Irrational and transcendental numbers are everywhere. Most numbers aren’t rational. Most numbers aren’t even algebraic. That’s a very strange notion: we can’t write most numbers down.
Even stranger, even though we know, per Cantor, that most numbers are transcendental, it’s incredibly difficult to prove that any particular number is transcendental. Most of them are, but we can’t even figure out which ones!
What does that mean? That our math-fu isn’t nearly as strong as we like to believe. Most numbers are beyond us. Here are some interesting numbers that we know are either irrational or transcendental:
The square root of 2: irrational, but algebraic