__ PEMDAS__ or Please Excuse My Dear Aunt Sally is what most students think of when they think of the Order of Operations. P.E.M.D.A.S. stands for Parenthesis, Exponents, Multiplication, Division, Addition, and then Subtraction. It refers to the order in which you perform those operations.

I recently found an article that was critical of the P.E.M.D.A.S. acronym stating that it doesn't always work. Of course this peaked my interest. Here is essentially what the article said.

What does 5 - 3 + 2 equal? If you answer 4, good job! You're absolutely correct. If you have $5, spend three, and then get another $2 - you have $4 - right? If you use P.E.M.D.A.S. they say you would add 3+2 and get 5, then the equation would be 5 - 5 = 0. A similar problem occurs with multiplication and division. When deciding which to do first, it is not Multiplication before Division. The rule is that you do equal operations left to right. So, instead of P.E.M.D.A.S. it should read P. then.E. then either M or D depending on which is listed first reading left to right, then either A. or S. depending on which is listed first reading left to right..

The problem isn't just with PEMDAS, there are other rule-driven examples where the rules don't always work. Don't get me started on F.O.I.L. (not a big fan of F.O.I.L.). It works, but only if you're multiplying pairs of terms.

The Order of Operations is extremely important. Basically, when you're building an equation, you do the higher level stuff first. When we are expanding or taking an equation apart, it's just the reverse. Let's look at an equation and it's order of operations.

7 + (3 x 2^{2} + 5) Before I add that 7, I work with what is inside the parenthesis. And the highest order of operation in the parenthesis is the 2^{2} which is 4. Now we're left with 7 + (3 x 4 + 5), you might note that this is an expression as opposed to an equation. We are not saying this is equal to anything. You might also note that I am using x for multiplication, not as a variable. O'k, still inside the parenthesis the next complexity is the multiplication. 3 x 4 = 12. So, now we have 7 + (12 + 5) . All we have left to do is add, and we'll get the same answer regardless of order - but we're supposed to add the 12 to the 5, because they are in the parenthesis. 7 + (17) = 24.