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.
Careful! If you keep going like this you might even justify renormalization at some point. :p
What does that mean? Something with Quantum physics?
Originally Posted by topsquark
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.