illusion. Or more precisely, it is defined in terms of multiplication. We define 
to mean 
.
The other important thing to notice is that addition and multiplication are binary operations, meant to be functions of two variables. So the expression 
is defined to mean 
.
But the problems with PEMDAS don’t stop with multiplication and addition. There are many other issues with the order of operations as well. We lie to students when we just tell them they’ll encounter those limited operations in tidy situations. Here’s a perfectly legitimate expression with which PEMDAS would flounder:
The reason the expression is so difficult for PEMDAS is because PEMDAS doesn’t acknowledge the existence of other functions or grouping symbols, and ignores the possibility of nested exponentiation.
As another somewhat related example, I often have Calculus students who evaluate the following expression incorrectly (because they misuse their calculator):
And some high school students at every level still struggle with evaluating these expressions:
The wikipedia article on Order of Operations is very clear on all this, but I wanted to just highlight some of the common misconceptions. For more, feel free to check out that article. I’m in no way recommending that PEMDAS be removed from math curriculum, but that it be used with great care. And by the way, the expression above evaluates to 1.
should be evaluated. We’re just unclear on how 2^3^4 should be evaluated.
, evaluate 
.
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