Best rational approximations of an irrational number

TL;DR

This paper develops theorems and algorithms for finding the best rational approximations of irrational numbers, utilizing the nearest integer concept, pigeonhole principle, and continued fractions to improve approximation quality.

Contribution

It introduces new theorems and algorithms for optimal rational approximations, enhancing classical methods with the use of the pigeonhole principle and continued fractions.

Findings

Derived improved algorithms for rational approximation

Provided theoretical insights into continued fractions

Enhanced understanding of approximation bounds

Abstract

The concept of nearest integer is used to derive theorems and algorithms for the best approximations of an irrational by rational numbers, which are improved with the pigeonhole principle and used to offer an informed presentation of the theory of continued fractions.