If you have been introduced to Complex Numbers, then you know that . Operations on give both real and complex results. For instance, and . But one of the surprising results is the value of .

In this post, we are going to show that is a real number. The proof is credited to Nick Benallo’s blog MathyNick. Nick has permitted me to include this beautiful proof in The Book.

**Theorem**: is a real number.

**Proof**

In the proof, we are going to use the Euler’s Formula. Using the Euler’s Formula,

.

Now, since

, raising both sides by gives us

Since, ,

, we have

This shows that is a real number.

The theorem says i**i=1 and the proof states that i**i=2.0787…

Yes, changed it already.