This is a continuation of an earlier post called, Division By Zero Exploration, which was a continuation of an even earlier post, Division By Zero.
In the exploration post, I concluded that, if dividing by zero is valid, then, “it is basically possible to make anything equal anything”. Further thought has lead to an additional conclusion: one does not necessarily equal one. This becomes evident from the basic properties of Q and 0, and specifically, infinity times zero equals one, or Q*0=1. If this holds true, then one does not necessarily equal one. Here we go.
What I mean is that: one may not equal one, all the time. This is because we can rewrite Q*0 = 1 an infinite amount of ways by adding additional zeros or Qs. ( At least, if 0*0 = 0 and Q*Q=Q). The way the expression is factored causes it to have a potential series of possible answers, some true, and others false.
There are two basic expansions, each which results in two different possible outcomes, for a potential of three answers. The first is Q*0*0. If it holds true that 0*0 always equals zero, then this should be a rational procedure. This results in two possible factorings: (Q*0)*0 and Q*(0*0). The first factoring equals 1, the second equals 0. The other basic expansion is Q*Q*0. This ends up resulting in either 1 or Q.
When we add in the x variable, Q*0*X, we get outcomes 1 or X.
Using this process, I go back to the previous example of a=b and start with a+b=b. From here, there are a lot of potential procedures, but the simplest path to a=b is two steps, I think.
Q*0*b resolves to 1 which resolves to Q*0*0
Q*0*0 resolves to 0
Of course, there are other outcomes from this procedure that are false. These are the other generated outcomes:
This leads to another conclusion, zero does not necessarily equal zero, nor does Q equal Q.
I think inventing math is my new hobby, comments are appreciated.