Why Do We Have the Order of Operations?

Date: 03/05/2006 at 13:31:40
From: Oliver
Subject: Why was the order of operations created, and who created it

I have always hated the order of operations.  It seems like a stupid 
way to do things.  Why was it made, and who and when was it made?  

I understand that it is a convention, but why don't people just 
simply calculate from left to right?  It seems much more logical.  If 
I had to answer scientific formulas, could I use the order of 
operations?  I mean, does jumping around from number to number work 
in real life equations?

Date: 03/05/2006 at 22:41:01
From: Doctor Peterson
Subject: Re: Why was the order of operations created, and who created it

Hi, Oliver.

You can find some thought on when and why the rules developed here:

  History of the Order of Operations
    http://mathforum.org/library/drmath/view/52582.html 

I can suggest a couple examples to illustrate why it seemed natural to
the early developers of algebraic notation.

First, think about one of the better known formulas:

  A = pi r^2

for the area of a circle. What does that mean?  It says to multiply pi
times the square of the radius.  Do you see how the order of 
operations is necessary to interpret even such a simple formula 
correctly?  If you just took it from left to right, it would tell you 
to first multiply pi times r, and then square the result, which is not 
what you need to do to get the area.  In order to make it read 
correctly, you would have to either be very careful about order, and 
write

  A = r^2 pi

or to use parentheses:

  A = pi (r^2)

Now consider a common type of expression in algebra, the polynomial.
This is a sum of terms, each of which can be a product of a number and
some power of the variable:

  3x^2 - 4x + 2

If we just went left to right, we couldn't write it that simply, but
would have to use parentheses something like this:

  3(x^2) - (4x) + 2

The standard order of operations is designed to make it easy to write
these types of expressions!

Finally, think about the properties of operations, such as the
commutative property of addition.  This lets us rewrite something of
the form

  a + b

as

  b + a

if it helps.  For example, we can rewrite

  2x + 3y + 4x

as

  2x + 4x + 3y

so that we can combine the x terms and have

  6x + 3y

But without the order of operations that says to do each
multiplication first, you just couldn't do that; you'd always have to
be very careful about the order of things, unless you used parentheses
all over the place.

So the order of operations really does make algebra easier, not
harder.  That's why it's used in ALL algebraic equations and formulas,
from the familiar to the technical.  Without such a universal 
standard, we couldn't be sure what anything meant; and with a 
different one, the most useful expressions would be the hardest to 
read and write, rather than the easiest (as is true in reality).

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/