We know Euler's formula $$e^{i \theta} = \cos(\theta) + i \sin(\theta)$$

Let say $$\theta=\frac{\pi}{2}$$

Then we will get $$e^{i \frac{\pi}{2}} = \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})$$

As we can see $$e^{i \frac{\pi}{2}} = i$$
Then we can wright... $$i^i = i^{e^{i \frac{\pi}{2}}}$$ or
$$i^i = (e^{i \frac{\pi}{2}})^i = e^{-\frac{\pi}{2}} \approx 0.208$$

As we can see the result is as Real number. It is not an imaginary number!

Did I compute it correctly?

I think that I made little mistake above instead of $\theta=\frac{\pi}{2}$ I should wrote $\theta=\frac{\pi}{2} +2\pi k $ , $ k\in N\cup {0}$

$$e^{i {(\frac{\pi}{2}+2\pi k})} = i$$
$$i^i = (e^{i (\frac{\pi}{2}+2\pi k)})^i = e^{-\frac{\pi}{2} - 2\pi k}$$
So...
$$k = 0 \implies e^{-\frac{\pi}{2}} \approx 0.208 $$
$$k = 1 \implies e^{-\frac{\pi}{2} - 2\pi} \approx 3.882*10^{-4} $$
$$k = 3 \implies e^{-\frac{\pi}{2} - 6\pi} \approx 1.354*10^{-9} $$

Is it true that: As we can see $ {i}^{i} $ has a lot of answers!?

I mean it has as many answers as many natural number exists!

Are all answers as a real number?