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 $\displaystyle <\mathbb{R},+>$, we will "add" two numbers to this group and redefine the addition operation to keep it a group.

Definition: "Hackerian space" is the set $\displaystyle \mathcal{H}=\mathbb{R}\cup \{-nan,+nan\}$

Definition: The binary operation,

$\displaystyle *:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ will be as follows:

$\displaystyle \forall a,b\in \mathbb{R} \, \, \, \, a*b=a+b$

$\displaystyle \forall a\in \mathcal{H} \not = \mp nan \, \, \, \, a*\pm nan=\pm nan*a=\pm nan$

$\displaystyle \forall a\in \mathcal{H} \, \, \, \, \, 0*a=0*a=0$

$\displaystyle \pm nan*\mp nan=\mp nan*\pm nan=0$

Theorem: Hackerian space is a group.

Proof: Trivial

We do obtain something interesting with this construction.

The "Blank set" $\displaystyle =\{-nan,0,nan\}$ is a normal subgroup of Hackerian space. And the interesting property is that,

$\displaystyle \boxed{\mathcal{H}/\mathbb{B}\simeq \mathbb{R}}$

Now there is a chance that $\displaystyle \mathcal{H}\simeq \mathbb{R}$ 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.*)

*)Conjecture.

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.