Proof bertrands postulate projection (2 résultats)

Taschenbuch. Etat : Neu. nach der Bestellung gedruckt Neuware - Printed after ordering - High Quality Content by WIKIPEDIA articles! Proofs of the famous mathematical result that the rational number 22 7 is greater than date back to antiquity. What follows is a one-line modern mathematical proof that 22 7 , requiring only elementary techniques from calculus. The purpose is not primarily to convince the reader that 22 7 is indeed bigger than ; systematic methods of computing the value of exist. Unlike some elementary proofs, the calculus-based proof presented here is straightforward; its elegance results from its connections to the theory of diophantine approximations. Stephen Lucas calls this proposition 'One of the more beautiful results related to approximating '. Julian Havil ends a discussion of continued fraction approximations of with the result, describing it as 'impossible to resist mentioning' in that context. If one knows that is approximately 3.14159, then it trivially follows that 22/7. But it takes much less work to show that 22/7 than to show that is approximately 3.14159.…

Taschenbuch. Etat : Neu. nach der Bestellung gedruckt Neuware - Printed after ordering - High Quality Content by WIKIPEDIA articles! In mathematics, Bertrand's postulate states that for each n 2 there is a prime p such that n p 2n. It was first proven by Pafnuty Chebyshev, and a short but advanced proof was given by Srinivasa Ramanujan. The gist of the following elementary but involved proof by contradiction is due to Paul Erd s; the basic idea of the proof is to show that a certain binomial coefficient needs to have a prime factor within the desired interval in order to be large enough. It is then shown by some extended computation that the second fact is inconsistent with the first one. Therefore Bertrand's postulate must hold. In order to present the main argument of the proof intelligibly, these lemmas and a few auxiliary claims are proved separately first.…