Proof that 22/7 exceeds pi
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- The title of this article is incorrect because of technical limitations. The correct title is proof that 22/7 exceeds π.
The rational number 22/7 is a widely used approximation of the transcendental value 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:
- <math>22/7 \approx 3.142857\dots\,<math>
- <math>\pi \approx 3.14159\dots\,<math>
Although many people know this numerical value of π from school, far fewer know how to compute it. 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.
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The idea
- <math>0<\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx=\frac{22}{7}-\pi.<math>
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:
- <math>0<\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx<math>
- <math>=\int_0^1\frac{x^4-4x^5+6x^6-4x^7+x^8}{1+x^2}\,dx<math>
- <math>=\int_0^1 \left(x^6-4x^5+5x^4-4x^2+4-\frac{4}{1+x^2}\right) \,dx<math>
- <math>=\left.\frac{x^7}{7}-\frac{2x^6}{3}+ x^5- \frac{4x^3}{3}+4x-4\arctan{x}\,\right|_0^1<math>
- <math>=\frac{1}{7}-\frac{2}{3}+1-\frac{4}{3}+4-\pi\ <math> (recall that arctan(1) = π/4)
- <math>=\frac{22}{7}-\pi.<math>
Therefore 22/7 > π.
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.
See also
External links
- The problems of the 1968 Putnam competition (http://www.kalva.demon.co.uk/putnam/putn68.html), with this proof listed as question A1.
Categories: Pi | Proofs