# What is the value of 0^0?

1. Jul 13, 2004

### Feynman

what is the value of 0^0?

2. Jul 13, 2004

### arildno

It's indeterminate, it depends on the way in which you "approach it"
For example, consider the function f(x) defined on positive numbers:
$$f(x)=0^{x}\to{f(x)}=0,x>0$$
Approaching $$0^{0}$$ by evaluating f at ever closer x's, clearly indicates that:
$$0^{0}=\lim_{x\to{0}}f(x)=0$$

Now, consider the function g(x):
$$g(x)=x^{0}\to{g}(x)=1,x>0$$

Using g in the limiting procedure, yields $$0^{0}=1$$

That is $$0^{0}$$ "by itself" is indeterminate

3. Jul 13, 2004

### Feynman

so it is a complex number?

4. Jul 13, 2004

### Feynman

if x < 0?????????????????????????///

5. Jul 13, 2004

### gazzo

You're not allowed to do that? Atleast not with the set of reals?

As a consequence of one of the multiplication axioms, by definition.

x^0 = 1
iff x =/= 0

there are websites devoted to the number zero and im sure somebody will quote one as usual :P

6. Jul 13, 2004

### gazzo

7. Jul 13, 2004

### Feynman

So if x=0????????????
What will do?
Gasso?

8. Jul 13, 2004

### gazzo

it's not allowed.

you'll be banished to astrology!

:P

9. Jul 13, 2004

### Feynman

So we can't calculate 0^0 like an limite
So what should we do?

10. Jul 13, 2004

### gazzo

why would you want to $$0^0$$ anyway?

Last edited: Jul 13, 2004
11. Jul 13, 2004

### Feynman

What do u mean Gasso?

12. Jul 13, 2004

### tomkeus

When I looked last time $$0^0$$ wasn't defined in maths so it cannot be calculated. It's somewhat similar to $$\frac{1}{0}$$. It cannot be calculated, but if some function is approaching it it may converge to some nuber, but it depends on the function.

13. Jul 13, 2004

### Feynman

It can calculate but using colex nombers like 1^n

14. Jul 13, 2004

### tomkeus

Then how much it is?? I'm very curious.

15. Jul 13, 2004

### Feynman

What do u mean tomkeus?

16. Jul 13, 2004

### rayjohn01

From my physics viewpoint zero is never zero but a small +/- dx , in this sense 0^0 also involves the non integer root of a negative number

17. Jul 13, 2004

### tomkeus

I just want to say that $$0^0$$ isn't real or complex valued like $$e^{i\phi}$$ or $$2^{15}$$. It's indefinite value.

18. Jul 13, 2004

### Feynman

No , i think that 0^0 has a value

19. Jul 13, 2004

### chroot

Staff Emeritus
No, Feynman, not in this universe. It has no specific value. Mathematicians call such objects indeterminate.

- Warren

20. Jul 13, 2004

### matt grime

Why must it be defined, Feynman? Just because you can write it and think that it looks like it ought ot be a number doesn't mean it is actually such. log(x) can be defined for all real positive x, and if you're prepared to learn some complex analysis for complex non-zero x too, that doesn't mean log(0) is defined.

21. Jul 13, 2004

### Njorl

So, 0^0 is indeterminate because:

lim x->0 of x^0 =1 and,
lim x->0 of 0^x=0.

assume z= y^x, y=f(x) is such that when y->0, x->0.

Can we construct a functional relationship between x and y that when they approach 0 as a limit, they produce a "z" that is some finite number between 0 and 1? Or are 0 and 1 the only allowable results.

Njorl

22. Jul 13, 2004

### rayjohn01

This is exactly why I raised the physical viewpoint -- I'm not a mathematician but it appears to me that they are always in trouble with 'zero'.
A typical example is Integration -- y = int ( f(x).dx ) . dx ----> 0
IF you ignore any closed form result and start with a numerical analyisis , it forces you to choose dx because the sum of zeros IS zero. So dx=0 does not make sense but it can be a small as you like.
Nature ( isn't that what maths tries to describe) does not deal in zeros even though some objects may be VERY small ( 10^-39 ) or so . Even worse than that nature keeps objects moving in such a way as you may not even know where they are !!

23. Jul 13, 2004

### Zurtex

$$0^0 = 0^10^{-1} = \frac{0}{0}$$

24. Jul 13, 2004

### CrankFan

25. Jul 13, 2004

### matt grime

no it would appear not

there are two dx's in there, is that what you really mean?

closed form of what? what numerical analysis?

finite or countably finite indexed sum that is, uncountable becomes moot, obviously

who said it did? dx isn't even a number

i'm sorry? you're confusing delta and d, it appears: dx is not a number, though on occasion by treating it as such it may yield useful applied results.

who knows, maths may be used to model naturally occuring phenomena, and it certainly does use zeroes: the cardinality of the set of elephants that are mice is zero.

hmm, don't think you want to introduce quantum mechanics, which is after all a mathematical model, and especially the uncertainty princple which is just a formal result of certain parts of analysis and integration, which you didn't appear to understand when you used it above.