Careful! If you keep going like this you might even justify renormalization at some point.
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.
I have solved my problem.
If you were reading what I wrote, I mentioned it would be quite stupid if Hackerian space is the same as the Real numbers (up to isomorphism). But the answer is simple. Hackerian space has a subgroup order 3,i.e. Blank space. However, R does not!
So we are truly dealing with a different algebraic structure.
But the sad thing is that we cannot turn R by introducing new made up numbers (inf, nan, ... ) into an algebriac structure where division is permitted. Because it needs to be a division ring. Which is violated by "division by zero". Thus, it is not possible to extend the reals into numbers division where zero divison is permitted.