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Rational numbers are algebraic so how are algebraic numbers reasonably far from each other? Algebraic numbers are dense in the real number line.
by syki 5y ago
Rational 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.