Root of power i from 1

Another very simple example of complex number problems - the root of power \(i\) from \(1\); \(\sqrt[i]{1} = ?\). For doing this, I will use Euler, or the polar form of complex numbers, and will use the fact that \(1 = e^{i2\pi n}\):

\[\sqrt[i]{1} = 1^{\frac{1}{i}} = 1^{\frac{i}{i\times i}} = 1^{-i} = e^{i\times(0 + 2\pi n)\times(-i)} = e^{2\pi n}\]

Therefore:

\[\sqrt[i]{1} = e^{2\pi n}\]

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Published: 2023-05-10 01:40:51
Updated: 2023-05-10 02:10:18