What is i^(Googol)?

Date: 05/27/2000 at 10:06:43
From: Grant Feest
Subject: Googol

What is the value of i raised to the googol power, where i is the 
square root of -1?

Thank you

Date: 05/29/2000 at 15:06:48
From: Doctor TWE
Subject: Re: Googol

Hi Grant - thanks for writing to Dr. Math.

Let's see,

     i^0 =                        1   any x^0 = 1, except x = 0
     i^1 =                        i   any x^1 = x
     i^2 =                       -1   by def; [sqrt(-1)]^2 = -1
     i^3 = i^2*i   = (-1)*i    = -i
     i^4 = i^2*i^2 = (-1)*(-1) =  1
     i^5 = i^4*i   = (1)i      =  i
     i^6 = i^4*i^2 = (1)*(-1)  = -1
     i^7 = i^6*i   = (-1)*i    = -i
     i^8 = i^6*i^2 = (-1)*-1   =  1

Do you see a pattern developing here? Now a googol is 10^100 (a one 
followed by 100 zeros.) Based on the above pattern, can you find a 
shortcut to finding i^(googol)?

I hope this helps. If you have any more questions, write back.

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