Division By Zero Exploration

This post isn’t so much about programming as much about an earlier post about Division By Zero. So, I thought, if the reciprocal operation to 1 / 0, 0 / 1, is valid, then maybe division by zero can be postponed and considered later. Similar to the imaginary concept, i or sqrt(-1).

Using just fraction concepts I managed to accomplish a lot of progress and, not being a mathematician, I don’t know how much merit there is to it, but here we go.

First, the assumptions:

$0 * 1 = \frac{0}{1}* \frac{1}{1} = \frac{0}{1} = 0$

$0 / 1 = \frac{0}{1} * \frac{1}{1} = \frac{0}{1} = 0$

$1 / 0 = \frac{1}{1} * \frac{1}{0} = \frac{1}{0} = Q$

$Q * Q = \frac{1}{0} * \frac{1}{0} = \frac{1}{0} = Q$

$Q / Q = \frac{1}{0} * \frac{0}{1} = \frac{0}{0}$

$0 / 0 = \frac{0}{1} * \frac{1}{0} = 0 * Q$

$0 * Q = 1 \text{ : } Q = \frac{1}{0}$

$1 / Q = \frac{1}{1} * \frac{0}{1} = \frac{0}{1} = 0$

$Q * 1 = \frac{1}{0} * \frac{1}{1} = \frac{1}{0} = Q$

$Q * x = \frac{1}{0} * \frac{x}{1} = Q * x$

$Q / x = \frac{1}{0} * \frac{1}{x} = \frac{1}{0} = Q$

$1 / 0 = Q$

$0 / 0 = 1$

OK, so now something that I found immediately useful to do with Q and seems to make sense, furthermore this is what lead me to think up Q.

$y= \frac{1}{x} \text{ ; x = 0 } \\\\Q = \frac{1}{x}\\\\\frac{1}{Q} = x \\\\ 0 = x$

Consider the following:

$\frac{0}{x} = 0 * x$

True, 0 = 0; now considering the reciprocal:

$\frac{x}{0} = x * Q$

how does this resolve?

$x = x * Q * 0 \\\\x = x$

What this leads to is another assumption: algebraically, the same way Q can be considered for later, so then, can Zero be considered later.

Consider:

$a = b\\\\a^2 = ba\\\\a^2-b^2 = ba-b^2\\\\(a-b)(a+b)=b(a-b)\\\\\frac{(a-b)}{(a-b)}(a+b)=b\frac{(a-b)}{(a-b)}\\\\(a+b)=b\\\\(\frac{a}{a}+\frac{b}{a})a=b\\\\0*(\frac{a}{a}+\frac{b}{a})a=0*b$

distribute zero once and then multiply both sides by Q (divide by zero )

$Q*0*a=Q*0*b\\\\a = b$

Now – I realize – if I had distributed the zero to the a instead, the answer would have come out incorrect.

Once it is possible to divide by zero and turn a zero into a one, it seems like it’s possible to produce lots of wrong answers, but some correct ones as well, it seems like it depends on trial and error, perhaps, or additional processing of some sort.

So, I’ve been messing around with this concept using my limited math skills, and it seems like sometimes Q works, and sometimes, it doesn’t work. It’s like sometimes there’s a good solution and an infinity of wrong ones, or the opposite, or some mix of random, maybe. I just don’t know. At the end of the day, sometimes it works, so maybe there is merit.

What I’ve concluded is that with this concept, a consequence, is that it’s possible to basically make anything equal anything, and that seems like an odd thing to consider.

Comments are appreciated.