Why can't we have a third number line for dividing by zero?

Question

Originally, taking the square root of a negative number wasn't possible, but then Rafael Bombelli added an imaginary number axis to allow that. Why, then, is there no 3rd dimensional number line to allow division by 0? If there was, you could create a unit z, where z is 1/0. Therefore, 5/0 could be 5z, etc., and multiplying by its reciprocal is a 90° rotation to the real number axis, just like squaring i causes a 90° rotation to the real axis.

Answer 1

This is good, creative thinking about number systems. But here's the hitch: you have to follow through with the consequences of your definitions. If $z = 1/0$ as you suggest, then it must be that $z \cdot 0 = 1$.

Why is this a problem? Well, you are probably already familiar with the idea that $a \cdot 0 = 0$ for any number $a$. Incidentally, this is a consequence of the fact that $0$ is the additive identity and that your operations of multiplication and addition satisfying a distributive law. To wit,
$$ a \cdot 0 = a \cdot 0 + a \cdot 0 - a \cdot 0 = a \cdot (0 + 0) - a \cdot 0 = a \cdot 0 - a \cdot 0 = 0 $$ With $a = z$, your special reciprocal of $0$, you now have that $z \cdot 0 = 1$ and $z \cdot 0 = 0$. Thus, $1 = 0$.

Once you establish that $1 = 0$ in your number system, then any number is equal to any other number! Consider two numbers $b$ and $c$, and calculate: $$ b = b + 0 = b + 0 \cdot (c-b) = b + 1 \cdot (c-b) = c. $$

All your numbers collapse into a singularity. Oh noes!

Answer 2

Interesting question! Although I’m not sure Rafael Bombelli created the imaginary axis since his life largely predates the widespread use of Cartesian coordinate systems.

Before postulating a division by zero axis, it may help to revisit why x/0 has consensus as an undefined concept. It’s not just a void of mathematics, its use leads to contradictions within our other established axioms:

It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if b ≠ 0 then the equation a / b = c is equivalent to a = b × c. Assuming that a / 0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0 / 0

Lots more contradictions in the wiki, over many different areas of math https://en.wikipedia.org/wiki/Division_by_zero

Which then comes around to, what are we hoping to postulate a new axis or number line to achieve? Exploratory math, sure go ahead, but I’d like to tickle your brain more on why you’d even want to do this.