As is usual when I start learning about a topic I started to fiddle around with the concept. A little bit of manipulation here and there goes a long way. Very soon I stumbled across a mystery which I managed to explain to myself. We all know that

*i*

^{2}= −1

*and*

*So I took this simple problem: i^5 and decided that it could be represented as (i^4)^(5/4). Since i^4 = 1 the expression can be simplified to 1^(5/4). 1^(5/4) = 1.*

*I was mystified. I could use this technique to prove that any power of i is equal to one. Had I discovered some disrepancy, some fallacy in mathematics? Not likely, I thought. So I set out to analyse my results.*

I wish to prove that i^(5) is i and not 1.

*Very soon I found the error in my calculations. I was just beginning to understand how quirky imaginary numbers can be. My mistake lay in assuming that 1^(5/4) = 1. After some thinking, I figured out that it was not. Here's what I did. I reasoned that*

*1^(5/4) = (1^5)^(1/4)*

*= (1^(1/2))^(1/2) Now since 1^(1/2) = +1 or -1*

*= 1^(1/2) or -1^(1/2)*

*= -1 or 1 or i or -i*

*So in the end I concluded that by breaking up the exponent of 5 to 4 and 5/4 I was actually introducing 3 new solutions into the problem. It is similar to a situation in which you have a linear equation and by squaring the equation you introduce new spurious solutions which were never part of the original problem. Since (i^4)^(5/4) is not equal to i^5, i^5 remains equal to i and only i. Q.E.D*

*Interesting isn't it?*