 Proof that π is irrational

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.

The proof that π is irrational was first established by Johann Lambert in 1768. His proof is somewhat complex and relies on properties of continued fractions, but I can provide a high-level overview of the concepts involved in proving the irrationality of π. ### Overview of Lambert's Proof 1. **Definitions**: A number is irrational if it cannot be expressed as a fraction of two integers.