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?
Well, if you are looking for a problem from your theorem, once you have that formula u² = u and you assume you are extending the reals by u, you get that u² - u = 0 which forces u = 0 or u = 1.
ReplyDeleteAnother big problem is that if you let u=1/0 then you get 0 = 0u = 0(1/0) =1 following usual rules for fraction canceling. Just my two cents.
Even though i is defined as the square root of -1, there are other properties (like algebraic closure) that make its existence plausible; whereas, dividing by 0 leads to bad problems almost instantly.
Good points, and the student involved encountered these as well. He was forced to make arbitrary rules, like 0u is neither 0 nor 1. I felt it was a good introduction to groups, which he has probably encountered by now.
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