# What is 0/0

1. Aug 2, 2013

### 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?

2. Aug 2, 2013

### Curious3141

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.

3. Aug 2, 2013

### 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.

4. Aug 2, 2013

### Staff: Mentor

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

5. Aug 2, 2013

### cdux

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

6. Aug 2, 2013

### junaid314159

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

7. Aug 2, 2013

### Staff: Mentor

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.

Last edited: Aug 2, 2013
8. Aug 2, 2013

### HallsofIvy

Staff Emeritus
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.

9. Aug 2, 2013

### Mandelbroth

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.

Last edited: Aug 2, 2013
10. Aug 2, 2013

### D H

Staff Emeritus
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.

11. Aug 2, 2013

### Mandelbroth

Yes. Pardon me. I meant to say indeterminate.

12. Aug 2, 2013

### Stephen Tashi

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.

13. Aug 2, 2013

### 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.

14. Aug 2, 2013

### micromass

Staff Emeritus
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.

15. Aug 2, 2013

### cdux

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?

16. Aug 2, 2013

### micromass

Staff Emeritus
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?

17. Aug 2, 2013

### 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.

18. Aug 2, 2013

### micromass

Staff Emeritus
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.

19. Aug 2, 2013

### 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'..

20. Aug 2, 2013

### Staff: Mentor

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.