What is i^i?

There is a very interesting question: what is the value of $i^i$, where $i$ is the unit imaginary number?

The answer is more interesting: $i^i = 0.2078795763507619...$. It is a REAL NUMBER!

How could we derive it?

First you need to know Euler's formula. Which states

$$
e^{ix} = cos(x) + i*sin(x)
$$

To construct the $i$ is really simple, just set $x = \pi/2$, we have

$$
e^{i*\pi/2} = i
$$

Now comes to the magic part. Raise both sides to power of $i$. The right side is $i^i$. The left side is now

$$
e^{i*i*\pi/2} = e^{-\pi/2} = 0.2078795763507619...
$$

However, this is not the end of the story. Notice $i$ has infinity amount of forms:

$$
e^{i*(2\pi n+\pi/2)} = i, n \in \mathbf{Z}
$$

As a result, $i^i$ has infinity different values:

$$
e^{-2\pi n-\pi/2}, n \in \mathbf{Z}
$$

So $i^i$ is not a number. It represents a set of numbers. And, I can't believe the crazy world any more!

References