The following definitions hold:

x/x = 1         ( x != 0 )
0/x = 0         ( x != 0 )
x/y = x * (1/y) ( y != 0 )
[download]

That's all there is to it. We disallow division by zero and dispense with long winded explanations.

Consider the following formulation that proves 2 = 1.

(1)              a = b            ( a != 0, b != 0 )
(2)             aa = ab
(3)        aa - bb = ab - bb
(4) (a - b)(a + b) = b(a - b)
(5)          a + b = b
(6)          b + b = b            (from a = b, (1))
(7)             2b = b
(8)              2 = b/b
(9)              2 = 1
[download]

This flaw in this "proof"? To get to step 5 we divide both sides by (a - b) (which, because a=b, (a-b) = 0).

You could formulate step 5 as:

(5)   ( ( a - b ) / ( a - b ) )( a + b ) =  ( ( a - b ) / ( a - b ) )b
[download]

It doesn't matter. Sure, it doesn't "make sense" to divide by 0. But there are several cases where you can decide to define that some specific expression that doesn't really "make sense" should be considered to have a specific value because doing so would simplify having to deal with some edge cases.

For example, 0**0 is generally defined to be 1 even though 0**X is 0 for all values of X > 0. Also, 0! (factorial) is defined to be 1. Because these "conventions allow us to extend definitions in different areas of mathematics that would otherwise require treating 0 as a special case."

I wish I could remember cases where continuous functions were the complete justification for such conventions (I know there is at least one such, but I can't think of it).

        - tye (but my friends call me... um... something)

An example where 0/0 really is defined and finite: $y = sin($c * $x) / $x; , considered continuous with $c constant, should give $y == $c where $x is zero.

After Compline,
Zaxo

For example, 0**0 is generally defined to be 1 even though 0**X is 0 for all values of X > 0. Also, 0! (factorial) is defined to be 1. Because these "conventions allow us to extend definitions in different areas of mathematics that would otherwise require treating 0 as a special case."

Seems to me that this is why we don't allow 0/0 = 1, or in fact, n/0 ever. Because, it doesn't work without making a lot of special cases. If 0/0 = 1, then the proof I gave before would prove that 2=1.

Here's a definition that we'd have to apply a special case to if 0/0 = 1 were allowed:

   0 * x = 0  (except when x = n/0)
[download]

If we simply say that n/0 is not a number, all these special cases go away.

I think the examples of continuous functions that seem to work when the denominator goes to 0 is conflating division by 0 with taking the limit as the denominator goes to 0. I'm not sure, as I'm no expert in math, but I believe that these are all examples of functions whose values are approximations represented by infinite series. In such cases, you would have to examine the infinite series to understand what's really going on.

I hereby resolve to read to the end any given post before leaping to try and understand the details.

After scribbling lines and points over both sides of several sheets of paper trying to picture this, I then resorted (as any ex-Mech.Eng. would) to a cardboard box, a pencil, a ruler and a pair of sissors.

I was going along great until I got to the bit about the cylinder. A loo roll center wasn't big enough (Have you ever tried to re-roll a pile of loo paper? Take it from me, don't bother!).

So, out to the garage, a hacksaw to a discarded carpet roll tube that has been sitting on the rafters (cos I thought it might be useful) and a 1 foot section later. A few judicious snips on the cardboard box allowed me to roughly center this over one axis (a corner of the box to us laymen). A quick re-labelling, lay the whole thing on its side. And the realisation dawns and finally it makes sense.

Those things that you mathematicians call curves, are (in this case) what any engineer would call straight lines. Now the description makes sense :^).

Then I read the last paragraph and "Ah! Now there's a good reason for not doing it".

Seriously though, thankyou for the excellent explanation!

(I'm nearly finished with almost explaining this)

That's okay Tye I almost finished nearly understanding it:^)