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Question Corner and Discussion Area

What is i to the Power of i?

Asked by Oliver Varban, student, Earl Haig Secondary School on March 6, 1997:
I am interested in knowing what i to the power of i is.
The first question to address is what it means to raise one complex number to the power of another. There is a basic definition of what it means to raise e to a complex power, as described in the answer to an earlier question. Therefore, if z is any complex number for which  (IMAGE) ,  (IMAGE) is a possible value for  (IMAGE) .

What are the possible values for z? Well, if we write z = a + bi, then  (IMAGE) . By de Moivre's theorem (explained in the answer to an earlier question),  (IMAGE) , so  (IMAGE) . This expression equals i exactly when a=0, cos(b)=0, and sin(b)=1. This occurs when  (IMAGE) for some integer n, so the possible values of z are  (IMAGE) .

Therefore, the values of  (IMAGE) are  (IMAGE)

for any integer n.

Note that there is more than one value for  (IMAGE) , just as 2 and -2 are both square roots of 4. (However, while the square roots of a number always have the same magnitude even if they differ in sign, the values of  (IMAGE) have different magnitudes). The principal value of  (IMAGE) would be  (IMAGE) --the case where n=0.

It's also interesting to note that all these values of  (IMAGE) are real numbers.

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