i to the power of i

Take a look at this equation, an imaginary number raised to an imaginary power. You have 5 seconds to calculate the results. Most people assume it's impossible or that it results in another imaginary number, but they're wrong.

This is a real number, but to prove it we have to look at what an imaginary number actually is. Now, this is the real number line. It contains everything you've ever counted.

Now, we all know that if you multiply a number by a negative one, it flips its position 180°. But, what if you only want to rotate 90°? Well, there's no real number that does that. So, we define one. We call it I. Multiplying by I rotates you 90° off the real line into the complex plane.

Now, what does it mean to raise a number to a power? Exponential growth is governed by Euler's number, E. When you raise E to an imaginary power, it traces a perfect circle. Euler's formula tells us that E to the power of I * pi completes a half circle, landing on -1. So, to land exactly on I, pointing straight up, we only need a quarter circle. E to the power of I * pi / 2 is exactly equal to I.

Now, we make the substitution. We take I to the power of I and replace the base with our new equation. We get E to the power of I * pi / 2, all raised to the power of I. When you raise a power to a power, you multiply the exponents. I * I is I squared, and I squared is -1. The imaginary component cancels itself out completely. We're left with E to the power of -pi / 2. So, there's no imaginary parts left, just a real number, 0.2078.