The reason that the result of a division by zero is
undefined is the fact that **any attempt at a definition
leads to a contradiction.**

To begin with, how do we define *division*? The
ratio *r* of two numbers *a* and *b*:

is that number *r* that satisfies

Well, if *b=0*, i.e., we are trying to divide by
zero, we have to find a number *r* such that

Now you could say that *r=infinity* satisfies (1).
That's a common way of putting things, but what's infinity?
It is not a number! Why not? Because if we treated it like
a number we'd run into contradictions. Ask for example what
we obtain when adding a number to infinity. The common
perception is that infinity plus any number is still
infinity. If that's so, then

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

I said above that we can't solve the equation (1) unless
*a=0*. So, **in that case**, what does it mean to
divide by zero?

Again, we run into contradictions if we attempt to assign
any number to *0/0*.

Let's call the result of *0/0*, *z*, if it
made sense. *z* would have to satisfy

That's OK as far as it goes, any number *z* satisfies
that equation. But it means that the result of *0/0*
could be anything. We could argue that it's 1, or 2, and
again we have a contradiction since *1* does not
equal *2.*

But perhaps there is a number *z* satisfying (2)
that's somehow special and we just have not identified it?
So here is a slightly more subtle approach. Division is a
continuous process. Suppose *b* and *c* are
both non-zero. Then, in a sense that can be made
precise.
the ratios *a/b* and *a/c* will be *close
* if *b* and *c* are close. A similar
statement applies to the numerator of a ratio (except that
*it* may be zero.)

So now assume that *0/0* has some meaningful
numerical value (whatever it may be - we don't know yet),
and consider a situation where both *a* and *b
* in the ratio *a/b* become smaller and smaller.
As they do the ratio should become closer and closer to the
unknown value of *0/0*.

There are many ways in which we can choose *a* and
*b* and let them become smaller. For example,
suppose that *a=b* throughout the process. For
example, we might pick

Since

for all choices of *a* we get the ratio 1 every time!
This suggests that *0/0* should equal 1. But we
could just as well pick

and let *a* be **twice as large** as *b*.
Then the ratio is always 2! So *0/0* should equal 2.
But we just said it should equal 1! In fact, by letting
*a* be *r* times as large as *b* we
could get any ratio *r* we please!

So again we run into contradictions, and therefore we are compelled to

It's a common strategy in teaching to simplify concepts when
they are first encountered. In other words, it's common for
your teacher **to lie** to you. I just did! Actually,
there *is* a way to make sense of the expression *
0/0*. The basic idea is to let both the numerator and
the denominator become smaller and smaller, and to make the
value of *0/0* dependent upon the way in which
numerator and denominator approach 0. This is explained
more thoroughly
here.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[17-Feb-1997]