My Coding > Mathematics > Complex numbers > One in power i

One in power i

How to calculate one in power i, or \(1^i\)? The answer is not that simple as you think!

We know that any complex number can be expressed in polar form as \(re^{i\theta}\), where \(r\) is the magnitude or modulus of the complex number, and \(\theta\) is the argument or angle between the positive real axis and the complex number in the complex plane. Therefore, we can write \(1\) as \(1e^{i(0 + 2\pi n)}\).

Now, let's raise \(1\) to the power of \(i\), i.e. \(1^i\). We can express this in polar form as \((1e^{i(0 + 2\pi n)})^i\).

\[1^i = (1e^{i(0 + 2\pi k)})^i = e^{0i^2 + 2\pi k i^2} = e^{−2\pi k}; k = 0, \pm1, \pm2, ...\]

Because the \(k\) value can be positive and negative, we can say, that:

\[e^{−2\pi k} \equiv e^{2\pi k} \]

This is the final ansver:

\[1^i = e^{2\pi k}; k = 0, \pm1, \pm2, ...\]

How to check that this is correct, please watch our video with explanation:


Published: 2023-05-09 00:14:20
Updated: 2023-05-09 00:22:51

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