# 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.