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Ivan Niven wrote a celebrated elementary proof of the Irrationality of pi. His one-page 1947 article turns a hypothetical rational expression for $\pi$ into an impossible integer integral.^[ref]

A factorial-normalized polynomial has integer-valued endpoint derivatives under the assumption $\pi =a/b$. Repeated integration by parts makes its integral against $\sin x$ an integer. Yet positivity and the factorial denominator make that same integral lie strictly between zero and one for sufficiently large degree. The contradiction demonstrates how carefully designed analytic estimates can establish an arithmetic fact.^[ref]

Ivan Niven's 1947 proof by contradiction constructs an integer strictly between zero and one.^[ref] Suppose $\pi =a/b$ for positive integers $a,b$. For a positive integer $n$, putEvery derivative of $f_n$ at zero is an integer: its coefficients have denominator $n!$, its lowest polynomial degree is $n$, and differentiation contributes the divisible factorial. The symmetry $f_n(\pi -x)=f_n(x)$ gives the same assertion at $\pi$.

Cancellation gives $F_n+F_n^{\prime \prime }=f_n$, henceThe integrand is positive in $(0,\pi )$, but $x(a-bx)=bx(\pi -x)\le b{\pi }^2/4$, soFor large $n$ this is impossible for an integer. Therefore Pi is an irrational number.