Is it possible to have a 3rd number system based on division by zero? Is it possible in mathematics to use a third number line based on division by zero; in addition to the real and imaginary number lines?
This is because some solutions blow up when there is a division by zero. Would it be possible to solve them with this new number line?
$\therefore$ on the z axis we would have $\frac{1}{0}$ , $\frac{2}{0}$ , $\frac{3}{0}$ , etc. where,  $p = \frac{1}{0}$ and $p \cdot 0 = 1$ .
division by zero graph
Is this a viable number system?
A similar question to this one: Is there a third dimension of numbers?

$\textbf {Rules of the p number system using 1/0 and $\infty$ }$
If $p = \frac{1}{0}$ and $p \cdot 0 = 1$. Then I suggest the following rules.
$$p - p = \frac{1}{0} - \frac{1}{0} = \frac{0 \cdot 1 - 0 \cdot 1}{0 \cdot 0} = \frac{0 \cdot (1 - 1)}{0 \cdot 0} = \frac{0}{0} = 1$$
problem lies here (if it is a problem?) $$\frac{0}{0} \neq \frac{0}{\delta x} = 0$$
where $\delta x \to 0$ (see previous answer), going by the rule of $\frac{0}{0} = 1$ .
$\underline{ RULES \, that \,appear\, to \,work}$
$$\frac{2}{3} + \frac{0}{0} = \frac{2 \cdot 0 + 3 \cdot 0}{3 \cdot 0} = \frac{5 \cdot 0}{3 \cdot 0} = \frac{5}{3}$$
$$p \cdot p = \frac{1}{0} \cdot \frac{1}{0} = \frac{1}{0^2} = p^2 = \infty \cdot \infty = \infty^2$$
$$p + p = \frac{1}{0} + \frac{1}{0} = \frac{0 \cdot 1 + 0 \cdot 1}{0 \cdot 0} = \frac{0 \cdot (1 + 1)}{0 \cdot 0} = \frac{2}{0} = 2p$$
$$p \cdot 0 = 1 \neq p \cdot (0 + 0)$$
$$p \cdot (0 + 0) = p \cdot 2 \cdot 0 = \frac{1}{0} \cdot 2 \cdot 0 = 2$$
$$0 + 0 + 0 \,+ ... = 3 \cdot 0 \,+ ...$$
