Which is more 'simple'?

Date: 09/24/2001 at 15:27:29
From: Pam Brislin
Subject: Which is more 'simple"?

My class is trying to figure out which is considered more simple (when 
simplifying): 2x + 2y (two terms and three operations) or 2(x + y)
(one term and two operations, but the higher operation of 
parentheses).

Thanks!
Pam

Date: 09/24/2001 at 22:19:41
From: Doctor Peterson
Subject: Re: Which is more 'simple"?

Hi, Pam.

I often wonder the same thing; when we get questions about 
simplification out of context, it's hard to be sure what form is 
expected.

My own sense of the matter is that "simplicity" is in the eye of the 
beholder - and more specifically, in the context of the problem. 
We simplify an expression for a reason: to make the next step easier. 
In some cases, the next step is just to evaluate the expression for 
particular values, and then the form with the fewest operations may 
be best (unless those operations involve division by an irrational 
number, in which case we prefer a rationalized denominator despite 
a larger number of operations). Other times, we are going to be doing 
something "additive," such as looking for a polynomial, and your 
first form is better; or we are going to be doing something 
"multiplicative," and the factored form is best.

If a student asked me for help simplifying this, and I chose to 
actually show the final answer rather than just pointing her in the 
general direction, I would probably choose the polynomial form; 
everything from the order of operations rules up is built around a 
preference for that form and an avoidance of parentheses, so without 
context I naturally tend in that direction. But I'll still wonder 
whether it's really right ...

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