[syki]

Rational numbers are algebraic so how are algebraic numbers reasonably far from each other? Algebraic numbers are dense in the real number line.

[pfortuny]

It is a specific statement by Liouville: if you can approximate a number "very well" using rational numbers, then it must be transcendental.

https://mathworld.wolfram.com/LiouvillesApproximationTheorem...

My statement above may be a bit confusing, though.

[syki]

They are using a different notion of “measure” than the standard notion of absolute value of the difference. Under the standard measure every number is within epsilon distance of a rational for any positive epsilon. Thank you for the clarification.

[pfortuny]

Yes, of course. Sorry. It is an asymptotic result, so the meaning of "distance" is very blurry in my statement.

I was replying to the previous comment which seemed to imply that knowledge.

[syki]

I’ve never seen this before so thanks for the links and clarification. I learned something new.