Proofs of the famous mathematical result that the rational number22⁄7 is greater than π date back to antiquity. What follows is a modern 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, the calculus-based proof presented here is straightforward;[1] 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 π".[2] Julian Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.[3]
The approximation has been known since antiquity. Archimedes wrote the first known proof that 22⁄7 is an overestimate in the 3rd century BC, 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 circumscribedregular polygon with 96 sides to the diameter of the circle. A more accurate approximation of π is 355/113.
The basic idea
The basic idea behind the proof can be expressed very succinctly:
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 powers of nonnegative real numbers. Since the integrand is positive, the integral from 0 to 1 is positive because the lower limit of integration is less than the upper limit of integration (0 < 1).
It remains to show that the integral in fact evaluates to the desired quantity:
(substitute one for x, then zero for x, and subtract them, use arctan(1) = π/4)
(addition)
Appearance in the Putnam Competition
The evaluation of this integral was the first problem in the 1968Putnam Competition.[4] 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.
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]
The above ideas can be generalized to get better approximations of π. More precisely, for every integer n ≥ 1,
where the middle integral evaluates to
involving π. The last sum also appears in Leibniz' formula for π. The correction term and error bound is given by
where the approximation (the tilde means that the quotient of both sides tends to one for large n) follows from Stirling's formula and shows the fast convergence of the integrals to π.
Calculation of these integrals
For all integers k ≥ 0 and ℓ ≥ 2 we have
Applying this formula recursively 2n times yields
Furthermore,
where the first equality holds, because the terms for 1 ≤ j ≤ 3n − 1 cancel, and the second equality arises from the index shift j → j + 1 in the first sum.
Integrating (*) from 0 to 1 using (**) and arctan(1) = π/4, we get the claimed equation involving π.
The results for n = 1 are given above. For n = 2 we get
and
hence 3.14159231 < π < 3.14159289, where the bold digits of the lower and upper bound are those of π. Similarly for n = 3,
with correction term and error bound
hence 3.14159265340 < π < 3.14159265387. The next step for n = 4 is
with
which gives 3.14159265358955 < π < 3.14159265358996.
References
^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.
^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."
^Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. pp. p. 96. ISBN 0-691-09983-9. {{cite book}}: |pages= has extra text (help)