Fra [[Shtetl-Optimized]], https://scottaaronson.blog/?p=8506#comments, comment 21: Martin Mertens #14: Greg Kuperberg has given me permission to share the following beautiful argument of his, which is a modification of an argument of Niven, which in turn is a modification of the original argument of Hermite. Let In = integral from 0 to π of xn (π-x)n / n! (sin x) dx Here are some of its values: I5 = 60π4 – 6720π2 + 60480 ~ 0.80 I6 = -2π6 + 1680π4 – 151200π2 + 1330560 ~ 0.31 I7 = -112π6 + 50400π4 – 3991680π2 + 34594560 ~ 0.10 I8 = 2π8 – 5040π6 + 1663200π4 – 121080960π2 + 1037836800 ~ 0.030 On the one hand, In = pn(π) is an integer polynomial in π of degree at most n, in fact π2. You can prove this with iterated integration by parts, which eats the n! denominator before producing any non-zero terms. On the other hand, the values start to fall rapidly. The integrand is positive and unimodal, and the value in the middle is (π/2)2n/n!. Thus, In converges to zero at a superexponential rate, more precisely at a factorial rate. It follows that π (moreover π2) cannot be a rational number a/b, because if pn(x) is any sequence of integer polynomials of degree n with pn(a/b) > 0, then pn(a/b) ≥ 1/bn. I.e., if pn(a/b) converges to zero from above, then the rate of convergence is at most exponential.