Dimostrazione che 22/7 è maggiore di π: differenze tra le versioni

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The [[rational number]] '''22/7''' is a widely used [[approximation]] of '''[[pi|π]]'''. It is a convergent in the simple [[continued fraction]] expansion of π. It is greater than π, as can be readily seen in the decimal expansion of these values:
Il [[numero razionale]] '''22/7''' è ampiamente usato cpme [[approssimazione]] di '''[[pi greco|π]]'''. Esso è una convergenza della semplice espansione in [[frazione continua]] di π. È maggiore di π,come si può notare dalla espasione decimale:



:<math>\frac{22}{7} \approx 3.142857\dots\,</math>
:<math>\frac{22}{7} \approx 3.142857\dots\,</math>


:<math>\pi \approx 3.141592\dots\,</math>
:<math>\pi \approx 3.141592\dots\,</math>



Although many people know this numerical value of &pi; from school, far fewer know how it is computed. What follows is a [[mathematical proof]] that 22/7 > &pi;. It is ''simple'' in that it is short and straightforward, and requires only an elementary understanding of [[calculus]].
Although many people know this numerical value of &pi; from school, far fewer know how it is computed. What follows is a [[mathematical proof]] that 22/7 > &pi;. It is ''simple'' in that it is short and straightforward, and requires only an elementary understanding of [[calculus]].

Versione delle 13:25, 16 feb 2006

Il numero razionale 22/7 è ampiamente usato cpme approssimazione di π. Esso è una convergenza della semplice espansione in frazione continua di π. È maggiore di π,come si può notare dalla espasione decimale:



Although many people know this numerical value of π from school, far fewer know how it is computed. What follows is a mathematical proof that 22/7 > π. It is simple in that it is short and straightforward, and requires only an elementary understanding of calculus.

The idea

Therefore 22/7 > π.

The details

That the integral is positive follows from the fact that the integrand is a quotient whose numerator and denominator are both nonnegative, being sums or products of even powers of real numbers. So the integral from 0 to 1 is positive.

It remains to show that the integral in fact evaluates to the desired quantity:

(recall that arctan(1) = π/4)

Appearance in the Putnam Competition

The evaluation of this integral was the first problem in the 1968 Putnam Competition. If it seems trivially routine for a Putnam Competition problem, one may perhaps surmise that its inclusion was motivated by the conjunction of the punch line (summarized by the title of this article) with the fairly nice pattern in the integral itself.

Many years earlier, the result was given in D. P. Dalzell, On 22/7, Journal of the London Mathematical Society 19 (1944) 133-134.

See also

External links