A proof of irrationality of π based on nested radicals with roots of 2

2025/11/26 by Abrarov, Sanjar M., Siddiqui, Rehan, Jagpal, Rajinder Kumar +1
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2511.21776

Abstract

In this work, we consider three theorems and a lemma to prove the irrationality of π. This approach is based on nested radicals with roots of 2 of kind c_k = √2 + c_{k - 1} and c₀ = 0. Sample computations showing how the rational approximation tends to π with increasing the integer k are presented.

Citations

• Application of a new iterative formula for computing π and nested radicals with roots of 2
• A well-motivated proof that pi is irrational
• How Archimedes showed that π is approximately equal to 22/7
• Error estimates for the Gregory-Leibniz series and the alternating harmonic series using Dalzell integrals

Related

• Reducing radicals in the spirit of Euclid
• Irrationality Measure of Pi
• A Proof of Irrationality of π by Contraposition
• Application of a new iterative formula for computing π and nested radicals with roots of 2
• A direct proof of the irrationality of tan²(r π)