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Thread: E to the i Pi

  1. #1 E to the i Pi 
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    Plain and simple...

    Why does e to the i pi equal negative one

    That is e to the power of i times pi equals negative one.

    My friend and I have been wondering.


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  3. #2  
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    Did you start from the definition of for ? Unless you know what the definition is, you are not likely to get very far at all. :?

    One way is to define as the complex power series , which is analytic on the whole complex plane. Well? That probably wonít help in the discussion between you and your friend, so letís try another one Ö

    An alternative and intuitively simpler definition is this: writing , where x and y are real, define .

    For , just substitute into the formula.


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  4. #3  
    . DrRocket's Avatar
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    Quote Originally Posted by JaneBennet
    Did you start from the definition of for ? Unless you know what the definition is, you are not likely to get very far at all. :?

    One way is to define as the complex power series , which is analytic on the whole complex plane. Well? That probably wonít help in the discussion between you and your friend, so letís try another one Ö

    An alternative and intuitively simpler definition is this: writing , where x and y are real, define .

    For , just substitute into the formula.
    The best way is to start with your power series and then substitute ix for z. Then notice that the power series naturally separates into a real and an imaginary part which are recognizable as the power series for cos(x) and i sin (x). You don't need the machinery of complex variables and analytic functions to do that.
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  5. #4  
    Forum Masters Degree bit4bit's Avatar
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    There is another post about this, where Serpicojr (and Magimaster) helped me go through the derivations. You might find it helpful:

    http://www.thescienceforum.com/viewt...umber+division
    Chance favours the prepared mind.
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  6. #5  
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    I didn't understand your explanations. I'll show what you guys said to my Calculus teacher, so your time spent answering wasn't pointless. Hopefully she'll be able to explain it to me.

    Thanks
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  7. #6  
    Forum Ph.D. Leszek Luchowski's Avatar
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    Quote Originally Posted by SteveC
    I didn't understand your explanations.
    The whole point is that you need to know what it means to raise a number to a complex (or pure imaginary) power. Once you get that, e^(i*pi)=-1 follows naturally.

    Cheers, Leszek.
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  8. #7  
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    Quote Originally Posted by Leszek Luchowski
    Quote Originally Posted by SteveC
    I didn't understand your explanations.
    The whole point is that you need to know what it means to raise a number to a complex (or pure imaginary) power. Once you get that, e^(i*pi)=-1 follows naturally.

    Cheers, Leszek.
    Actually it helps to have defeined e^z via the power series before you try to define arbitrary complex numbers (or arbitrary real numbers for that matter) to a complex power. The reason is that to do the general case you need the notion of a logarithm, which takes some work in the complex case, but the exponential function is easily and globally defined by the power series.
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