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"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

"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 4y ago
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 4y ago
  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 4y ago
    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 4y ago
    Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.
henry2023 4y ago
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,_%CE%B4)-definition_of_limit https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...
- deleted 4y ago
  [deleted]
- topaz0 4y ago
  "_____ 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 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.
ndsipa_pomu 4y ago
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 4y ago
  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 4y ago
    There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/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.

"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 4y ago

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 4y ago
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 4y ago
  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 4y ago
  Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.

henry2023 4y ago

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,_%CE%B4)-definition_of_limit https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...

deleted 4y ago
[deleted]
topaz0 4y ago
"_____ 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 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.

ndsipa_pomu 4y ago

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 4y ago
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 4y ago
  There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/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.