We can’t graph here, this is bat country! *Complex* bat country.

…well ok, let’s stop and take a look anyway. But no graph.

You’ve seen this jewel of mathematics. It’s Euler’s identity.

It comes from the more general expression right below this paragraph, which is today’s Sunday Function. You might wonder where this expression comes from. It’s a long story, but if you want to plow through it, I commend you.

Set θ = π, and Euler’s identity pops right out. But let’s set θ = π/2 instead and see what happens:

In other words, we’ve just figured out a new way to write the imaginary number i:

Now let’s raise that to the *i*th power.

“Wait, what? What does it even mean to raise a complex number to a complex power? Something like 2^{3} is easy, it means 2*2*2 = 8. Fractional and negative exponents are a little weirder since they involve roots and division, but conceptually they’re not that much of a stretch. But an imaginary number to an imaginary power? That’s crazy talk!”

Don’t worry. Just follow the rules and be consistent, and even this problem will fall to your skill. Write i^{i}, but for the lower i use that expression we just derived:

Now invoke one or two high school exponent rules:

And as the last step we can rewrite that term on the right:

And that, dear friends, amazes me even more than Euler’s identity. It is almost too numinous for words.

I can tell that you mathematicians are about to fire up your keyboards and note that my exponential expression for i is not unique due to the fact that I have failed to pin down the possibility of a 2πn phase in the exponent. True, but we can worry about that later. For the moment it’s enough just to be awed by the power of writing a complex number as an exponential.