Order of Operations

Date: Mon, 21 Aug 95 20:37:50 
From: Forman Howes
Subject: ORDER OF OPERATIONS

What is the reason for the order of operations in a math equation?  Is it just
convention to do multiplication and division before addition and subtraction or
is there a deeper reason for this.  If there is a deeper reason, and I suspect
there is, please include in your answer parentheses and exponents.
Thanks, Forman

Date: 8/27/95 at 13:56:22
From: Doctor Ken
Subject: Re: ORDER OF OPERATIONS

Hello!

Well, the concept of "order of operations" is really one that's not inherent
to the structure of mathematics, but rather to mathematical notation.  
What I mean by that is that order of operations refers to which operations
should be performed in what order, but it doesn't actually dictate anything
(nor is it dictated by) the operations themselves.  So in a sense, it's
just convention.  

So if that's true, we should be able to use different "orders of operations"
and come up with a perfectly consistent mathematical system.  And in fact, 
we can.  Here's an example of the same expression being expressed in three
different notation systems, resulting in three different orders of
operation.

5 + 7 x (3-2)

3 - 2 x 7 + 5

5 7 3 2 - x +

The first expression is the standard one that most people use when writing
things down.  You do the 3-2 first, then multiply by 7, then add 5.

The second expression is what you would key into a normal calculator (the
kind you might find in a cereal box or something, not some fancy one).  
You do the subtraction first, then when you hit the times key it takes that
answer as imput to the next operation, and so on.

The third expression is written in the order in which you'd key it into my
Hewlett-Packard fancy calculator.  First you give it all the numbers, then
you tell it what to do with them.  So the operations say "subtract the 
two preceding numbers, then take the result of that and multiply it by the
number preceding it, then take the result and add it to the preceding
number."  

Also, we could invent our own order of operations in which we adding takes
a higher precedence than multiplying.  So if we wanted to convert
(3+4) x (7-9) x 3 + 6 to the new notation, we'd write it as
(3 + 4 x 7 - 9 x 3) + 6.

So the notation tells you which operations to do first, not the underlying
mathematics.  Of course, there are some pretty good reasons for doing it
the normal way: it's much less awkward than the other methods.  For 
instance, how would exponentiation fit into the scheme of the new notation?

-Doctor Ken,  The Geometry Forum