When i^n Will Be i, -i, or 1

Date: 9/10/96 at 15:57:29
From: Anonymous
Subject: When i^n will be i, -i, or 1

Raising "i" to a power runs in cycles. i to the first is i, i to the 
second is -1, i to the third is -i, and i to the fourth is 1. And then 
the cycle starts over. I was taught that to find the value of i, no 
matter the exponent, divide the exponent by 4 and the remainder will 
be the equivalent exponent to make finding the value of i easier. 

For example, raise i to the 11th. Now divide 11 by 4, and you get a 
remainder of 3. Now, i to the 3rd (from the remainder) is -i and 
that's your answer. Now, My teacher doesn't always like to explain why 
things like this happen and sometimes gets mad if we do, so I'm asking 
you. Why does this remainder method work? What's the theory behind it? 

Date: 9/10/96 at 19:0:47
From: Doctor Tom
Subject: Re: When i^n will be i, -i, or 1

I think you're not far from seeing the answer.  Instead of stopping
at i^4, let's make a longer list:

i  -1  -i   1   i  -1  -i   1   i  -1  -i   1   i  -1  -i   1 ...

The list above shows i to the first, second, third, ... powers.

Notice that it repeats forever, because once you get back to 1,
multiplying by the next i starts the pattern over again.  So since the 
pattern is 4 steps long, every set of four steps gets you back to 
where you were.  Going 3 steps is the same as going 7 steps or 11 
steps or 15 steps, since each is just 4 more than the previous.

Now think about what the "remainder after dividing by 4" means.  It
means that 4 went in some number of times, and there was some left
over. But every time 4 went in, that's just a complete loop in the 
cycle, so you can ignore it - it brings you back to where you started.  
The only thing that matters is the remainder.

To make it completely clear, look at your example -- i^11.  Well,
11 = 2*4 + 3.  The quotient is 2, and the remainder is 3.  The 
quotient just says you went completely through the cycle twice, and 
then there were 3 left over.  So it's the same as i^3 = -i.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 9/11/96 at 15:7:22
From: Xavier
Subject: Re: When i^n will be i, -i, or 1

Thank you very much Dr. Math! Today, another student asked the teacher 
the same question I asked you and I explained it better then the 
teacher, thanks to your help! 

Ryan