Why does PEMDAS work?

Date: 7/9/96 at 14:29:37
From: Anonymous
Subject: Why PEMDAS?

Dear Dr.Math,

I am having trouble with a few problems that involve PEMDAS 
(Parentheses, Exponents, Multiplication, Divison, Addition,
Subtraction). The real question I have is why does this order of 
operations really work?

Thanks a lot, 
Kathleen 

Date: 7/9/96 at 17:45:7
From: Doctor Ceeks
Subject: Re: Why PEMDAS?

Hi,

When you ask "why does this order of operations really work?", you
are asking something which suggests a slight misunderstanding about
math.

The order of operations in which one is to interpret a mathematical
expression such as "2+3 X 5" is a convention. This means that a long 
time ago, people just decided that the order in which operations 
should be performed should be such and so.  It has nothing to do with 
magic or logic.  It's just that some people decided to adopt a way, 
and it has stuck ever since.  It just makes communication a lot 
easier.

It's just like asking, "why is it that when I say 'cow', people tend
to conjure up an image of a beast in a field, possibly with spots?"
Well, it's because a long time ago, someone decided to call that beast
a cow. Now, everyone who speaks English knows this and so if you want
to make someone think of the big beast, you just have to say "cow" to
them...how efficient! It has nothing to do with the poor little
grass eating creature...it just has to do with what people have agreed
upon to make communication with each other a little easier.

So the answer to your question is that it works because you're just
learning how to interpret math formulas the way people have long 
decided to interpret them.

Now, if you're having trouble learning the order of operations, then
you have to remember that you're not having a problem with mathematics
(unless you're getting wrong answers because you're adding 
incorrectly). What you're having trouble with is learning a 
convention, like learning the grammar of a foreign language. Because 
it's just a convention, learning it is just a matter of practice, 
practice, practice. Start slowly and carefully. Eventually you'll 
have it memorized.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/