Zero-Factor Theorem

Date: 02/25/2002 at 09:02:53
From: DeSheng Zhu
Subject: Zero-Factor Theorem 

There is a Zero-Factor Theorem in my textbook, _College Algebra_
(Gustafson/Frisk).

   If a and b are real numbers, and if ab = 0, then a = 0 or b = 0.  

Why the restriction 'a and b are real numbers'? Are there any two 
unreal numbers whose product is zero but where neither itself is zero?

Date: 02/25/2002 at 11:55:28
From: Doctor Paul
Subject: Re: Zero-Factor Theorem 

You have to know a bit of modern algebra, but there are algebraic 
structures in which the product of two nonzero elements is zero.  In 
the ring of integers mod 6, 2*3 = 6 = 0.

In the ring of 2x2 matrices, 

[0 1] [0 1]
[0 0] [0 0]

= [0 0]
  [0 0]

The idea is that in an Integral Domain the so called zero factor 
theorem holds. The defining characteristic of an integral domain is 
that if x and y are both nonzero then xy is nonzero as well. Thus in 
an integral domain, if xy = 0 then x = 0 or y = 0. 

Most of the algebraic structures with which you are familiar will be 
integral domains - the integers and the real numbers are two such 
examples. But the integers mod 6 and the ring of 2x2 matrices are 
not integral domains, since I have demonstrated that the product 
of two nonzero elements can in fact be the zero element.

I hope this helps. Please write back if you'd like to talk about this 
more.

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