Saturday, 31 July 2010

An 'i' for an 'i' !!!

I'm learning about complex and imaginary numbers right now. Now, the previous sentence probably throws my less mathematically inclined readers into a state of utter confusion. I'm afraid it just can't be helped because this little mathematical curiosity is so interesting that I find myself unable to delay the moment of revelation even to give a short overview about what complex and imaginary numbers are. If you are very keen on understanding this post I suggest you brush up by clicking here and here.

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
i^{4n} = 1\,
i^{4n+1} = i\,
i^{4n+2} = -1\,
and 
i^{4n+3} = -i.\,


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?
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