# i to the i-th power

Ever wondered what the imaginary unit i raised to itself is equal to? Is it going to be a complex number? Or a real one? Let’s find out!

$\dpi{100}&space;\large&space;i^i=?$

Let’s first define i:

$\dpi{100}&space;\large&space;i=\sqrt{-1}$

Therefore,

$\dpi{100}&space;\large&space;i^i=\left&space;(\sqrt{-1}&space;\right&space;)^i=\left&space;(&space;-1&space;\right&space;)^{\frac{1}{2}i}$

There is an identity in maths which is in my opinion one of the most beautiful equations that exist, and that is Euler’s identity:

$\dpi{100}&space;\large&space;e^{i\pi}+1=0$

This means

$\dpi{100}&space;\large&space;e^{i\pi}=-1$

At this point we have two expressions:

$\dpi{100}&space;\large&space;i^i=\left&space;(&space;-1&space;\right&space;)^{\frac{1}{2}i}$ $\dpi{100}&space;\large&space;e^{i\pi}=-1$

We can use the second one to replace the -1 in the first:

$\dpi{100}&space;\large&space;i^i=\left&space;(&space;e^{i\pi}\right&space;)^{\frac{1}{2}i}=e^{i\pi\cdot&space;\frac{1}{2}i}=e^{-\frac{\pi}{2}}$

We no longer have the i because i times i equals -1.

Finally,

$\dpi{100}&space;\large&space;i^i=e^{-\frac{\pi}{2}}$

Isn’t this beautiful? This is a real number! If you use a calculator you will find that this is equal to 0.2078795… and what’s cool is that we have an irrational number raised to another irrational number!

Now you should be able to find the imaginary square root if i. I will talk about that in another post, but if you want you can leave the answer in the comments!

$\dpi{100}&space;\large&space;\sqrt[i]{i}=?$

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