## Back to Mth Roots

As mentioned in §3.4, there are different numbers which satisfy when is a positive integer. That is, the th root of , which is written as , is not unique--there are of them. How do we find them all? The answer is to consider*complex*numbers in

*polar form*. By Euler's Identity, which we just proved, any number, real or complex, can be written in polar form as

*e.g.*, . When , we get the same thing as when . When , we get the same thing as when , and so on, so there are only distinct cases. Thus, we may define the th th-root of as

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Roots of Unity

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e^(j theta)