Proof that 22/7 exceeds π
|Part of a series of articles on the|
|mathematical constant π|
Proofs of the famous mathematical result that the rational number 22/7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques from calculus, has attracted attention in modern mathematics due to its mathematical elegance and its connections to the theory of diophantine approximations. Stephen Lucas calls this proof “one of the more beautiful results related to approximating π”. Julian Havil ends a discussion of continued fraction approximations of π with the result, describing it as “impossible to resist mentioning” in that context.
The purpose of the proof is not primarily to convince its readers that 22/7 (or 31⁄7) is indeed bigger than π; systematic methods of computing the value of π exist. If one knows that π is approximately 3.14159, then it trivially follows that π < 22/7, which is approximately 3.142857. But it takes much less work to show that π < 22/7 by the method used in this proof than to show that π is approximately 3.14159.
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:
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 may not have been the first to use that 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. Another rational approximation of π that is far more accurate is 355/113.
The proof can be expressed very succinctly:
Therefore, 22/7 > π.
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. This integral has also been used in the entrance examinations for the Indian Institutes of Technology.
Details of evaluation of the integral
That the integral is positive follows from the fact that the integrand is non-negative, being a quotient involving only sums and products of powers of non-negative real numbers. In addition, one can easily check that the integrand is strictly positive for at least one point in the range of integration, say at 1/2. Since the integrand is continuous at that point and non-negative elsewhere, the integral from 0 to 1 must be strictly positive.
It remains to show that the integral in fact evaluates to the desired quantity:
(See polynomial long division.)
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:
Thus we have
Proof that 355/113 exceeds π
where the first six digits after the decimal point agree with those of π. Substituting 1 for x in the denominator, we get the lower bound
substituting 0 for x in the denominator, we get twice this value as an upper bound, hence
In decimal expansion, this means 3.141 592 57 < π < 3.141 592 74, where the bold digits of the lower and upper bound are those of π.
The above ideas can be generalized to get better approximations of π; see also Backhouse (1995) and Lucas (2005) (in both references, however, no calculations are given). For explicit calculations, consider, for every integer n ≥ 1,
where the middle integral evaluates to
where the approximation (the tilde means that the quotient of both sides tends to one for large n) of the central binomial coefficient follows from Stirling's formula and shows the fast convergence of the integrals to π.
The results for n = 1 are given above. For n = 2 we get
hence 3.141 592 31 < π < 3.141 592 89, 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 3.141 592 653 40 < π < 3.141 592 653 87. The next step for n = 4 is
which gives 3.141 592 653 589 55 < π < 3.141 592 653 589 96.
- Approximations of π
- Chronology of computation of π
- Proof that π is irrational
- Lindemann–Weierstrass theorem (proof that π is transcendental)
- List of topics related to π
- Lucas, Stephen (2005), "Integral proofs that 355/113 > π" (PDF), Australian Mathematical Society Gazette, 32 (4): 263–266, MR 2176249, Zbl 1181.11077
- Havil, Julian (2003), Gamma. Exploring Euler's Constant, Princeton, NJ: Princeton University Press, p. 96, ISBN 0-691-09983-9, MR 1968276, Zbl 1023.11001
- Alexanderson, Gerald L.; Klosinski, Leonard F.; Larson, Loren C. (editors) (1985), The William Lowell Putnam Mathematical Competition: Problems and Solutions: 1965–1984, Washington, D.C.: The Mathematical Association of America, ISBN 0-88385-463-5, Zbl 0584.00003
- 2010 IIT Joint Entrance Exam, question 38 on page 15 of the mathematics section.
- Dalzell, D. P. (1944), "On 22/7", Journal of the London Mathematical Society, 19 (75 Part 3): 133–134, MR 0013425, Zbl 0060.15306, doi:10.1112/jlms/19.75_part_3.133.
- Dalzell, D. P. (1971), "On 22/7 and 355/113", Eureka; the Archimedeans' Journal, 34: 10–13, ISSN 0071-2248.
- Backhouse, Nigel (July 1995), "Note 79.36, Pancake functions and approximations to π", The Mathematical Gazette, 79 (485): 371–374, JSTOR 3618318