Ordering Exponents and Variables

Date: 04/08/2000 at 14:53:25
From: Cherie
Subject: Descending Order and Variables

1. Is there a rule for putting terms in descending order if the  
   variables have the same exponent?

2. What about negative exponents and descending order? (Is there a 
   connection?)

Sincerely,
Cherie

Date: 04/12/2000 at 09:12:01
From: Doctor Peterson
Subject: Re: Descending Order and Variables

Hi, Cherie.

First, I'm assuming you are referring to polynomials, and probably to 
polynomials of more than one variable. The second question is odd, 
then, since polynomials can't have negative exponents, but I suppose 
you might try to extend the concept.

Second, there is no rule about writing polynomials in descending 
order. It is conventional to do so when there is only one variable, 
because it makes it easier to operate on polynomials (compare, add, 
multiply) using methods similar to those for working with numbers 
written with their place value in descending order. With two 
variables, there may be a most common way to order the terms, but no 
real convention; with three, it becomes impossible in general.

Third, I can't tell what you mean by "if the variables have the same 
exponent." Does this refer to two variables in one term having the 
same exponent, or to two terms having the same exponent for all 
variables, or to two terms having the same exponent for one variable 
and different exponents for another variable? 

I have to assume that you are looking simply for a rule for writing 
polynomials in two variables; the most common convention, I think, is 
to first order all terms by their total degree (the sum of the 
exponents), and then within each group to order them by their degree 
in one of the variables. There's no strong reason for doing this, as 
there is for a single variable, so it's hard to consider it a rule. 
You can see this style illustrated (though not prescribed) in Eric 
Weisstein's World of Mathematics: Polynomial:

   http://mathworld.wolfram.com/Polynomial.html   

Fourth, as I said, negative exponents don't exist in a polynomial, so 
there's no real convention here; if you have them, go ahead and keep 
sorting them in descending order. I don't see what connection this has 
to the other question.

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