Proof that 22 over 7 exceeds π

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Proofs of the famous mathematical result that the rational number 22⁄7 is greater than π date back to antiquity.

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:

\frac{22}{7} \approx 3.142857\dots\,
\pi \approx 3.141592\dots\,

The approximation has been known since antiquity. Archimedes wrote the first known proof that 22⁄7 is an overestimate in the 3rd century BCE, 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.

The approximation is known as Yuelü (Traditional Chinese: 約率; Simplified Chinese: 约率; pinyin: yuē lǜ; literally "rough (approximation) ratio")[1] in China, given by Zu Chongzhi together with another approximation Milü (355⁄113) in the 5th century CE. What follows is a different mathematical proof that 22⁄7 > π, requiring only elementary techniques from calculus. The purpose is not primarily to convince the reader that 22⁄7 is indeed bigger than π; systematic methods of computing the value of π exist. Unlike some elementary proofs, it is straightforward;[2] its elegance results from its connections to the theory of diophantine approximations. Lucas calls this proposition "One of the more beautiful results related to approximating π".[3] Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.[4]

Contents

[edit] The basic idea

The basic idea behind the proof can be expressed very succinctly:

0<\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx=\frac{22}{7}-\pi.
Therefore 22⁄7 > π.

[edit] 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 powers of nonnegative real numbers. Since the integrand is positive, the integral from 0 to 1 is positive because 0 < 1.

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

0\, <\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx
=\int_0^1\frac{x^4-4x^5+6x^6-4x^7+x^8}{1+x^2}\,dx (expanded terms in numerator)
=\int_0^1 \left(x^6-4x^5+5x^4-4x^2+4-\frac{4}{1+x^2}\right) \,dx (performed polynomial long division, an important aspect of formulating algebraic geometry)
=\left.\frac{x^7}{7}-\frac{2x^6}{3}+ x^5- \frac{4x^3}{3}+4x-4\arctan{x}\,\right|_0^1 (definite integration)
=\frac{1}{7}-\frac{2}{3}+1-\frac{4}{3}+4-\pi\ (substitute one for x, then zero for x, and subtract them—arctan(1) = π/4)
=\frac{22}{7}-\pi. (addition)

[edit] 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.[citation needed]

[edit] Quick upper and lower bounds

In Dalzell (1944), 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:[5]

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

Thus we have

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

Perhaps no other method of calculating π to nearly three decimal places is both so quick and so elementary. Also see Dalzell (1971)[6].

[edit] References

  1. ^ 韩雪涛. "数学科普：常识性谬误流传令人忧", 中华读书报, 2001-08-29. Retrieved on October 6, 2006.
    Although another name Shulü (Chinese: 疏率; pinyin: shū lǜ) also used, it is suggested that the use of the name was improper.
  2. ^ Contrast Hardy, G. H. and E. M. Wright, chapter 22, on the elementary proof of the prime number theorem.
    (1938). An Introduction to the Theory of Numbers, Oxford University Press, USA; 5 edition (April 17, 1980) ISBN 0198531710.
  3. ^ Lucas, Stephen. "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."
  4. ^ Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press, p. 96. ISBN 0-691-09983-9. 
  5. ^ Dalzell, D. P. (1944). "On 22/7", Journal of the London Mathematical Society 19, pages 133–134.
  6. ^ Dalzell, D. P. (1971). "On 22/7 and 355/113", Eureka; the Archimedeans' Journal, volume 34, pages 10–13.

[edit] See also

[edit] External links

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