Proof that 22/7 Exceeds p

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.