Mathematics and Statistics Vol. 10(3), pp. 477 - 485
DOI: 10.13189/ms.2022.100303
Reprint (PDF) (404Kb)

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

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

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[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). APA Format:
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.