A Simple Proof that π is Irrational

Abstract

Let π=a/b, the quotient of positive integers. We define the polynomials

$$\begin{gathered} f(x) = \frac{{{x^n}{{(a - bx)}^n}}}{{n!}} \hfill \\ F(x) = f(x) - {f^{(2)}}(x) + {f^{(4)}} - ... + {( - 1)^n}{f^{(2n)}}(x) \hfill \\ \end{gathered} $$

the positive integer n being specified later. Since n!f(x)has integral coefficients and terms in x of degree not less than n, f(x) and its derivatives f ⁽ⁱ⁾(x) have integral values for x=0; also for x=π=a/b, since f(x) =f(a/b–x). By elementary calculus we have

$$\frac{d}{{dx}}\{ F'(x)\sin x - F(x)\cos x\} = F''(x)\sin x + F(x)\sin x = f(x)\sin x$$

and

$$\int_0^\pi {f(x)\sin xdx} = \left[ {F'(x)\sin x - F(x)\cos x} \right]_0^\pi F(x) + F(0)$$