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Name: John A.
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Question:
What is the equation that describes the following cyclical behavior of the number "i" ?
i^1 = i
i^2 = -1
i^3 = -i
i^4 = i
and then the cycle repeats,
i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1
and so on ...

Obviously, there must be some sort of equation that deals with this regularity -- and since it has to do with "i" and a regular pattern of exponents, I can only guess that the number "e" must figure in there somewhere.



Replies:
John,

The cyclic property of "i" is not based on e. It is very similar to the cylclic properties of "1" and "-1".

1^1=1
1^2=1
1^3=1
and so on...
Also,
(-1)^1=-1
(-1)^2=1
(-1)^3=-1
(-1)^4=1
and so on...

In all of these cases, the actual SIZE, or magnitude, of the number never increases. There are a variety of other numbers that will do such a thing, but they require trigonometry to express them. What will work is
{ cos(360deg/N)+i*sin(360deg/N) }, where N is any integer.

"1" as a base has a cycle of 1: for N=1, the formula yields 1+0=1. "-1" has a cycle of 2: for N=2, the formula yields -1+0=-1. "i" has a cycle of 4: for N=4, the formula yields 0+i=i. If you want to assign a mathematical quality to the repetivity, it is trigonometric.

Dr. Ken Mellendorf
Illinois Central College


The symbol "i" is used to denote the complex number that is equivalent to the square root of minus one. The number "i" is really not defined in the real number system, but is well defined in an extension of the real number system called complex variables. The term "complex" does not mean "complicated" in this context. The name "complex" is an accident of history. Complex variables just obey different algebraic rules than our usual real numbers.

However, if you associate "i" with the "SQRT(-1)" you can see that "i"^2 = [SQRT(-1)] x [SQRT(-1)] = -1; "i"^3 = "i" x "i"^2 = [ SQRT(-1)] x -1 = -"i"; but note, what you have above is not entirely correct: "i"^4 = "i"^2 x "i"^2 = (-1)x(-1)= +1 Using these relations you can generate all the powers of "i".

The number "e" does play an important role in complex variable theory, but not directly in the problem of finding the integral powers of "i". A very famous relation between the very important algebraic quantities: "e", "pi", "i", and "1" is the remarkable identity: e^(pi*i) = -1.

There is a book about the number "i" you may find interesting: "An Imaginary Tale: The story of "i" " written by Paul J. Nahin.

Vince Calder



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