Closure Axiom

Date: 06/12/97 at 19:01:11
From: Stacy Stojakovic
Subject: Closure Axiom - What does it really mean?

In my algebra book, the closure axiom states:

The set of real numbers is closed under addition and multiplication. 

That is,
 
  1. x+y is a unique real number
  2. xy is a unique real number

I have re-examined this axiom many times and I am still confused.

Could you please explain it to me?

Date: 06/13/97 at 08:29:49
From: Doctor Anthony
Subject: Re: Closure Axiom - What does it really mean?

The idea of 'closure' is actually very simple. If you add together 
two whole numbers, you will always get another whole number. If you 
multiply two whole numbers, you will get a whole number as a result.  
So we say that whole numbers (integers) are 'closed' under the 
operations of addition and multiplication.

What about division?  Well 12 divided by 2 is 6, which is a whole 
number, so in this case we get a whole number result. But 12 divided 
by 5 = 2 and 2/5, so now we have moved out of the field of whole 
numbers. If we divide two whole numbers we cannot guarantee that the 
result will still be a whole number. So the set of whole numbers is 
not closed under the operation of division.

Positive whole numbers are closed under addition - you always get a 
positive whole number in the result. But they are not closed under 
subtraction, since, for example,  4 - 9 = -5  and -5 is not a positive 
whole number.

To decide whether a set of numbers is closed under some operation or 
other, look for cases where the result is no longer in the set you 
started with.

In the case of real numbers, which include positive, negative, 
fractional, and irrational (like sqrt(2)) numbers, the operations of 
addition, multiplication, division and subtraction are all closed (apart
from division by zero which is not defined). But taking square roots is
not closed because if, for example, we try sqrt(-5), we no longer get a
real number as a result. In fact, we have moved into the realm of
complex numbers.

The idea of 'closure' does not apply only to operations on sets of 
numbers. We can have operations on vectors and matrices, for example, 
which might or might not be closed.  

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