Is It Possible That x/0 is Not Really Undefined?

Date: 06/14/2004 at 04:16:42
From: Stephane
Subject: Can we not define x/0 in the example of the complex number i

As we are all told, n = 1/0 is undefined, since no number n exists 
that, when multiplied by 0, gives the result 1 (i.e., n*0 = 1).  

However, drawing an example from the theory of complex numbers, there 
also exists no obvious number i that, when squared, gives the result 
-1.  The square root of -1 would, in past centuries, have been 
described as nonsense or undefined.  

Mathematicians nonetheless define just such a number, enlarging the
known numbers from the real to the complex, and use the number i 
successfully in many real-world calculations.

Can we be certain that, for any nonzero x, x/0 is actually undefined 
(i.e. unable to be ascribed any actual value or meaning), or is it 
possible that an important number and new number class has so far
escaped discovery?

Regrettably, I do not have the required mathematical training to flesh 
this proposition out substantially, except to suggest that if, by some 
novel number definition (say, j = 1/0), an additional "dimension" of 
numbers can be defined, perhaps this new, larger, set also encompasses 
the complex numbers, as these encompass the real numbers.  To put it 
another way, if we can think of the "real number line" being extended 
into a "complex number plane" by the definition of i = sqrt(-1), 
perhaps the complex plane can be extended into an even larger, 
"ulterior volume" by the definition of a number j = 1/0.

Recognizing that all this is probably just idle speculation, I 
appreciate the time and effort anyone is willing to devote to 
providing an answer.

Date: 06/14/2004 at 08:26:44
From: Doctor Mitteldorf
Subject: Re: Can we not define x/0 in the example of the complex number i

Dear Stephane,

This is a good, thoughtful question, and there's a satisfying answer
to it.  You don't have to have an extensive math background to 
appreciate the answer, just some basic algebra and the confidence to 
apply it.

The definition i = sqrt(-1) is such a rich mathematical construct 
because of two facts.  First, the definition is self-consistent.  It 
does not introduce any contradictions into the system we already have 
in place.  Second, it doesn't lead to more numbers that need defining, 
but provides solutions (in the form of complex numbers) to all 
polynomial equations.  For example, you don't need to define a new 
number for sqrt(-i), because the square root of -i can be calculated 
in terms of the complex numbers (using 1 and i) that we have already 
defined.  (You can verify that sqrt(-i) = (1-i)/sqrt(2) by multiplying 
this number by itself and seeing that you get -i as an answer.)

The problem with defining j = 1/0 is that it doesn't pass the first
test I mentioned.  The mathematics that results is no longer
self-consistent, and it's therefore not very interesting.  Here's an
example of an inconsistency:

  By definition 1/0 = j.  Multiply both sides by 2, and you 
  find that 2/0 = 2j.

The problem comes when we multiply both sides through by 0.  We find
in the first case that 0 * j = 1 and in the second case that    
0 * 2j = 2.  So far, so good.  But multiplication is supposed to be
"associative".  If you multiply (a*b) * c you should get the same 
answer as a * (b*c).  For example, we can write 2*3*4 without 
specifying which multiplication is done first, because (2*3)*4 = 6*4 
= 24, and 2*(3*4) = 2*12 = 24.  We get the same answer both ways.

Now try this with 0 * 2 * j, and you'll see what I mean about 
introducing inconsistency.

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

Date: 06/14/2004 at 20:44:27
From: Stephane
Subject: Thank you (Can we not define x/0 in the example of the
complex number i)

I see your point, and why we can't just arbitrarily define such a 
number.  Thank you for taking the time to answer my question.