Understanding the Paradox of 0 Divided by 0: Is it 0 or Undefined?

[chiro]

Assume there exists a mathematician that defines 0/0 to be something and he begins an article by writing something like "In this paper, we define 0/0 to be 0." Assume farther that the paper was actually published and (this next one is far more likely than the last one) every mathematician picked it up to read it at the exact same time. There would be a measurable event on the Richter Scale as the article was simultaneously thrown into the trash can by nearly every mathematician reading it.

Now, this is, of course, a mere theory of mine. And I am not a mathematician (yet), but I would imagine this would happen.

Well you could use the Micromass-PhysicsForums theorem to show that the Richter-Scale is defined over the reals and does converge to the dirac delta function evaluated at that point in time.

You might have to use a few other results, but I think you're on to something here.

 

[arildno]

Assume there exists a mathematician that defines 0/0 to be something and he begins an article by writing something like "In this paper, we define 0/0 to be 0." Assume farther that the paper was actually published and (this next one is far more likely than the last one) every mathematician picked it up to read it at the exact same time. There would be a measurable event on the Richter Scale as the article was simultaneously thrown into the trash can by nearly every mathematician reading it.

Now, this is, of course, a mere theory of mine. And I am not a mathematician (yet), but I would imagine this would happen.

 

[Hurkyl]

For the record, I'd like to offer up three examples where a mathematician is doing something other than arithmetic of real numbers, where 0/0 can be usefully defined.
The first I encountered in the book Concrete Mathematics which introduces the concept of a strong zero. A strong zero multiplied by any expression results in zero -- even if that other expression doesn't make sense! So, in particular,

\mathbf{0} \cdot \frac{1}{0} = \mathbf{0}​

where I've used boldface for the strong zero. The application was the manipulation of series: they introduced a bracket operator [ \cdot ] on propositions by

[P] = \begin{cases} \mathbf{0} & \neg P \\ 1 & P \end{cases}​

The typical use of this bracket is in a summation, such as

H(n) = \sum_{k} [1 \leq k \leq n] \frac{1}{k}​

where the sum is over all integers, but the [] expression is used to control which terms actually contribute. This is a surprisingly useful tool for doing computations with sums. With this typical usage, it's not hard to see why the strong zero semantics makes sense.

The other example I've encountered is that of a wheel: a variation on the usual arithmetic axioms designed so that inversion is a total operation. The theory of wheels has been fleshed out to some extent, and the basic identities clearly illuminates some of the sacrifices one has to make so that division by zero makes sense; e.g. the distributive law no longer works and must be replaced with

xz + yz = (x+y)z + 0z​

even the idea of multiplicative inverse breaks down:

x/x = 1 + 0x/x​

The final example is that sometimes, people do calculations not with functions, but instead with partial functions. Suppose x is a real variable. If you are working with functions, then x/x is an illegal expression: you're only allowed to divide by nonzero things. However, if you are doing arithmetic with partial functions, then x/x is a partially-defined real number. (Defined on the domain of all nonzero x)

In this arithmetic, 0/0 is an "empty" real-valued constant. It has no value. It's essentially nothing more than a variation on the notion of "undefined" that we can use in expressions. The relation 0/0=1 is neither true nor false: it is the empty truth value. Again, it's the analog of undefined.

 

[Nano-Passion]

For the record, I'd like to offer up three examples where a mathematician is doing something other than arithmetic of real numbers, where 0/0 can be usefully defined.



The first I encountered in the book Concrete Mathematics which introduces the concept of a strong zero. A strong zero multiplied by any expression results in zero -- even if that other expression doesn't make sense! So, in particular,

\mathbf{0} \cdot \frac{1}{0} = \mathbf{0}​

where I've used boldface for the strong zero. The application was the manipulation of series: they introduced a bracket operator [ \cdot ] on propositions by

[P] = \begin{cases} \mathbf{0} & \neg P \\ 1 & P \end{cases}​

The typical use of this bracket is in a summation, such as

H(n) = \sum_{k} [1 \leq k \leq n] \frac{1}{k}​

where the sum is over all integers, but the [] expression is used to control which terms actually contribute. This is a surprisingly useful tool for doing computations with sums. With this typical usage, it's not hard to see why the strong zero semantics makes sense.




The other example I've encountered is that of a wheel: a variation on the usual arithmetic axioms designed so that inversion is a total operation. The theory of wheels has been fleshed out to some extent, and the basic identities clearly illuminates some of the sacrifices one has to make so that division by zero makes sense; e.g. the distributive law no longer works and must be replaced with

xz + yz = (x+y)z + 0z​

even the idea of multiplicative inverse breaks down:

x/x = 1 + 0x/x​

The final example is that sometimes, people do calculations not with functions, but instead with partial functions. Suppose x is a real variable. If you are working with functions, then x/x is an illegal expression: you're only allowed to divide by nonzero things. However, if you are doing arithmetic with partial functions, then x/x is a partially-defined real number. (Defined on the domain of all nonzero x)

In this arithmetic, 0/0 is an "empty" real-valued constant. It has no value. It's essentially nothing more than a variation on the notion of "undefined" that we can use in expressions. The relation 0/0=1 is neither true nor false: it is the empty truth value. Again, it's the analog of undefined.

Thanks for the post, it reminds me how truly versatile mathematics is.

Do you have an idea of where wheel theory is applicable?

 

[Hurkyl]

Thanks for the post, it reminds me how truly versatile mathematics is.

Do you have an idea of where wheel theory is applicable?

I'm not really sure if it has found practical application yet; it is relatively new, and in my initial estimation, in the niche where it is most likely to be useful it has some well-established competitors that do a "good enough" job.

e.g. I've seen a few situations where projective coordinates are used that a wheel can describe slightly better. But the deficiency of projective coordinates is mild enough that there isn't really demand for a better description.

 

[Nano-Passion]

I'm not really sure if it has found practical application yet; it is relatively new, and in my initial estimation, in the niche where it is most likely to be useful it has some well-established competitors that do a "good enough" job.

e.g. I've seen a few situations where projective coordinates are used that a wheel can describe slightly better. But the deficiency of projective coordinates is mild enough that there isn't really demand for a better description.

Okay, so what are some applications of projective coordinates?