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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.
by pfortuny 5y ago
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.html https://mathworld.wolfram.com/LiouvillesApproximationTheorem...
My statement above may be a bit confusing, though.
- syki 5y agoThey 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 5y agoYes, 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 5y agoI’ve never seen this before so thanks for the links and clarification. I learned something new.