Order of operations

Why do we have order of operations? Well it is pretty simple--so we can all read the same expression in the same way. It is all about communicating to one another effectively in the language of mathematics.

if What wrote a in sentence words the I of order any?

It would be really hard to understand what I meant--but not impossible. I bet with some effort you could figure out my sentence. And that is because English has some rules that we "just know" for the most part. The rest we call grammar.

We learned the basics of grammar long, long ago when we were just little kids. And mathematics is the same way. Unfortunately, we don't "speak" much math to little kids so there isn't much in the way of the basics of the mathematical language in our heads. We have to learn it the "formal way."

But that is ok. The rules are very simple--way more simple than the rules of the English language. And PEMDAS sums it all up. I have a post explaining what that means and how to use it here.

But the long and the short of it is that PEMDAS is a set of simple rules that keep us all speaking the same language.

Here is an example.

in the box

That is a nice prepositional phrase. We never see it written "box the in." There is an order that we MUST follow here. But after we do, we can put that phrase in many locations relative the thing that is "in the box."

Put the sweater in the box.
Put the sweater that I love in the box.
In the box goes the sweater.
Going in the box is the sweater.

Some sentences are better than others but you can understand the meaning of all of them.

Now, in mathematics.

(3 + 4)

This is the same as the phrase "in the box." I can move it around in an expression.

8*(3+4)
(3+4)*8
(3+4)/8
(3+4)^5
(3+4) + (5+6)

When we follow know the order of operations, we can construct very complex and terribly organized equations, yet all will be able to read it. Of course we strive to make our equations simple and beautiful--but that is were experience comes into play.

The order of operations is simply part of the grammar of mathematics making a jumble of symbols and numbers make coherent concepts.