Sure, I remember the discussion in first-year algebra where the teacher showed positively that division by 0 was meaningless and had to be disallowed because if allowed, it presented all kinds of paradoxes and impossibilities (e.g., 3=4).
But I recall thinking at the time that it seemed to be a glaring *hole* in the system. How can we arbitrarily disallow some function? For instance, why not disallow multiplication by 17, or not allow an odd number to be added to an even number... that sort of thing?
As a sort of parallel example: Extracting the square root of -1 is obviously impossible and nonsensical. Nevertheless, there is an entire branch of mathematics devoted to it. Complex numbers (using "i" to represent the square root of -1) are useful in many real-world applications
Now why can't we do a similar thing with another "impossible" operation and value? Let's call "one divided by zero" (1/0) "p" and for instance, 7/0 is 7p. Having split off the special part of this as p (for "preposterous") number, can we now come up with a set of operations and a system of mathematics for making useful calculations?
I apologize that this may sound like it's related to some crackpot "theory of existence" but I assure you that I am not postulating anything -- I'm simply looking for a clearly-worded and understandable explanation of the situation. I am no mathematician -- I struggled but was able to bluff my way through a high school Trig class many years ago. So be gentle :-)
I'd also be interested in discussing related topics. For instance, if such a system has been (or could be) developed, what are (or would be) the implications?
Another oddball thought: What if instead of 0 we just substitute 0.0000001? For instance,
7/0.0000001
is a perfectly valid operation. It seems to me that it would be very nearly the same answer as 7/0, but just not as accurate. What's wrong with my thinking on that?
