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

[wilsonb]

I think you completely missed the point of my post, which was that the correct term for 0/0 is that it is indeterminate rather than undefined. 1/0 is undefined, but 0/0 is indeterminate.

0/0 cannot be given meaning, period. 1/0 can be given a meaning in various contexts. a/0 is ∞ on the projective real number line for all non-zero a. In complex analysis, a/0 (with a≠0) is sometimes treated as complex infinity, a number whose magnitude is greater than any real number but whose argument is indeterminate.

In layman terms, in any value at all, there can be an infinite amount of 0s. Hence there can also be infinite 0s in a single 0. hence indeterminate.

 

[pwsnafu]

In layman terms, in any value at all, there can be an infinite amount of 0s.

Hence there can also be infinite 0s in a single 0. hence indeterminate.

That's not what "indeterminate" means. It's a technical term. It has a definition and is used only for that definition. You seem to be trying to rationalize the word choice. Definitions don't work like that.

PS: If we take your logic as is, then you are arguing all questions whose answer is "zero" is indeterminate. Or (worse) all finite answers are indeterminate. Neither is correct.

 

[mathwonk]

arrggggh.

 

[Anti-Crackpot]

Mathematical systems must be contradiction-free.

Can't help but wonder if Godel would agree with this statement as opposed, for instance, to the following statement:

"Sufficiently complex mathematical systems cannot be contradiction-free."

via Wikipedia...
"Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic."
http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

 

[arildno]

very simple.
All theorem built from axioms, and mathematics axiom is what we often say "difficult' to prove,

An utter misconception.

How do you "prove" that a poker hand can have 5, and only 5 cards?

It is laid out as a rule of the game, and in like manner, maths is a game where we pick whichever rules we want to play with. Those rules are called "axioms"

Obviously, we may construct as many maths games we want. Just like we can invent new card games, by laying down some new set of rules.

 

[pwsnafu]

Can't help but wonder if Godel would agree with this statement as opposed, for instance, to the following statement:

"Sufficiently complex mathematical systems cannot be contradiction-free."

Nonsense. If you are going to bring up the incompleteness theorems, at least learn them. The first theorem says you can't be contradiction free and be complete at the same time (at least for anything which includes PA). It says nothing about just being consistent. We prioritize consistency over completeness.

 

[micromass]

Can't help but wonder if Godel would agree with this statement as opposed, for instance, to the following statement:

"Sufficiently complex mathematical systems cannot be contradiction-free."

via Wikipedia...
"Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic."
http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

Godel would certainly agree with DH. Godels incompleteness theorems state (more or less):

1) Our current mathematical system can never be shown to be contradiction-free.
2) A mathematical system can never be AND complete AND contradiction-free.

(1) is certainly problematic, but it does not mean that there are actually contradictions in current math. We just can't prove it. So we more or less accept it on faith.

 

[arildno]

And, to add to micromass:
Jus because there MIGHT be some contradiction deeply embedded,as yet unrecognized bu us, in our preferred mathematical system, we need not worry too much about it, since if we DO notice it, it might well be easily remedied, by, for example, adding some pedantic little detail in an axiom formulation that doesn't do anything else than preventing just that contradiction from happening.
All previous results that (1) did NOT depend upon the flawed axiom to begin with, and (2) won't depend upon the new axiom would remain unaffected, and perhaps practically all the results which DID use the flawed axiom to begin with.

The contradiction might be more lethal than that, of course, but that it should be so is no implication that follows from the fact that we do not know if our current system is free of contradictions.

 

[D H]

Mathematical systems must be contradiction-free.

Can't help but wonder if Godel would agree with this statement as opposed, for instance, to the following statement:

"Sufficiently complex mathematical systems cannot be contradiction-free."

As others have noted, that is not what Godel's theorems say. What they do say is that in in any sufficiently complex mathematical system, (a) there will exist statements written using the constructs of the system that can neither be proved nor disproved using the constructs of that system, and (b) that the system is mathematically consistent is one of those statements that cannot be proved or disproved.

In other words, mathematical systems of sufficient complexity cannot be both consistent and complete. Consistency is essential, but completeness is not; it's just a nice thing to have. Mathematicians have given up on completeness (Hilbert's second problem) and assume consistency.

 

[wilsonb]

I think you completely missed the point of my post, which was that the correct term for 0/0 is that it is indeterminate rather than undefined. 1/0 is undefined, but 0/0 is indeterminate.

0/0 cannot be given meaning, period. 1/0 can be given a meaning in various contexts. a/0 is ∞ on the projective real number line for all non-zero a. In complex analysis, a/0 (with a≠0) is sometimes treated as complex infinity, a number whose magnitude is greater than any real number but whose argument is indeterminate.

You can switch em using calculus.
and 0/0 is just a product statement, it might have came from some complicated f(x) = h(x)/g(x)
where one simply got 0/0, when computing the limit.

undefined is a statement, where you get R/0, where R is any real number.
0/0 or anyother form of 0^n/0^n, when u compute limits of f(x) as a whole,
which give rise to the method of L'Hopital, when computing limit of f(x), when h(x)/g(x) gives you 0/0.

its 2 different things, one is the axiom & another is taking limits and tend to 0/0.
Additional Reference: Steward J' Calculus 7th ed.

 

[micromass]

You can switch em using calculus.
and 0/0 is just a product statement, it might have came from some complicated f(x) = h(x)/g(x)
where one simply got 0/0, when computing the limit.

undefined is a statement, where you get R/0, where R is any real number.
0/0 or anyother form of 0^n/0^n, when u compute limits of f(x) as a whole,
which give rise to the method of L'Hopital, when computing limit of f(x), when h(x)/g(x) gives you 0/0.

its 2 different things, one is the axiom & another is taking limits and tend to 0/0.
Additional Reference: Steward J' Calculus 7th ed.

No. You are totally missing the point of limits. In limits, we never get the operation 0/0.

If we have numbers 0 and 0, then its division 0/0 is not defined. This has nothing to do with limits or with functions.

If you have functions, then you can perhaps deal with $\lim_{x\rightarrow 0}\frac{\sin(x)}{x}$. But, you do NOT divide by 0 here. The limit means that if you get close to 0, then the expression sin(x)/x gets close to 1. You NEVER evaluate the function sin(x)/x in 0. So you NEVER deal with 0/0.

 

[wilsonb]

Mathematics are not just about the calculations. The calculations are to solve a particular situation, a problem statement, in which this mere question does not have.
This question lacks the problem statement we need to determine how we can solve it mathematically.

To put it in a sense where 0 = nothing, so 0 / 0 is definitely 0. There is nothing to divide it from and with, but the statement of 0 = nothing made it possible as nothing is required to divide nothing.

In short, UNDEFINED means it is undetermined UNLESS a problem statement is issued. Therefore, undefined can be anything as long as you fulfill the requisite and criteria of the conditions.

 

[micromass]

To put it in a sense where 0 = nothing, so 0 / 0 is definitely 0.

No.

In short, UNDEFINED means it is undetermined UNLESS a problem statement is issued.

No. Not in this case at least.

 

[Mark44]

Mathematics are not just about the calculations. The calculations are to solve a particular situation, a problem statement, in which this mere question does not have.
This question lacks the problem statement we need to determine how we can solve it mathematically.

No problem statement is needed, other than the main idea of this thread, which is what does 0/0 mean?

To put it in a sense where 0 = nothing, so 0 / 0 is definitely 0. There is nothing to divide it from and with, but the statement of 0 = nothing made it possible as nothing is required to divide nothing.

There is really nothing more complicated here than the arithmetic involved in the division of two numbers. It has been mentioned before in this thread that the division operation requires two input numbers, but an important point has been omitted: from division we require exactly one result. We require a single answer from all of the other arithmetic operations - why should division be any different?

The argument that 0/0 = 0 arises incorrectly from the fact that division and multiplication are inverse operations. If a/b = c, then a = b * c. This is true as long as b ≠ 0.

If we insist that 0/0 = 0 makes sense because 0 (the denominator) * 0 (the quotient) = 0 (the numerator), then we should also accept 0/0 = 2, because 0 * 2 = 0. Since we have gotten two different answers (and infinitely more are possible), this is a violation of the commonsense requirement that division produce a single result.

The upshot is that dividing by 0 is never defined, period.

[STRIKE]In short, UNDEFINED means it is undetermined UNLESS a problem statement is issued. Therefore, undefined can be anything as long as you fulfill the requisite and criteria of the conditions.[/STRIKE]

 

[Robert1986]

I just got an idea for my dissertation: I'm going to study the patterns and cycles by which threads like this get started on these forums. I can't wait until the next .999999... thread comes along. Then perhaps a thread from an arrogant ignoramus who is convinced that there is something fundamentally wrong about the real line.

 

[micromass]

I just got an idea for my dissertation: I'm going to study the patterns and cycles by which threads like this get started on these forums. I can't wait until the next .999999... thread comes along. Then perhaps a thread from an arrogant ignoramus who is convinced that there is something fundamentally wrong about the real line.

 

[Matt Benesi]

I just got an idea for my dissertation: I'm going to study the patterns and cycles by which threads like this get started on these forums. I can't wait until the next .999999... thread comes along.

Mind sammich.

 

[wilsonb]

No problem statement is needed, other than the main idea of this thread, which is what does 0/0 mean?

There is really nothing more complicated here than the arithmetic involved in the division of two numbers. It has been mentioned before in this thread that the division operation requires two input numbers, but an important point has been omitted: from division we require exactly one result. We require a single answer from all of the other arithmetic operations - why should division be any different?

The argument that 0/0 = 0 arises incorrectly from the fact that division and multiplication are inverse operations. If a/b = c, then a = b * c. This is true as long as b ≠ 0.

If we insist that 0/0 = 0 makes sense because 0 (the denominator) * 0 (the quotient) = 0 (the numerator), then we should also accept 0/0 = 2, because 0 * 2 = 0. Since we have gotten two different answers (and infinitely more are possible), this is a violation of the commonsense requirement that division produce a single result.

The upshot is that dividing by 0 is never defined, period.

The example I've given is merely the example for which a statement, 0 = nothing, is introduced into the calculation, which would provide a rectification on what the calculation is for. If the that statement is absent from this calculation, then your means would be true.
As I am typing this, I realized I have made an error, though I refuse to erase the top.
I was trying to implement algebraic argument to the 0 / 0 context, which in the end I believe, would be a mathematical fallacy. Forgive me on that.
I was initially planning to introduce a statement as to treat 0 as an unknown in which one could substitute for another for the cause of the current calculation, but after exercising the calculation from different aspects, I realized I was led to a spurious proof.

 

[micromass]

The example I've given is merely the example for which a statement, 0 = nothing, is introduced into the calculation, which would provide a rectification on what the calculation is for. If the that statement is absent from this calculation, then your means would be true.
As I am typing this, I realized I have made an error, though I refuse to erase the top.
I was trying to implement algebraic argument to the 0 / 0 context, which in the end I believe, would be a mathematical fallacy. Forgive me on that.
I was initially planning to introduce a statement as to treat 0 as an unknown in which one could substitute for another for the cause of the current calculation, but after exercising the calculation from different aspects, I realized I was led to a spurious proof.

I am sorry, but your posts show a very blatant misunderstanding of basic mathematics. I suggest you pick up a good math book and work through it.

I'll explain it once and for all:
0/0 is not defined because we choose it to be undefined. We could define it if we wanted to, but we choose not to. We have very good reasons for this.

First, let's define what division actually means: we say that n/m=p if and only if p is the unique number satisfying mp=n. The reason we choose not to define 0/0 is because there is no unique number p such that 0p=0. All number satisfy! We want / to be a function: that is, every input must give a unique output. This is not satisfied, so we rather choose not to define 0/0.

There is no way to prove that 0/0=0, because this would just be a definition. You can't prove definitions.
There is no context what-so-ever in which 0/0=0. Math works perfectly fine with not defining 0/0. So does computer science by the way: no plane ever fell from the sky because 0/0 has not been defined.

Arguing about 0/0 is pointless, since you're just arguing a definition. You can agree or disagree with a definition, sure. But the fact remains that 99.999999...% of the mathematicians choose to let 0/0 be undefined.

 

[arildno]

According to micromass, there must either A) exist at least 100 million professional mathematicians in the world, or B) EVERY one of them choose to let 0/0 be undefined.

Somehow, I doubt that..

 

[chiro]

According to micromass, there must either A) exist at least 100 million professional mathematicians in the world, or B) EVERY one of them choose to let 0/0 be undefined.

Somehow, I doubt that..

Where do you get 100 million from? (Just curious).

 

[arildno]

Where do you get 100 million from? (Just curious).

He wrote about, at the very least 99.999999%.

Now, how big must the population of mathematicians be in order for that percentage of the mathematicians to come from the division between two integers?

(I absolutely refuse to accept the existence of any non-integral mathematicians..)

 

[D H]

He wrote about, at the very least 99.999999%.

He wrote about 99.999999...%, which is another way of saying 100%.

 

[arildno]

He wrote about 99.999999...%, which is another way of saying 100%.

That is the B) option I mentioned originally. I doubt the validity of that assertion, as well.

 

[Robert1986]

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.

 

[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?