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Math Help - 0!

  1. #1
    Senior Member pacman's Avatar
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    0!

    Why is 0! = 1?
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  2. #2
    ynj
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    it is specially defined.
    Since 1!=1*0!.
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  3. #3
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    A silly poem I wrote while in college:



    Man has wondered
    Since time immemorial
    Why 1 is the value
    Of 0!.

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  4. #4
    Super Member dhiab's Avatar
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    Hello 0! is a convention without demonstration.
    Same  2^0 = 1 <br />
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by pacman View Post
    Why is 0! = 1?
    We can show this is the case using calculus (if this is a bit advanced for you, please bear with me )

    The Gamma Function is defined by \Gamma\left(x\right)=\int_0^{\infty}e^{-t}t^{x-1}\,dt for x>0

    If n\in\mathbb{N}, the Gamma Function has a special property: \Gamma\left(n\right)=\left(n-1\right)!.

    So it follows that {\color{blue}\Gamma\left(1\right)}=\left(1-1\right)!=\color{red}\boxed{0!}

    By definition, {\color{blue}\Gamma\left(1\right)}=\int_0^{\infty}  e^{-t}t^{1-1}\,dt=\int_0^{\infty}e^{-t}\,dt=\lim_{b\to\infty}\left.\left[-e^{-t}\right]\right|_0^{b} =-\lim_{b\to\infty}e^{-b}-\left(-e^{-0}\right)=0-(-1)=\color{red}\boxed{1}.

    Thus, {\color{blue}\Gamma\left(1\right)}=\color{red}\box  ed{0!=1}.
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  6. #6
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    I tried figuring out the Gamma function from Wikipedia a few weeks ago...but didn't understand it that well. Do you know any website where I can find a proper introduction to it? Or should I just wait until they cover it in one of my future college courses....once I start college?

    Thanks.
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  7. #7
    A riddle wrapped in an enigma
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    Quote Originally Posted by pacman View Post
    Why is 0! = 1?
    Hi pacman,

    I posted this in another thread, but I'll repost it here.

    See if this helps.

    n! is defined as 1 * 2 * 3 * . . . * n


    And with a little manipulation, we can show that 0! = 1 by demonstrating that...

    1! = 1
    2! = 1! * 2
    3! = 2! * 3
    4! = 3! * 4
    etc.

    Reversing that, we can achieve...

    3! = 4!/4
    2! = 3!/3
    1! = 2!/2
    0! = 1!/1 = 1
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  8. #8
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by RobLikesBrunch View Post
    I tried figuring out the Gamma function from Wikipedia a few weeks ago...but didn't understand it that well. Do you know any website where I can find a proper introduction to it? Or should I just wait until they cover it in one of my future college courses....once I start college?

    Thanks.
    See if this helps you: The Gamma Function
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  9. #9
    Super Member Random Variable's Avatar
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    Let's say that 0! is equal to something else, say 2.

    then the taylor series of  \cos x expanded about x=0 would be  \frac{1}{2} - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + ...

    which would imply that  \cos (0) = \frac{1}{2}

    and all hell would break loose
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  10. #10
    Senior Member pacman's Avatar
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    random variable, can you elaborate further? thanks!
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  11. #11
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by pacman View Post
    random variable, can you elaborate further? thanks!
    When you evaluate the taylor expansion of cosine at zero, you get \cos\left(0\right)=\frac{1}{0!}.

    When he said all hell would break loose, he was commenting on the fact that if 0! was some other value than 1, then \cos\left(0\right)=\frac{1}{\text{other value}}, which is absurd since we know from trig that \cos\left(0\right)={\color{red}1}. This would then force 0!=1.
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  12. #12
    Senior Member pacman's Avatar
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    ahhh, that is much CLEARER now. Thanks Cris
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  13. #13
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    So many different ways! I like this.

    The way I was taught was using the fact that that there are ^{n}C_k ways of picking k items from a set of n, where

    ^{n}C_k = \frac{n!}{(n-k)!k!} .

    If we think about it, it should be that there is only 1 way of choosing no items from a set of n items, therefore ^{n}C_0 = 1 and so,

    \frac{n!}{(n-0)!0!} = 1 \Rightarrow \frac{1}{0!} = 1  \Rightarrow  0! = 1

    Some of the other explanations are nicer I think, but there's my two cents!
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  14. #14
    Senior Member pacman's Avatar
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    Pomp, i like your way of obtaining 0! = 1 through Combination.
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  15. #15
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    If 0! was equal to 0, then n! would = 0,
    since we'd have to assume that ! represents 0*1*2......*n
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