A few years ago I was teaching complex numbers in an algebra 2 class. It took students awhile to grasp the idea of i. I told them it was defined as the square root of negative one.
"But you can't square root a negative."
"That's why we have to create a new definition - we want to see what results from this definition," I said.
One student was particularly enamored by this idea, and I suggested he look at the other forbidden operation (besides square rooting a negative): dividing by zero. He defined 1/0 as u, perhaps for "undefined," but called numbers involving u "blank numbers," since a calculator would return a blank screen after the error.
Over several days I prodded this student to create theorems about blank numbers, and to avoid contradictions by adding "special rules." I have since lost his work, and don't remember the stumbling block that kept him from continuing.
Perhaps you can find it. Let 1/0 = u.
Thm. 1: u² = u
Where does it go from here?