Revisiting Fibonacci Numbers through a Computational Experiment

Sergei Abramovich
School of Education and Professional Studies, State University of New York at Potsdam, NY, US

Gennady A. Leonov
Faculty of Mathematics and Mechanics, St Petersburg State University; Saint-Petersburg, Russian Federation

Series: Education in a Competitive and Globalizing World
BISAC: MAT021000

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The material of this book stems from the idea of integrating a classic concept of Fibonacci numbers with commonly available digital tools including a computer spreadsheet, Maple, Wolfram Alpha, and the graphing calculator. This integration made it possible to introduce a number of new concepts such as: Generalized golden ratios in the form of cycles represented by the strings of real numbers; Fibonacci-like polynomials the roots of which define those cycles’ dependence on a parameter; the directions of the cycles described in combinatorial terms of permutations with rises, as the parameter changes on the number line; Fibonacci sieves of order k; (r, k)-sections of Fibonacci numbers; and polynomial generalizations of Cassini’s, Catalan’s, and other identities for Fibonacci numbers.

The development of these concepts was motivated by considering the difference equation f_(n+1)=af_n+bf_(n-1),f_0=f_1=1, and, by taking advantage of capabilities of the modern-day digital tools, exploring the behavior of the ratios f_(n+1)/f_n as n increases. The initial use of a spreadsheet can demonstrate that, depending on the values of a and b, the ratios can either be attracted by a number (known as the Golden Ratio in the case a = b = 1) or by the strings of numbers (cycles) of different lengths. In general, difference equations, both linear and non-linear ones serve as mathematical models in radio engineering, communication, and computer architecture research. In mathematics education, commonly available digital tools enable the introduction of mathematical complexity of the behavior of these models to different groups of students through the modern-day combination of argument and computation.

The book promotes experimental mathematics techniques which, in the digital age, integrate intuition, insight, the development of mathematical models, conjecturing, and various ways of justification of conjectures. The notion of technology-immune/technology-enabled problem solving is introduced as an educational analogue of the notion of experimental mathematics. In the spirit of John Dewey, the book provides many collateral learning opportunities enabled by experimental mathematics techniques. Likewise, in the spirit of George Pólya, the book champions carrying out computer experimentation with mathematical concepts before offering their formal demonstration.
The book can be used in secondary mathematics teacher education programs, in undergraduate mathematics courses for students majoring in mathematics, computer science, electrical and mechanical engineering, as well as in other mathematical programs that study difference equations in the broad context of discrete mathematics.
(Imprint: Nova)

Preface

Chapter 1. Theoretical Background: Fibonacci Numbers as a Framework for Information vs. Explanaition Cognitive Paradigm

Chapter 2. From Fibonacci Numbers to Fibonacci-Like Polynomials

Chapter 3. Different Approaches to the Development of Binet’s Formulas

Chapter 4. Fibonacci Sieves and Their Representation through Difference Equations

Chapter 5. Tite Explorations of Generalized Golden Ratios

Chapter 6. Exploring Cycles Using a Combination of Digital Tools

Chapter 7. Method of Iterations and Fibonacci-Like Polynomials

Chapter 8. Identities for Fibonacci-Like Polynomials

Chapter 9. Uncovering Hidden Patterns in the Oscillations of Generalized Golden Ratios

References

"One of the most fascinating aspects of this book is the discovery of cycles formed by the orbits of a very simple non-linear recursive equation. An experimental nature of some results formulated by Abramovich and Leonov can serve as a motivation for rigorous mathematics research." - Nikolay Kuznetsov, Saint-Petersburg State University, Russia

"While there is an abundance of popular and scholarly literature on Fibonacci numbers, the book offers a markedly new perspective on this classic concept made possible by the use of modern computational tools. The authors did their best in demonstrating an educational side of experimental mathematics." - Pekka Neittaanmäki, University of Jyväskylä, Finland

"Abramovich and Leonov have written a uniquely valuable book for secondary teachers and students. They begin with the simple sequence 1, 1, 2, 3, … and expand it into a universe of ever more sophisticated and fascinating mathematical ideas. A remarkable feat." - Pat Thompson, Arizona State University, USA

"As a mathematics educator, I really appreciate the creative work of Gennady Leonov and Sergei Abramovich. It stimulates the design of mathematically substantial learning environments about Fibonacci numbers across the whole curriculum." - Erich Ch. Wittmann, Technical University of Dortmund, Germany

"Abramovich & Leonov’s book aims to explore Fibonacci numbers through the tools of the digital age. These range from graphing calculators to computer algebra packages enabling students from secondary mathematics to undergraduate level over a range of science and engineering disciplines to appreciate some of their properties and possibly discover some ‘new’ concepts." Martin C. Harrison, Loughborough University, United Kingdom

Keywords: Fibonacci numbers, experimental mathematics, computational experiment, cycles, generalized golden ratios, spreadsheets, Maple, Wolfram Alpha

The material presented in this book can be used in secondary mathematics teacher education programs, undergraduate mathematics courses for students majoring in mathematics, computer science, electrical and mechanical engineering, as well as in other undergraduate mathematics programs that study difference equations in the broad context of discrete mathematics.

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