Frequency of Rational Fractions on [0, 1]

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.