A simple proof that 22/7 exceeds pi Guide, Meaning , Facts, Information and Description
The rational number 22/7 is a common approximation of the transcendental value π. Its value is greater than π, as can be readily seen in the initial, established decimal places for these values:
22/7 ≈ 3.142857... π ≈ 3.14159...What follows is a mathematical proof of the contention that 22/7 > π. It is called simple because it is short and straightforward, and requires only an introductory-level understanding of calculus. In contrast, much higher levels of calculus are needed to obtain rigorous knowledge of π's value beyond the common approximations that are typically used.
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2 The details 3 Appearance in the Putnam Competition 4 See also |
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:
The evaluation of this integral was once one of the problems set in the 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.The idea
The details
Now we see the difference between 22/7 and π is greater than zero, so 22/7 > π.Appearance in the Putnam Competition