Ford circles, continued fractions, and best approximation of the second kind

TL;DR

This paper provides an elementary geometric proof using Ford circles to show that the convergents of a continued fraction are exactly the best approximations of the second kind for a real number.

Contribution

It introduces a geometric proof connecting Ford circles with continued fractions and best approximations, offering a new perspective on classical number theory results.

Findings

Convergents of continued fractions are best approximations of the second kind.

Elementary geometric proof using Ford circles.

Clarifies the relationship between continued fractions and approximation quality.

Abstract

We give an elementary geometric proof using Ford circles that the convergents of the continued fraction expansion of a real number $\alpha$ coincide with the rationals that are best approximations of the second kind of $\alpha$.