What is the Result of 0/0? Exploring Infinity

[cdux]

If both numbers approach 0 but one does not know their exact state, doesn't that mean the result can be either 1 or anything around it up to -∞and +∞? Does that in turn mean that 0/0 = Everything?

 

[Curious3141]

If both numbers approach 0 but one does not know their exact state, doesn't that mean the result can be either 1 or anything around it up to -∞ and +∞? Does that in turn mean that 0/0 = Everything?

It means it's undefined when written simply as $\displaystyle \frac{0}{0}$. "Undefined" means we do not assign any value to it.

In analysis, limits of the form $\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)}$ where $\displaystyle f(a) = g(a) = 0$ may be 0, ∞, -∞, any real number at all, or undefined. It simply depends on the exact definition of the functions.

 

[cdux]

My full reasoning:

"0/0" via a/b: a, b tend to 0

1. The two numbers are independent hence they can be approaching zero but no one knows if they are the same or not

2. That means their result can be either 1 or anything around it.

3. i.e. it can be between -oo and +oo

i.e. 0/0 = Anything.

hence 0/0 is buzzing around 1.

 

[Nugatory]

It means that 0/0 is undefined. You can write the symbols on a piece of paper, but they don't mean anything.

 

[cdux]

I could further describe it as a cloud of numbers around 1. I wonder if it has Quantum Mechanical implications.

 

[junaid314159]

Another term that is often used to describe $\frac{0}{0}$ is indeterminate.

 

[Nugatory]

I could further describe it as a cloud of numbers around 1. I wonder if it has Quantum Mechanical implications.

No, not even slightly. This stuff you hear about how quantum mechanics replaces the notion of a point particle with a "cloud"... it's a very hand-wavey oversimplification of stuff that actually has a very precise mathematical description using precise mathematical concepts with no inherent fuzziness.

The problem with trying to find a meaning for $0/0$ is that it doesn't have any meaning to find. Sure, you can write it down on a piece of paper, but not everything written down on paper has to have a meaning. When you write $0/0$, you're basically saying "Hey - $0$ is a mathematical symbol; $/$ is a mathematical symbol; let's put them side by side and ask what it means". You could just as well ask what the value of $)+4$ should be - I got that the same way, by combining mathematical symbols in a way that makes no sense. The only difference is that it's more obvious that $)+4$ can't be expected to mean anything; $0/0$ wants to trick you into thinking that it might mean something.

Go back and reread Curious3141's post above - he's pointing out the right way of proceeding when you find yourself thinking that you're looking at a $0/0$ situation. Also, if your math is up to it (somewhere around the second or third year of high-school algebra is enough background to give this a try) you might want to look at the mathematical concept of the "derivative of a function"- this is one of the most important and practically useful applications of these techniques.

 

[HallsofIvy]

If both numbers approach 0 but one does not know their exact state, doesn't that mean the result can be either 1 or anything around it up to -∞and +∞? Does that in turn mean that 0/0 = Everything?

What was your purpose in posting this? You ask a question but then ignore all responses. Your "mystical" ideas on what 0/0 means have nothing to do with mathematics.

 

[Mandelbroth]

If both numbers approach 0 but one does not know their exact state, doesn't that mean the result can be either 1 or anything around it up to -∞and +∞? Does that in turn mean that 0/0 = Everything?

We can't define it uniquely. Any number $x$ satisfies $0x=0$, so there isn't a unique value to give for $\frac{0}{0}$. It has nothing to do with a "cloud" or "buzzing." It's indeterminate.

 

[D H]

We can't define it. Any number $x$ satisfies $0x=0$, so there isn't a unique value to give for $\frac{0}{0}$. Thus, it's undefined. It has nothing to do with a "cloud" or "buzzing." It's undefined.

Better: It's indeterminate, not undefined.

Compare $0x=1$ with your $0x=0$. Every number $x$ satisfies the latter expression, but none satisfies the former. Thus $\frac 1 0$ is undefined but $\frac 0 0$ is intedeterminate.

 

[Mandelbroth]

Better: It's indeterminate, not undefined.

Compare $0x=1$ with your $0x=0$. Every number $x$ satisfies the latter expression, but none satisfies the former. Thus $\frac 1 0$ is undefined but $\frac 0 0$ is indeterminate.

Yes. Pardon me. I meant to say indeterminate.

 

[Stephen Tashi]

If both numbers approach 0 but one does not know their exact state, doesn't that mean the result can be either 1 or anything around it up to -∞and +∞? Does that in turn mean that 0/0 = Everything?

In another thread you say that you try to give a physical meaning to mathematical ideas. This is an incorrect approach to mathematics and your thinking about 0/0 illustrates this. You are assuming the symbols 0/0 mean something "real" and you are attempting to investigate this supposedly real phenomena by thinking about some process. This is what can be called a "Platonic" approach to math. You think that mathematical things have some existence before we bother to say precisely what those things are.

Many people have such feeling about triangles, the integers, the real numbers, finite groups etc. Where Platonic reality stops and the feeling that mathematical things are "just definitions" varies from person to person. However, from a point of view of learning math, you'll get completely screwed up if you substitute your own private ideas about what mathematical things are for their formal definitions - or lack thereof. People who do this build up their own fantasy worlds that never quite agree with mathematics.

 

[cdux]

Who told you it's not wrong? Next time try to answer with no insults.

Just because someone postulates something it doesn't mean he has a Messiah complex and you have to attack him.

 

[micromass]

I see no insults here. Please report anything you see as an insult and do not respond in the thread about them. Let's keep this on topic.

Also, cdux, you seem ot have your mind already made up. Did you ask this because you didn't know the answer but wanted input from other people? Or did you make this to advance your own theory?

 

[cdux]

I see no insults here. Please report anything you see as an insult and do not respond in the thread about them. Let's keep this on topic.

Also, cdux, you seem ot have your mind already made up. Did you ask this because you didn't know the answer but wanted input from other people? Or did you make this to advance your own theory?

For the second time, I have no Messiah complexes. An idea can be expressed explicitly, do I really have to add tons of text apologizing "It may be wrong guys, help me", to avoid being treated like having Narcissist Personality Disorder?

 

[micromass]

For the second time, I have no Messiah complexes. An idea can be expressed explicitly, do I really have to add tons of text apologizing "It may be wrong guys, help me", to avoid being treated like having Narcissist Personality Disorder?

Nobody said this about you, except you.

So, of the answers you were given, do you think they were helpful? Any more concerns about them?

 

[cdux]

They are helpful, though to be honest it's mainly on being rigorous, which is helpful of course.

It's still sane to say that 0 x Anything = 0 in those conditions so 0 / 0 = Anything pans out.

But excuse me while I don't look very closely to responses that are basically sarcastic.

 

[micromass]

They are helpful, though to be honest it's mainly on being rigorous, which is helpful of course.

It's still sane to say that 0 x Anything = 0 in those conditions so 0 / 0 = Anything pans out.

But excuse me while I don't look very closely to responses that are basically sarcastic.

I really see no sarcastic responses here, but never mind.

Anyway, your point of view is that you see division as a multi-valued function. This is a perfectly ok point of view. And math can be developed that way and it gives us a very elegant theory. But you should realize that this is not the convention that mathematics takes. It could take that convention, no problem with it. But it doesn't.

So while there is no problem doing things the way you do. You should keep in mind that all other people do it differently.

 

[cdux]

I know, I really should be more rigorous concerning mathematics.

I have this bad habit of overanalysing it and doing little math.

I guess I'll have to google 'mathematical rigor'..

 

[Mark44]

They are helpful, though to be honest it's mainly on being rigorous, which is helpful of course.

It's still sane to say that 0 x Anything = 0 in those conditions so 0 / 0 = Anything pans out.

No, your latter equation does NOT pan out. Division by zero is not defined, so you can't get from 0*x = 0 to 0/0 = x.

 

[cdux]

you can't get from 0*x = 0 to 0/0 = x.

I didn't. The reasoning was different.

 

[Nugatory]

It's still sane to say that 0 x Anything = 0 in those conditions so 0 / 0 = Anything pans out.

No, your latter equation does NOT pan out. Division by zero is not defined, so you can't get from 0*x = 0 to 0/0 = x.

I didn't. The reasoning was different.

I have to say I'm having some trouble understanding your position here. If you're using the $/$ symbol as the division operator, then I don't understand how you can say that an expression containing $/0$ does not involve division by zero. Nor do I know of a way to get from $Ax=B$ to $x=B/A$ that doesn't assume that division by $A$ is a valid operation.

But this is all a bit beside the point... Curious3141 pointed you in the right direction way back in post #2 and I tried another constructive response in the last paragraph of post #7 of this thread. Have you followed up on these? It's really hard to improve on the current machinery for handling $0/0$ problems if you don't know what that machinery is and what it already does.

 

[Mark44]

you can't get from 0*x = 0 to 0/0 = x.

I didn't. The reasoning was different.

This is what you said:

It's still sane to say that 0 x Anything = 0 in those conditions so 0 / 0 = Anything pans out.

I paraphrased what you wrote (Anything --> x). It doesn't matter what your reasoning is. You started with 0x = 0 and concluded that 0/0 = x, so you apparently divided both sides of the first equation by zero. This is not valid to do.

 

[cdux]

This is what you said:


I paraphrased what you wrote (Anything --> x). It doesn't matter what your reasoning is. You started with 0x = 0 and concluded that 0/0 = x, so you apparently divided both sides of the first equation by zero. This is not valid to do.

Jesus, read the 2nd reply of the thread. Or even the OP.

 

[Stephen Tashi]

I think the thread would benefit from clearly addressing the question of whether "0/0" has a definition or not. This depends on the context for using the symbols "0/0".

Some books may take the trouble to define "indeterminate forms" of limits and "0/0" is one of these.

However, there is no definition for "0/0" as a number. Trying to "prove" that "0/0" is NOT a number or that it is an "indeterminate" number (whatever that may be) by using a series of mathematical arguments is doomed to failure since you must begin by invoking the properties of something that has no definition.

If we interpret "a/b" as notation for an algorithm that says "To compute a/b = c find the unique number c such that a = bc and we interpret "0/0" as an abbreviation for setting a = 0 and b = 0 in that algorithm, then we can rationally discuss whether that algorithm can product a unique output.

 

[cdux]

If we interpret "a/b" as notation for an algorithm that says "To compute a/b = c find the unique number c such that a = bc and we interpret "0/0" as an abbreviation for setting a = 0 and b = 0 in that algorithm, then we can rationally discuss whether that algorithm can product a unique output.

Is it not possible to produce a fuzzy output? My understanding is that a/b with them independently approaching zero, will produce a fuzzy 'anything' number around 1. PS. If they are equal it will produce 1.

 

[cdux]

A weird postulation this may produce is that if that 'anything' has a tendency to be closer to 1 rather that infinities then it might point towards why physical numbers tend to not be infinite.

 

[Enigman]

Isn't this thread slightly redundant? That was the oppinion when I began reading...now I have a headache with all the talk about algos, anythings, psychological complexes and what-nots..…

 

[Nugatory]

Is it not possible to produce a fuzzy output? My understanding is that a/b with them independently approaching zero, will produce a fuzzy 'anything' number around 1. PS. If they are equal it will produce 1.

That process does not produce a fuzzy output, it produces an exact result. What that result will be depends on exactly how $a$ and $b$ approach zero - this is the point that curious3141 made way back in post #2 where he describes how to assign a value to $a/b$ when $a$ and $b$ are independently approaching zero.
(BTW, the value need not be especially near one; depending on how $a$ and $b$ approach zero, we can get anything between $-∞$ and $+∞$.)

 

[cdux]

That process does not produce a fuzzy output, it produces an exact result. What that result will be depends on exactly how $a$ and $b$ approach zero - this is the point that curious3141 made way back in post #2 where he describes how to assign a value to $a/b$ when $a$ and $b$ are independently approaching zero.
(BTW, the value need not be especially near one; depending on how $a$ and $b$ approach zero, we can get anything between $-∞$ and $+∞$.)

Both approach towards zero so they will both tend to be two very small numbers. This points to a similarity. They are not arbitrary, they are specifically very small. Hence it comes to reason that while the result can be anything between -∞ and +∞, they will have a bias towards 1 rather than ∞.

The same would be true for ∞/∞.

 

[Mentallic]

Both approach towards zero so they will both tend to be two very small numbers. This points to a similarity. They are not arbitrary, they are specifically very small. Hence it comes to reason that while the result can be anything between -∞ and +∞, they will have a bias towards 1 rather than ∞.

If you're happy to avoid any Mathematical rigour and just throw ideas out there, then I'll have my take on it too.

I could also argue that since both numbers are tending to 0, and "many" indeterminate forms of 0/0 are far from 1, being either $\pm\infty$ then this is evidence that points towards it having a bias towards infinity rather than 1. I'm imagining that the only reason you're seeing a bias towards 1 is because x/x=1 for every other real value of x. Or, how about we take another approach? If we consider the limits that are non-negative, then the value for 0/0 must be $[0,\infty)$ and since if we consider

$$\lim_{(a,b)\to 0}\frac{a}{b}$$

then the value is only equal to 1 when a approaches 0 at the same rate as b, so there are going to be equally many values of a that approach zero faster than b (giving a value of < 1) as there are going to be values of a approaching 0 slower than b (giving a value > 1), so does this mean 1 is equidistant between 0 and $\infty$?

*** There is of course no mathematical rigour in these statement I've made, and I absolutely do not stand by them.

 

[Mark44]

Is it not possible to produce a fuzzy output?

No, and you have been told this repeatedly. Division, when it is defined, produces a single result. Furthermore, division by zero is undefined.

My understanding is that a/b with them independently approaching zero, will produce a fuzzy 'anything' number around 1.

Then your understanding is flawed.
These three limits are all of the [0/0] indeterminate form, but the limit values are wildly different.
$$\lim_{x \to 0} \frac{x}{x^2} \text{does not exist} $$
$$\lim_{x \to 0} \frac{x^2}{x} = 0$$
$$\lim_{x \to 0} \frac{x}{x} = 1$$

What you seem to be missing is that even though both numerator and denominator are approaching zero, how quickly one or the other is approaching zero is the determining factor.

PS. If they are equal it will produce 1.

If they both approach zero at the same rate, the limit will be zero.

A weird postulation this may produce is that if that 'anything' has a tendency to be closer to 1 rather that infinities then it might point towards why physical numbers tend to not be infinite.

Nonsense.

Both approach towards zero so they will both tend to be two very small numbers.

Yes, of course. That's what "approaching zero" means, but again, what's important is how quickly one or the other (or both) are approaching zero.

This points to a similarity. They are not arbitrary, they are specifically very small. Hence it comes to reason that while the result can be anything between -∞ and +∞, they will have a bias towards 1 rather than ∞.

No.

The same would be true for ∞/∞.

No, absolutely not, and for the same reason I gave above. The important consideration is not that both numbers are getting arbitrarily large, but rather, how quickly one or the other (or both) is getting large.

 

[Nugatory]

Both approach towards zero so they will both tend to be two very small numbers. This points to a similarity. They are not arbitrary, they are specifically very small. Hence it comes to reason that while the result can be anything between -∞ and +∞, they will have a bias towards 1 rather than ∞.

Here is one specific and very important (it's the basis for all calculus, and it's been known and well understood since the 17th century) example:$$\lim_{a\to0}\frac{f(x+a)-f(x)}{a}$$

It should be clear that both the numerator and the denominator are approaching zero. However, the value of this expression doesn't have a "bias towards 1" - it is the slope of the graph of $f$ at the point $x$ (the $a$ disappears everywhere when you take the limit) and it's only going to have a "bias towards 1" if $f$ is specifically the function $f(x)=x$. Trivially, it has the value $A$ when $f$ is the function $f(x)=Ax$ and there's no reason why $A$ should be anywhere near one. It gets even more interesting and even less biased towards one if $f$ is a more interesting function (try it with $f(x)=2x^2$ as an exercise).

As I said, this application of the $0/0$ machinery was discovered along with differential calculus in the 17th century. Until you've worked through it, you're restricting your mathematical understanding to the state of the art - four centuries ago.

 

[micromass]

I think it's time to close this. Many posters have given clear and correct answers. It's up to the OP to decide whether he wants to listen or not.