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Math Help - complex nos problem

  1. #1
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    complex nos problem

    which is greater

    i^i or i^-i

    i^i^i or i^(pi/2)
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  2. #2
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    Can complex numbers be ordered?
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  3. #3
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    Quote Originally Posted by prasum View Post
    which is greater

    i^i or i^-i

    i^i^i or i^(pi/2)
    Enter these expressions as Google searches if you want them evaluated. For example, a search for i^i will get the response i^i = 0.207879576. In fact, i^i and i^-i are real numbers, so it makes sense to ask which is greater. But i^(i^i) and i^(pi/2) are complex. As pickslides points out, there is no natural sense in which one is greater than the other.
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  4. #4
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    Quote Originally Posted by pickslides View Post
    Can complex numbers be ordered?
    Yes, e.g. by the lexiographic order. However, there is no ordered complex field.
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  5. #5
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    i^i = \left(e^{\frac{\pi}{2}i}\right)^i

     = e^{\frac{\pi}{2}i^2}

     = e^{-\frac{\pi}{2}}.


    Meanwhile i^{-i} = (i^i)^{-1}

     = \left(e^{-\frac{\pi}{2}}\right)^{-1}

     = e^{\frac{\pi}{2}}.


    Clearly e^{\frac{\pi}{2}} > e^{-\frac{\pi}{2}}, so i^{-i} > i^i.
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