A Simple Proof That Pi Is Rational
Fermat's Library | A simple proof that π is irrational annotated/explained version.
A SIMPLE PROOF THAT TT IS IRRATIONAL IVAN NIVEN Let 7T = a/6, the quotient of positive integers. We define the poly-nomials xn(a — bx)n F(x) = ƒ<» - fW(x) +/(4)0) + (- l)n/(2n)0)> the positive integer n being specified later. Since nlf(x) has integral coefficients and terms in x of degree not less than n, f(x) and its derivatives ƒ{j)(x) have integral values for # = 0; also for x*=ir~a/b, since ƒ(x) =f(a/b—x). By elementary calculus we have d — \F'(x) sin x — F(x) cos x\ = F"(x) sin x + F(x) sin x = f(x) sin # dx and (1) ! ƒ(» sin xdx = [F'(a) sin x - i?(» cos #]0* = F(v) + F(0). J o Now F(TT)-{-F(fi) is aninteger, since ƒ (#(?r) and/(j,)(0) are integers. But for 0<x<7T, 7rnan 0 < ƒ(#) sin x < j n\ so that the integral in (1) is positive, but arbitrarily small for n suffi-ciently large. Thus (1) is false, and so is our assumption that TT is rational. PURDUE UNIVERSITY Received by the editors November 26, 1946, and, in revised form, December 20, 1946. 509 