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Thread: Sequence of powers of i and Pi gives 1

  1. #1 Sequence of powers of i and Pi gives 1 
    Forum Freshman
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    This is probably a trivial problem for an experienced mathematician...

    I was playing with the Google calculater and got this (for me) surprising result. All other combinations produce much 'uglier' outcomes.

    i^(pi^(i^(pi^(i^(pi^(i^pi)))))) = 1

    Why is this sequence of powers of Pi and i equal to 1 ?

    Thanks!


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  3. #2  
    Forum Professor serpicojr's Avatar
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    It's actually not. The problem is calculators only give approximations. If you calculate:

    i^(π^(i^(π^(i^π)))

    You get:

    -4.96013383 × 10<sup>42</sup> - 1.54189139 × 10<sup>42</sup> i

    This is a complex number whose real part is negative and very large in magnitude. So since:

    |e<sup>x+iy</sup>| = e<sup>x</sup>

    π to the above number is a very small complex number. So small, in fact, that Google calculates it to be 0. Then anything to the 0th power is defined to be 1.


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  4. #3  
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    Thanks. Got it. Thought in that direction as well.

    I also found:

    i^(pi^(i^(pi^(pi^(i^pi))))) = 0

    Assume this is the same problem, but you now raise a number close to zero to the power pi.
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  5. #4  
    Forum Professor serpicojr's Avatar
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    It's the same thing. In this case, it's easier to see this can't be right: a<sup>b</sup> ≠ 0 unless a = 0.
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