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What is the difference between "any interval" and "close to a given rational"? Rationals are dense, after all.
by Evbn 14y ago
What is the difference between "any interval" and "close to a given rational"? Rationals are dense, after all.
- ColinWright 14y agoThe difference is in the way it's being described, and how that description feeds in to the subsequent definition and proof. The submitted item says: take an interval, here's how to construct a transcendental inside your chosen interval. The proof that gjm11 gives says: Here's a proof that if something is unreasonably close to a rational then it's not algebraic, and here's how to construct something like that. The concept of "unreasonably close" then has to be expressed in terms of sizes of denominators, and hence is a relative term. That forms a key part of the proof.