[amitport]

"it becomes possible to "prove" that any number is equal to any other number."

There are multiple ways to define what division by zero means. Which definition leads to this outcome? How?

[afiori]

let ϴ = 0/0 then 1*ϴ = ϴ = 0/0 = (0*0)/0 = 0*(0/0) = 0*ϴ it follows 1 = 0 and thus x = x * 1 = x * 0 = 0 = y * 0 = y * 1 = y for all x and y

[xigoi]

This is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.

[afiori]

True, but the point of giving a "value" to 0/0 is to use it somehow.

For example in the context of limits you define a whole lot of number like values like 0+ or 0- that are useful wrt operations on limits.

I was trying to give an example of how ℝ ∪ {Θ} has almost no advantages compared to just ℝ

[tshaddox]

Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.

[ndsipa_pomu]

The most common definition of division being the inverse of multiplication.

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

[amitport]

Thanks, this definition does seem problematic. In any case, it is not the only possible definition and in a/0=c, c does not have to be defined as a real number. We can define it as similarly to complex number with new rules that do not collide with existing reals.

[ndsipa_pomu]

There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/Riemann_sphere) which does define division by zero, but sacrifices the numbers forming a field under addition and multiplication.

[henry2023]

Division by zero is not defined anywhere on math.

The closest thing you'd get to it is to

1. define a limit (lim x->a of ƒ(x) exists if and only if given any ε > 0 there exists a δ > 0 such that ...)[1].

2. chose a function ƒ(x) such that on a given "a", ƒ(a) = ƒ(a)/0.

3. prove that the limit exists and is finite.

Now if we defined division by zero it would look like this:

Axiom: For every element x of the real numbers there exists a x' in the real numbers such that x/0 = x'

I advise you to play with this new "rule" to see if it leads to something interesting. Hint: try to prove that 1/0 = 2/0

[1]: https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...

[topaz0]

"_____ is not defined anywhere in math"

is a kind of sentence that is almost never true, and even if it were, it would be impossible to prove that someone hadn't jotted a valid definition on a napkin somewhere. In this case it is certainly not true (as others have mentioned: https://en.wikipedia.org/wiki/Riemann_sphere ). Now, specifying a definition for division by zero does require you to be careful about how the other operations extend to this new number, but there are perfectly consistent (and useful!) ways to do so.