Proof that 22 over 7 exceeds π

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The rational number 22/7 is a widely used Diophantine approximation of 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 expansions of these values:

<math>\frac{22}{7} \approx 3.142857\dots\,</math>
<math>\pi \approx 3.141592\dots\,</math>

The approximation has been known since antiquity. Archimedes wrote the first known proof that 22/7 is an overestimate, although he did not necessarily invent the approximation. His proof proceeds by showing that 22/7 is greater than the ratio of the perimeter of a circumscribed regular polygon with 96 sides to the diameter of the circle.

What follows is a different mathematical proof that 22/7 > π. It is short and straightforward, and requires only an elementary understanding of calculus.

The purpose is not to convince the reader that 22/7 is indeed bigger than π. Rather, the great elegance and simplicity of the argument may serve to make the reader suspect that this is the tip of a deeper iceberg of understanding of the theory. Lucas (cited below) calls this proposition "One of the more beautiful results related to approximating π". Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.

Contents

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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>

Therefore 22/7 > π.

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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\,</math> <math><\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>
_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>

Polynomial long division, an important aspect of formulating algebraic geometry, was used between the second and third lines.

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Appearance in the Putnam Competition

The evaluation of this integral was the first problem in the 1968 Putnam Competition. It is easier than most Putnam Competition problems, but the competition often features seemingly obscure problems that turn out to refer to something very familiar.

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Quick upper and lower bounds

In Dalzell (1944) (see References below), it is pointed out that if 1 is substituted for x in the denominator, one gets a lower bound on the integral, and if 0 is substituted for x in the denominator, one gets an upper bound:

<math>{1 \over 1260} < \int_0^1 {x^4 (1-x)^4 \over 1+x^2}\,dx < {1 \over 630}.</math>

Thus we have

<math>{22 \over 7} - {1 \over 630} < \pi < {22 \over 7} - {1 \over 1260}.</math>

Perhaps no other method of calculating π to nearly three decimal places is both so quick and so elementary.

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References

  • D. P. Dalzell, "On 22/7", Journal of the London Mathematical Society 19 (1944) 133–134.
  • D. P. Dalzell, "On 22/7 and 355/113", Eureka; the Archimedeans' Journal, volume 34, pages 10—13, 1971.
  • Stephen Lucas, "Integral proofs that 355/113 > π", Australian Mathematical Society Gazette, volume 32, number 4, pages 263—266. This paper begins by calling this proposition "One of the more beautiful results related to approximating π".
  • {{cite book 
| first = Julian
| last = Havil
| year = 2003
| title = Gamma: Exploring Euler's Constant
| publisher = Princeton University Press
| id = ISBN 0-691-09983-9
| pages = 96

}}

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See also

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External links

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