Properties of Subtraction

Date: 04/08/97 at 17:22:35
From: kim Sebastian
Subject: Properties of Subtraction

1) Is subtraction commutative? How can this be justified?
2) Is subtraction associative? and how can this be justified?

Thanks,
   Mrs. Sebastian

Date: 04/09/97 at 15:01:08
From: Doctor Mike
Subject: Re: Properties of Subtraction

Dear Mrs. Sebastian,
  
This is an interesting question, and the answer is subtle, but
there IS an answer.  It's sort of a "yes and no" answer.  Here goes.
  
1. Subtraction is NOT commutative.  If you evaluate 3-2 and 2-3
   you get +1 and -1, respectively.  NOT the same thing.
  
2. Subtraction is NOT associative.  If you evaluate (3-2)-1 and
   3-(2-1) you get 0 and 2, respectively.  NOT the same thing.
  
3. However, **addition** is commutative for all numbers, even
   including negative numbers.  An example related to (1.) is:
   
          3 + -2  =  -2 + 3  =  +1
  
4. Also, **addition** is associative for all numbers, even the
   negative numbers.  An example related to (2.) above is: 
  
          (3 + -2) + -1  =  3 + (-2 + -1)  =  0   
  
   To see this better, evaluate what is inside parentheses, getting
   
          (  1  ) + -1  =  3 + (  -3  )  =  0 
  
I hope this helps.  Feel free to write back if you're not yet
completely comfortable with this.  I'm glad you want to get it.

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