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Thread: What is 1^i?

  1. #1 What is 1^i? 
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    If the complex number i is defined as i=sqrt(-1), then what is the value of 1^i?


    K. Srinivasa Ramanujam
    M.S (by Research) Scholar,
    http://ramanujamblog.blogspot.com
    India
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  3. #2  
    Forum Freshman Draculogenes's Avatar
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    1^-1^1/2 .... in short, it's imaginary


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  4. #3  
    Forum Isotope Zelos's Avatar
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    1=e^(0)
    1^i = (e^(0))^i = e^(0*i) = e^(0)=1
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

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  5. #4  
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    = 1^(-1/2)

    1^x = 1.

    *This is wrong, i thought i had the exponent rules in my head and all worked out. I didn't.
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  6. #5  
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    Depends on your branch of logarithm, 1^i is multivalued.

    arg(1)=2*pi*n

    for any integer n. So log(1)=2*pi*n*i, the choice of n corresponds to a choice of the bracnh of arg (equivalently a choice of branch of log), and a corresponding branch of z^i.

    1^i=exp(i*log(1))=exp(i*2*pi*n*i)=exp(-2*pi*n)

    depending on which branch you are on.

    though some texts will adopt the convention that a positive real base corresponds to the choice of arg that has arg of a positive real equaling zero.
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  7. #6  
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    Quote Originally Posted by LeavingQuietly
    = 1^(-1/2)

    1^x = 1.
    I don't understand your answer, can you explain that again? Thanks



    Quote Originally Posted by Zelos
    1=e^(0)
    1^i = (e^(0))^i = e^(0*i) = e^(0)=1
    I agreed with you :-D
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  8. #7  
    Forum Ph.D. william's Avatar
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    I used
    a<sup>b</sup> = e<sup>b ln(a)</sup>

    => 1

    cheers
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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  9. #8 Uh 
    New Member dmerthe's Avatar
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    1 to the power of anything is 1. 1 is just an identity element, remember?


    1^bananas = 1
    Raffiniert ist der Herrgott, aber boschaft ist er nicht
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  10. #9 Re: Uh 
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by dmerthe
    1 to the power of anything is 1. 1 is just an identity element, remember?


    1^bananas = 1
    Almost, 1 to the power of any real anything is always 1. Exponents are not that simple for complex numbers (as Schmoe mentioned) as the logarithm is multivalued.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  11. #10  
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    i got + or - 1

    let x = 1^i

    x = 1 ^ (-1^(1/2))
    x^2 = 1 ^ (-1)
    x^2 = 1/1 = 1
    x = +/- 1

    I'm not sure if it is right though
    so yeh just check it to be sure but i know mathematically i didnt make mistakes its just i dont know if the negative should be part of the answer
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  12. #11  
    Forum Professor river_rat's Avatar
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    Jako, you cant just square equations with out checking the answers - more importantly squaring the equation does not get rid of the problem in the exponent. You should have

    x^2 = 1<sup>2 i</sup>
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  13. #12  
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    yep soz i made that mistake i'll try again
    thnx for pointing it out its a highschool maths thing that stupid me
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  14. #13  
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    Quote Originally Posted by axlarry
    Quote Originally Posted by LeavingQuietly
    = 1^(-1/2)

    1^x = 1.
    I don't understand your answer, can you explain that again? Thanks



    Quote Originally Posted by Zelos
    1=e^(0)
    1^i = (e^(0))^i = e^(0*i) = e^(0)=1
    I agreed with you :-D
    I was wrong, that is how I explain it.
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  15. #14  
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    It is just plain 1

    let x= 1^i
    ln(x) = ln(1^i) ln on both sides
    ln(x) =i * ln(1) ln(a^b)=b * ln(a)
    e^ln(x) = e^[i*ln(1)] e on both sides
    x = [e^i]^ln1 a^(b*c)= (a^b)^c
    x =(e^i)^0 ln(1) = 0
    x =1 a^0 = 1
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  16. #15  
    Forum Professor river_rat's Avatar
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    Jaco - you taking logs in the complex plane and treating them as single valued functions instead of multivalued functions.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  17. #16  
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    so i gotto go and try it again
    this is good i love it when people keep on showing me just my mistakes and i gotto go and find the correction myself
    thanks river rat
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