Divide by 0 Undefined?

Date: 9/10/96 at 3:9:16
From: Anonymous
Subject: Divide by 0 Undefined?

When something is divided by 0, why is the answer undefined?

Date: 9/10/96 at 15:25:36
From: Doctor Robert
Subject: Re: Divide by 0 Undefined?

The reason is related to the associated multiplication question.  If 
you divide 6 by 3 the answer is 2 because 2 times 3 IS 6.  If you 
divide 6 by zero, then you are asking the question, "What number times 
zero gives 6?" The answer to that one, of course, is no number, for we 
know that zero times any real number is zero not 6.  So we say that 
division by zero is undefined, for it is not consistent with division 
by other numbers.

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

Date: 9/10/96 at 20:6:47
From: Doctor Tom
Subject: Re: Divide by 0 Undefined?

It's because there's just no sensible way to define it.

For example, we could say that 1/0 = 5.  But there's a rule
in arithmetic that a(b/a) = b, and if 1/0 = 5, 0(1/0) = 0*5 = 0
doesn't work, so you could never use the rule.  If you changed
every rule to specifically say that it doesn't work for zero
in the denominator, what's the point of making 1/0 = 5 in the
first place?  You can't use any rules on it.

But maybe you're thinking of saying that 1/0 = infinity.  Well
then, what's "infinity"?  How does it work in all the other
equations?  

Does infinity - infinity = 0?  
Does 1 + infinity = infinity?

If so, the associative rule doesn't work, since (a+b)+c = a+(b+c)
will not always work:

1 + (infinity - infinity) = 1 + 0 = 1, but
(1 + infinity) - infinity = infinity - infinity = 0.

You can try to make up a good set of rules, but it always leads
to nonsense, so to avoid all the trouble we just say that it
doesn't make sense to divide by zero.

What happens if you add apples to oranges?  It just doesn't make
sense, so the easiest thing is just to say that it doesn't make
sense, or, as a mathematician would say, "it is undefined".

Maybe that's the best way to look at it.  When, in mathematics,
you see a statement like "operation XYZ is undefined", you should
translate it in your head to "operation XYZ doesn't make sense".

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