Mathematics and Statistics 10(3): 477-485, 2022 http://www.hrpub.org DOI: 10.13189/ms.2022.100303 On Some Properties of Fabulous Fraction Tree A. Dinesh Kumar 1,* , R. Sivaraman 2 1 Department of Mathematics, Khadir Mohideen College (Affiliated to Bharathidasan University), Adhirampattinam, Tamil Nadu, India 2 Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai, India Received February 28, 2022; Revised March 26, 2022; Accepted April 29, 2022 Cite This Paper in the following Citation Styles (a): [1]A. Dinesh Kumar, R. Sivaraman , "On Some Properties of Fabulous Fraction Tree," Mathematics and Statistics, Vol. 10, No. 3, pp. 477-485, 2022. DOI: 10.13189/ms.2022.100303. (b):A. Dinesh Kumar, R. Sivaraman(2022).On Some Properties of Fabulous Fraction Tree. Mathematics and Statistics, 10(3), 477-485. DOI: 10.13189/ms.2022.100303. Copyright©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Among several properties that real numbers possess, this paper deals with the exciting formation of positive rational numbers constructed in the form of a Tree, in which every number has two branches to the left and right from the root number. This tree possesses all positive rational numbers. Hence it consists of infinite numbers. We call this tree “Fraction Tree”. We will formally introduce the Fraction Tree and discuss several fascinating properties including proving the one-one correspondence between natural numbers and the entries of the Fraction Tree. In this paper, we shall provide the connection between the entries of the fraction tree and Fibonacci numbers through some specified paths. We have also provided ideas relating the terms of the Fraction Tree with that of continued fractions. Five interesting theorems related to the entries of the Fraction Tree are proved in this paper. The simple rule that is used to construct the Fraction Tree enables us to prove many mathematical properties in this paper. In this sense, one can witness the simplicity and beauty of making deep mathematics through simple and elegant formulations. The Fraction Tree discussed in this paper which is technically called Stern-Brocot Tree has profound applications in Science as diverse as in clock manufacturing in the early days. In particular, Brocot used the entries of the Fraction Tree to decide the gear ratios of mechanical clocks used several decades ago. A simple construction rule provides us with a mathematical structure that is worthy of so many properties and applications. This is the real beauty and charm of mathematics. Keywords Fraction Tree, Levels of the Tree, Binary Expansions, One – One Correspondence, Fibonacci Sequence, Continued Fractions 1. Introduction The concept of Fraction Tree was first discovered independently by Moritz Stern in 1858 and AchilleBrocot in 1861. While Stern was a German Number Theorist, Brocot was a French clockmaker. So, the Fraction Tree which we discuss in this paper is known as “Stern-Brocot Tree” in mathematics literature. The Fraction Tree possesses tremendous beauty in the realm of mathematics and was equipped with several unexpected applications both in mathematics and with other branches of science. We will discuss several aspects of this fascinating Fraction Tree and discuss a few of its applications. For knowing more about Fraction Trees see [1–4]. 2. Definition Let a, b> 0 be integers. Then will be a Fraction (either proper fraction or mixed fraction). In fact, such a number will be a positive rational number. Starting with we generate two new numbers and called left child and right child of respectively. (2.1) We can view Figure 1, to understand this creation rule more precisely. a b a b a a b + a b b + a b