Zero as Denominator

Date: 10/22/97 at 15:02:18
From: V. Amburgey
Subject: Zero as denominator

Why can't zero be in the denominator for rational numbers?

My students, preservice elementary education teachers, have asked that 
this question be submitted to you. We have discussed this using the 
models of "parts of a whole region" model and the "division model."  
In discussing models for rational numbers, we have also discussed the 
"parts of a set" model for rational numbers. My students want to know 
why 0/0 can't be another name for one when using an empty set as 
their whole for the "part of a set" model. 

We look forward to your response.   Thanks.

Date: 10/22/97 at 19:55:06
From: Doctor Tom
Subject: Re: Zero as denominator

The way formal mathematics works is that you make up rules for a
system and then study that system. In principle, you can study ANY 
system this way, but in practice, useful systems get a lot more study 
than non-useful systems.

For the sake of this discussion, let's just consider the set of
rational numbers with +, -, x and /. The standard definition says that 
the operation of division is simply undefined if the denominator is 
zero.

There's nothing on earth to prevent you from using mathematical 
methods to look at systems where division by zero makes sense, either 
sometimes (like only when the numerator is zero) or always. What I'll 
try to show below is that when you do this, you lose so much that 
adding the definition for division by zero is not worth it.

For example, how shall I define 1/0? If you say "infinity," then
there's suddenly a new object, "infinity," in your system, and you'd 
better tell me how it behaves. What's 1+infinity, for example? If it's 
infinity, then you can no longer use the rule that if you subtract the 
same thing from both sides of an equation the results will be equal:

  1 + infinity = infinity

Subtract infinity from both sides, and 1 = 0.  So now if I ask you to 
solve this:

    x + 1 = 7

you can't just subtract 1 from each side and get x = 6, because there 
is no general rule. If you modify the rule to apply only to 
subtracting infinity, it's still no good. Is infinity-infinity equal 
to zero? If not, what is it?  Is subtraction now only sometimes 
defined? If it is zero, then since 1+infinity=infinity,
1+infinity-infinity = infinity-infinity - I just replace one of
the infinities in infinity-infinity=infinity-infinity by 1+infinity,
and again I get 1 = 0.

I can go on and on, but no matter how you decide to define 1/0, you 
make one or many of the basic laws of arithmetic fail, and you 
suddenly find that your system has lost the power to do almost 
anything.

People have looked for good ways to add 1/0 to the rationals (or even 
just 0/0), and in every case, the new system's rules are so weakened 
that it is not good for anything useful.

Geometers have had better luck. If you look at the line of real 
numbers and add an imaginary infinity to sort of "close the ends," you 
get a very servicable system called linear projective geometry. But 
this geometry has none of the addition or multiplication properties, 
and only deals with "neighborhoods" of points.

It can be very instructional to try various ways to define 1/0 to the 
rationals (leaving all the other operations as-is), and see what basic 
laws of arithmetic have to be re-written. Be sure to look at the 
commutative, associative, and distributive laws, the existence of 
inverses, closure of operations, et cetera.
 
I think division by zero has always been confusing because there is a 
lot to it. The books simply weasel out of it by saying, "Division by 
zero is not defined," and the instant knee-jerk response is, "Well, 
just define it."

It's in trying to find a "reasonable" definition that all the ugly 
problems come up.

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