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Math Help - [SOLVED] Power of Zero

  1. #1
    Gini
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    [SOLVED] Power of Zero

    Something I should remember but can't! What do numbers to the power of Zero equal? To the power of one they are themselves and squared, cubed etc is obvious but zero?? Anyone remember
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Gini
    Something I should remember but can't! What do numbers to the power of Zero equal? To the power of one they are themselves and squared, cubed etc is obvious but zero?? Anyone remember
    it is equal to 1
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by malaygoel
    it is equal to 1
    Except for 0^0, which is traditionaly undefined.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack
    Except for 0^0, which is traditionaly undefined.

    RonL
    Sometimes it is useful for Infinite Series to define that to be one.
    In England that is not the style though.
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  5. #5
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by ThePerfectHacker
    Sometimes it is useful for Infinite Series to define that to be one.
    In England that is not the style though.
    I don't see how that could ever be the style because  0^0 isn't undefined because of tradition, it's because of the rules of powers.

     a^3 \cdot a^{-3} = a^{(3-3)} = a^0 = 1
    or it could be written as...
     a^3 \cdot a^{-3} = \frac{a^3}{a^3} = 1

    so  0^1 \cdot 0^{-1} = 0^{(1-1)} = 0^0 = undefined
    or it could be written as...
    so  0^1 \cdot 0^{-1} = \frac{0^1}{0^1} = \frac{0}{0} =  undefined
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  6. #6
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    Quote Originally Posted by Quick
    tradition
    The tradition is that the bases are always positive numbers.

    This is my 14th Post!!!
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  7. #7
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    Quote Originally Posted by Quick
    I don't see how that could ever be the style because  0^0 isn't undefined because of tradition, it's because of the rules of powers.

     a^3 \cdot a^{-3} = a^{(3-3)} = a^0 = 1
    or it could be written as...
     a^3 \cdot a^{-3} = \frac{a^3}{a^3} = 1

    so  0^1 \cdot 0^{-1} = 0^{(1-1)} = 0^0 = undefined
    or it could be written as...
    so  0^1 \cdot 0^{-1} = \frac{0^1}{0^1} = \frac{0}{0} =  undefined
    How to define to 0^0 is an old controversy. This page has a good discussion and claims "Consensus has recently been built around setting the value of 0^0 = 1."
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  8. #8
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by JakeD
    How to define to 0^0 is an old controversy. This page has a good discussion and claims "Consensus has recently been built around setting the value of 0^0 = 1."
    It seems like they want it to be  0^0=1 for a lot of reasons, but they give no proof that  0^0=1 , although it might be convenient, I don't think it's very mathmatical.
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by Quick
    It seems like they want it to be  0^0=1 for a lot of reasons, but they give no proof that  0^0=1 , although it might be convenient, I don't think it's very mathmatical.
    There can be no proof, it will be defined to have a particular value
    if/when the convention of it being undefined is changed.

    RonL
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  10. #10
    Grand Panjandrum
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    Quote Originally Posted by JakeD
    How to define to 0^0 is an old controversy.
    It appears to be still controversial, which means that unless an
    author defines what they want it to mean, it is ambiguous and
    hence undefined

    This page has a good discussion and claims "Consensus has recently been built around setting the value of 0^0 = 1."
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  11. #11
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack
    It appears to be still controversial, which means that unless an
    author defines what they want it to mean, it is ambiguous and
    hence undefined
    I can't see why it could EVER be defined as one or the other since:
    \lim_{x \to 0}0^x \to 0

    and

    \lim_{y \to 0}y^0 \to 1

    As the limits do not agree we can't say that 0^0 is defined. What confuses me is why there is even discussion about defining it to be one or the other??

    -Dan
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  12. #12
    TD!
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    Have you thought of letting x approach 0 as base and power symmetrically?

    \mathop {\lim }\limits_{x \to 0} x^x  = 1

    This should make the "convention" at least more plausible.
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  13. #13
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by topsquark
    I can't see why it could EVER be defined as one or the other since:
    \lim_{x \to 0}0^x \to 0

    and

    \lim_{y \to 0}y^0 \to 1

    As the limits do not agree we can't say that 0^0 is defined. What confuses me is why there is even discussion about defining it to be one or the other??

    -Dan
    I would like to point out that when something can't be defined it becomes UNDEFINED

    I also think this post should be moved to chat room or miscellaneous.
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