Irrational Exponents

Let's kick things up a notch. It's time to introduce irrational exponents. You'll never need another Red Bull again if you can get addicted to these bad boys. What does 2π mean, for example? We can't possibly multiply π copies of 2 together, can we? Even if we could, wouldn't that get awfully messy?

While we can't write out all the digits of π, we can approximate π using decimals:

π ≈ 3 (not a great approximation)
π ≈ 3.1
π ≈ 3.14
π ≈ 3.141 and so on.

We can use these approximations of π to approximate 2π.

2π ≈ 23
2π ≈ 23.1
2π ≈ 23.14
2π ≈ 23.141

Since our approximations of π are rational numbers, we can find 23, 23.1, and so on. We'll never be able to write down all the digits of 2π, in the same way that we'll never be able to write down all the digits of π. Don't beat yourself up about it. Do what mathematicians do. Throw your hands in the air, give up on the thought of arriving at an exact answer, and PART-AY!

Uh, estimate. That last one was supposed to be "estimate."