What is i^i ?

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!

Mandelbrot Set