What is i to the power of i?

When you first learnt that $\boxed{i=\sqrt{-1}}$ , you have entered the mysterious world of complex numbers.

A mystifying question would be to ask, what is i to the power of i? Is it a complex number?

The surprising answer is that $i^i$ is a real number! Let us explain it here:

The key step is to use Euler’s formula: $\boxed{e^{i\pi}=-1}$ . This has been voted as the most beautiful equation in mathematics by many people.

Then, $i=\sqrt{-1}=(-1)^{1/2}={(e^{i\pi})}^{1/2}=e^{i\pi /2}$

Hence, $i^i=e^{i^2\pi /2}=e^{-\pi /2}\approx 0.208$

It is really amazing that an imaginary number to the power of an imaginary number gives a real number, isn’t it? Leave your comments below!

Tag: i to the power of i