Division by zero.

Yeah i know it's an age old topic but i just had a couple shower thoughts about division by 0 and I just wanted to ask a few questions about said topic.

1. If we assume that any system yx = z all 3 of them real numbers has a solution then I'd like to have a closer look at the case of 0x = 0. If we're looking at the system in this way without solving for x by rearranging the terms and just examine all the values x can take for the system to be satisfied wouldn't we just get x = {set of complex numbers}? So would, by now rearranging the terms, 0/0 not be equal to the set of all complex numbers? So 0/0 would just be the entirety of the complex plane.

2. Furthermore the question remains what about non-zero numbers divided by zero. Well what if we say the system 0x = 1 has the solution 'j'. I'm aware 1/0 effectively has 2 solutions -infinity and +infinity and is thus different from sqrt(-1) which doesn't have two differing values and is just impossible to determine with the set of real numbers. But still we simply can't determine a proper, unique solution for 0x = 1 with the reals so it just feels kinda natural to introduce yet another set of numbers that satisfy this equation and introduce an axis similar to that of the imaginary values of the complex plane, by assuming that 'j' satisfies the field axioms just like the real numbers do (this is also what Mathematicians assumed when introducing the imaginary set of 'i').

I just want to know why all this (probably) doesn't work.

Solution 1:

If we're looking at the system in this way without solving for x by rearranging the terms and just examine all the values x can take for the system to be satisfied wouldn't we just get x = {set of complex numbers}?

Well, for any suitably defined algebraic structure, we could say a lot more than just complex numbers. Quaternions, octonions, matrices, functions...

So would, by now rearranging the terms, 0/0 not be equal to the set of all complex numbers? So 0/0 would just be the entirety of the complex plane.

The thing is, expressions that result from arithmetic are supposed to a single, unique value. $1+1=2$ in the reals and under the usual definition of addition, for instance. It's never $3$, or $\pi$, or whatever.

So if $0/0$ could be defined as the entirety of the complex plane, you'd be saying that, for instance, $0/0 = 1$ is valid. And $0/0 = 2$, or $0/0 = z$ for any $z \in \Bbb C$.

But then $1=0/0=2$ and $1=2$, which is a very big problem!

Well what if we say the system 0x = 1 has the solution 'j'.

Generally, you should more properly show the existence of such a solution first before looking into this. Granted you could argue against this with the case of something like $i^2 = -1$, but just something to be careful about.

I'm aware 1/0 effectively has 2 solutions -infinity and +infinity

If we limit ourselves to the real numbers, it has no solutions, as $\infty,-\infty$ are not real numbers.

What you're trying to say is that

$$\lim_{x \to 0^\pm} \frac 1 x = \pm \infty$$

but that doesn't mean $\pm \infty$ are real numbers. Indeed, they don't obey some of the laws of real-number arithmetic.

But still we simply can't determine a proper, unique solution for 0x = 1 with the reals

In fact, one doesn't exist in the reals.

it just feels kinda natural to introduce yet another set of numbers that satisfy this equation and introduce an axis similar to that of the imaginary values of the complex plane, by assuming that 'j' satisfies the field axioms just like the real numbers do (this is also what Mathematicians assumed when introducing the imaginary set of 'i').

I suppose you technically can do that, but then the question becomes whether there is merit in doing so.