Frequency of Rational Fractions on [0, 1]

TL;DR

This paper explores the distribution of rational fractions in [0, 1], analyzing Farey sequences and continued fractions, revealing patterns, symmetries, and connections to Fibonacci-Lucas sequences through statistical visualization.

Contribution

It introduces a new statistical approach to visualize and analyze Farey sequences and their relation to continued fractions and Fibonacci-Lucas sequences.

Findings

Confirmed patterns and symmetries in the frequency distribution of rational fractions.

Clarified the internal mechanism of Farey sequence generation.

Revealed connections between statistical patterns and Fibonacci-Lucas sequences.

Abstract

In this paper, the authors design a trial to count rational ratios on the interval [0, 1], and plot a normalized frequency statistical graph. Patterns, symmetry and co-linear properties reflected in the graph are confirmed. The main objective is to present a new view of Farey sequence and to explain the inner principle of its procedure. In addition, we compare Farey sequence and Continued fraction in terms of numerical approximation track and clarify the internal reason why we iteratively choose mediant as the next suitable approximation for the first time. Besides, all sorts of Fibonacci-Lucas sequences emerge from the statistical graph.