4 ms·
Exactly: algebraic numbers, despite not being periodic, are in general "reasonably far from each other", and especially from rationals. I guess the problem can
by pfortuny 5y ago
Exactly: algebraic numbers, despite not being periodic, are in general "reasonably far from each other", and especially from rationals.
I guess the problem can be solved using what you say, certainly.
It is only transcendentals that can be "too near" each other, and near rationals (this is Liouville's result, which was improved later on, in a specific case the one you say).
- syki 5y agoRational numbers are algebraic so how are algebraic numbers reasonably far from each other? Algebraic numbers are dense in the real number line.
- pfortuny 5y agoIt 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.