Real Numbers Closed under Division

Date: 09/24/2002 at 19:41:38
From: Dave
Subject: Are real numbers closed under division?

Real numbers are not closed under division because we can't divide 
by 0.  Is this correct?

Date: 09/24/2002 at 22:53:23
From: Doctor Mike
Subject: Re: Are real numbers closed under division?

Hi Dave,
 
You have a good point. It's good to keep alert and thinking about what 
you hear in math classes.  
  
What you may have missed (or perhaps it was not actually said in so 
many words) is that the division

   X/0 

is officially "not defined," so technically, division of real numbers 
IS closed with respect to division. And the reason division was 
carefully defined the way it was is so the non-closure would not 
happen.
  
By the way, usually the main operations for real numbers are 
multiplication and addition. The operations of subtraction and 
division are by-products of the other two operations. 
  
The way this usually happens for subtraction is that -a is used to 
represent the additive inverse of a, such that  a + (-a) = 0.
  
The way this usually happens for division is that 1/a is used to 
represent the multiplicative inverse of a, such that  a * (1/a) = 1.  
HOWEVER, and this is important, zero does not have a multiplicative 
inverse, so 1/0 isn't defined at all.
  
Then subtraction is defined as  a - b = a + (-b) and division is 
defined as a / b = a * (1/b) if b is not zero.    

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