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What is the value of 0^0?

  1. Jul 13, 2004 #1
    what is the value of 0^0?
  2. jcsd
  3. Jul 13, 2004 #2


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

    Now, consider the function g(x):

    Using g in the limiting procedure, yields [tex]0^{0}=1[/tex]

    That is [tex]0^{0}[/tex] "by itself" is indeterminate
  4. Jul 13, 2004 #3
    so it is a complex number?
  5. Jul 13, 2004 #4
    if x < 0?????????????????????????///
  6. Jul 13, 2004 #5
    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
  7. Jul 13, 2004 #6
  8. Jul 13, 2004 #7
    So if x=0????????????
    What will do?
  9. Jul 13, 2004 #8
    it's not allowed.

    you'll be banished to astrology!

  10. Jul 13, 2004 #9
    So we can't calculate 0^0 like an limite
    So what should we do?
  11. Jul 13, 2004 #10
    why would you want to [tex]0^0[/tex] anyway?
    Last edited: Jul 13, 2004
  12. Jul 13, 2004 #11
    What do u mean Gasso?
  13. Jul 13, 2004 #12
    When I looked last time [tex]0^0[/tex] wasn't defined in maths so it cannot be calculated. It's somewhat similar to [tex]\frac{1}{0}[/tex]. It cannot be calculated, but if some function is approaching it it may converge to some nuber, but it depends on the function.
  14. Jul 13, 2004 #13
    It can calculate but using colex nombers like 1^n
  15. Jul 13, 2004 #14
    Then how much it is?? I'm very curious.
  16. Jul 13, 2004 #15
    What do u mean tomkeus?
  17. Jul 13, 2004 #16
    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
  18. Jul 13, 2004 #17
    I just want to say that [tex]0^0[/tex] isn't real or complex valued like [tex]e^{i\phi}[/tex] or [tex]2^{15}[/tex]. It's indefinite value.
  19. Jul 13, 2004 #18
    No , i think that 0^0 has a value
  20. Jul 13, 2004 #19


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

    - Warren
  21. Jul 13, 2004 #20

    matt grime

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    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.
  22. Jul 13, 2004 #21


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    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.

  23. Jul 13, 2004 #22
    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 !!
  24. Jul 13, 2004 #23


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    Think about it this way:

    [tex]0^0 = 0^10^{-1} = \frac{0}{0}[/tex]
  25. Jul 13, 2004 #24
  26. Jul 13, 2004 #25

    matt grime

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    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.
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