This was the question:

Order of Operations

I am having a very tough time in math and I want to know what are the steps of Orders of Operations. My teacher taught me. I told her that I did not understand the problem. She didn’t answer my question. So would you please give me an example of some of those problems?

Professor Homunculus' answer:

I don't know what kinds of specific questions you want, but you probably know that the order of operations is normally taught as:

• Parenthesis,
• Exponents, (like squares, cubes, square roots, cube roots, etc.)
• Multiplication,
• Division,
• Addition,
• Subtraction,

Which they usually call PEMDAS, and have you memorize it with the mind-numbing mnemonic "Please Excuse My Dear Aunt Sally."

What it means is that, say, if you have 4 + 2³/2 for example, you would have to do the 2³ first. And that would give you 8.

Now the equation is 4 + 8/2 . Much easier to do, nicht wahr?

In PEMDAS, division comes before addition, so you would have to do the 8/2 first, which gives you 4, making the equation now simply 4 + 4.

I think any teacher who doesn't understand that things like the order of operations are difficult for students, unless they have it explained to them well and to the satisfaction of each student, needs to go back to flipping burgers.

So, to rectify that situation for intelligent but neglected students like you, and help good teachers who just never had a chance to learn this stuff well themselves, I came up with the following. If you actually read it and think about it, it should clear up not only the order of operations, but why we need stuff like that in math.

Here goes:

Notes on PEMDAS: (Please Excuse My Dear Aunt Sally)

I have no Idea why they don’t explain this stuff in more detail in most math classes. They give you the rule, and expect you to learn it by rote without explaining it well, and adding the hints. And the explanation and the hints are so simple!

Here’s the deal:

First of all, let’s clear up the AS and the MD. A means addition and S means subtraction. Does the order of those two matter? Is there a difference between the answers to 4 + 7 - 3 and 7 - 3 + 4? Of course not. So why do they write "AS" and not "SA"? The answer is, "they could have"! But they had to write it in some order, so they picked "AS". It does not mean that you could not do the subtraction first. It is just a convention they decided upon, for convenience’s sake. Otherwise you could get something like PEDMSA.

This is something that will come up often in math, (or anything else for that matter). You should understand what convention means, and why conventions are used, so you don’t have to get frustrated by them. Unfortunately, it is seldom part of a school’s math curriculum to have you discover what they are. "Because that’s the way it is!" is one of the most ignorant and disrespectful things a teacher can say to you. Don’t fault them too hard for it, though; the odds are they had a jerk for a teacher, too.

Pondering point:
If you had understood about expediencies and conventions when you were young, would it have saved you a lot of frustration trying to use things you didn’t have a deeper understanding of at the beginning?

The same thing that goes for "AS" and "SA" goes for "MD" and "DM."
Multiplication and division are not interchangeable, as far as their order of operation goes. Why? Try it like you did addition and subtraction before.

I’ll wait.

Did you come up with something like:

Consider the expression: 20/5 x 4
If you divide first (left to right precedence), you get 16.
If you multiply first, you get 1

?

An observant reader did. She corrected my erroneous first version of this page. (Ευαριστω, Nancy!)

You will probably notice that in "PEMDAS", multiplication and addition both come before their "reverse" operations, division and subtraction. I believe that this is also convenient to remember that, at least in the acronym, the "destructive" operations come after the more "constructive" operations.

What about brackets? They are written like this []. Where do they fit in the mix?

Brackets come before all of these things. By the same convention that gave us parenthesis, we got brackets as a sort of extrapolation of the parenthesis rule. They just come in handy for some things, and we needed them, so we invented them. I do not think Moses came down from the mountain with them, they are just handy; that’s all. Maybe I am wrong about their origins, but they are still used before all the operations of PEMDAS.

Why don’t they teach us BPEMDAS, then?

My theory:
Mathematicians use what is convenient and effective to represent reality to the best of their ability. It is a hard science, and very difficult to bluff at. On the other hand, teachers can use what is expedient. BPEMDAS is an ugly acronym, and requires some serious tweaking. "What the heck, let’s just call it PEMDAS!"

Doesn’t it make you feel safer to know that they seldom use math teachers to do math that anyone’s life depends on, like rocket-science or bridge-building?

Penultimate PEMDAS note for today:
But Prison’s Even More Dumb And Stupid, as in "Math class sucks, but prison’s even more dumb and stupid".

Ultimate PEMDAS note for today:
There are things called “Curly Braces” that take precedence over all of the other things. They look like this {}. You don’t see them too often in the real world or basic math classes, though.

I hope that helps,

Brian

Math Mojo is part of Magic and Learning - a company that uses methods of magicians to teach thinking skills.