I am going to try to define numbers which will permit division by zero. Furthermore, I am going to try to keep the numbers behaving as well as possible. I was working on this problem. So far I have addition of these numbers complete. Here they are.
Consider the group , we will "add" two numbers to this group and redefine the addition operation to keep it a group.
Definition: "Hackerian space" is the set
Definition: The binary operation,
will be as follows:
Theorem: Hackerian space is a group.
We do obtain something interesting with this construction.
The "Blank set" is a normal subgroup of Hackerian space. And the interesting property is that,
Now there is a chance that because they have the same cardinality. But the boxed property shows that it is isomorphic by a non-zero factor group. I hope that shows that they are distinct algebra structures but I am not sure.*)
We have constructed a new group which is distinct by isomorphism from the reals if the conjecture is true. We can now try to procede to define multiplication.