Imaginary Numbers Raised to Imaginary Numbers

Date: 12/29/2001 at 23:10:01
From: Dan
Subject: Imaginary numbers raised to imaginary numbers

Hello. I was fooling around with my TI-89 graphing calculator, and I 
input i^i, because of my infinite curiosity. The calculator returned  
e^(-pi/2). I went to your FAQ and read all I could about imaginary 
numbers and e, but it didn't really explain how or why i^i produced 
that. If you could explain, it would be a great help. 

Thanks a lot!
Dan

Date: 12/30/2001 at 04:51:59
From: Doctor Anthony
Subject: Re: Imaginary numbers raised to imaginary numbers

We can write

  i = [cos(pi/2) + i.sin(pi/2)]

    = e^(i.pi/2)

and raising both sides to power i we get

 i^i = [e^(i.pi/2)]^i

     = e^(i^2.pi/2)

     = e^(-pi/2)

     = 1/e^(pi/2)

     = 0.2078795

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/