More importantly is what happens when you graph it: the limit of 1/x as x approaches zero is discontiguous. It's positive infinity when descending on the positive numbers, but negative infinity when ascending from the negatives. No one value can represent both!
Let's assume that the set of integers is Z_\inf. K? We can now define negative numbers as the 1's compliment of the number plus 1. 1 = 999...9998. then plus 1 = 999...9999. This plus 1 results in an infinite carry out, and the value 0. Awesome.
Now, let's look at 1/0, we see that from the right it's approaching \inf from the bottom, while we see that from the left, it's approaching \inf from the top. Now, at 0, obviously these two will be coincident, because we're working in Z_\inf, that value is the same value. Namely, -\inf = \inf. But that doesn't make sense, only 0 can be it's own negative!
But we've already known for a long time about Z_n where n is even, -(-128) in Z_256 is -128. -(-65536) in Z_2^16 = -65536. So, there's no trouble in making -\inf = \inf ...
Basically, 1/0 grows so fast that it manages to wrap around the entire infinite series of numbers. Which is exactly what it does...