A proof of irrationality of $\pi$ based on the nested radicals with roots of $2$

TL;DR

This paper proves the irrationality of π using nested radicals with roots of 2, providing sample computations of rational approximations approaching π as the radical depth increases.

Contribution

It introduces a novel proof of π's irrationality based on nested radicals with roots of 2, which is a new approach compared to classical proofs.

Findings

Nested radicals with roots of 2 approximate π increasingly closely.

The proof establishes the irrationality of π through these nested radical structures.

Sample computations demonstrate the convergence of rational approximations to π.

Abstract

In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing the integer $k$ are presented.