Vasile Berinde

Exploring, Investigating and Discovering in Mathematics

Springer Basel A G

Author: Vasile Berinde Department of Mathematics and Computer Science North University of Baia Mare Str. Victoriei Nr. 76 4800 Baia Mare Romania e-mail: vberinde@ubm.ro

2000 Mathematics Subject Classification: 97050; 97C30, 97U40

A CIP catalogue record for this book is available from the Ubrary of Congress, Washington D.C., USA

Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen NationaJbibliografie; detaillierte bibliograflsche Daten slnd Im Internet ilber <http://dnb.ddb.de> abrufbar.

ISBN 978-3-7643-7019-0 ISBN 978-3-0348-7889-0 (eBook) DOI 10.1007/978-3-0348-7889-0

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ISBN 978-3-7643-7019-0

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In the memory of my father

Preface

We present here the English version of the Romanian first edition (V. Berinde: Ex­plorare, investigare si descoperire in matematica, Editura Efemeride, Baia Mare, 2001). There are no major changes. Only a few printing errors were corrected. When transcribing Romanian names or denominations we did not use the diacrit­ical marks.

Our purpose is to provide an introduction to creative problem solving tech­niques with particular emphasis on how to develop inventive skills in students. We present an array of 24 carefully selected themes that range over all the main chapters in elementary mathematics: arithmetic, algebra, geometry, analysis as well as applied mathematics. Main goal is to offer a systematic illustration of how to organize the natural transition from problem solving activity toward exploring, investigating and discovering new facts and results.

The book is addressed mainly to students, young mathematicians, and teach­ers, involved or/and actively working in mathematics competitions and training gifted people. It collects many valuable techniques for solving various classes of difficult problems and, simultaneously, offers a comprehensive introduction to cre­ating new problems.

The book should also be of interest to anybody who is in any way connected to mathematics or interested in the creative process and in mathematics as an art.

Among the particular features of the book we mention: fully creatively worked-out source problem(s), for each theme; emphasis on teaching natural ex­ploration and investigation skills, starting from a source problem and its solution, that finally lead readers to discover new interesting and challenging problems, that invite further explorations and so on; the exposition is at an elementary level but it opens the way to both elementary and higher mathematics research work.

All 24 themes included in the book contain new information and original results, presented as much as possible in the framework of their natural creation process. The heuristic approach we used, based on generalization, generality and algorithmicity principles, as parts of the whole creative process, is briefly described in the Addendum. All these themes, as a whole, contribute to shaping the readers' inventive behaviour.

viii Preface

At the first level of reading, the book offers a significant number of acces­sible methods for solving difficult (sometimes very difficult) problems. The book assumes only a basic knowledge of elementary mathematics and provides worked solutions to over 100 problems, and hints or clear ideas of solving over 150 problems and exercises, more or less related to the first ones.

At the second level of reading, the book aims to develop creative and primary research skills in readers. In this respect, it can fill a serious gap in mathematics education: after many years of predominantly non-creative problem solving activ­ity, graduate students are suddenly expected to do their first research work - a task for which they have not been properly prepared. In the author's hopes, the present book could be an ideal starting point for learning how to perform such research work.

If desired, this book can be used for a lecture course on Methodology of Mathematical Research / Discovering in Mathematics with little or no further preparation.

I should like to thank my daughters, Madalina and Ruxandra, who worked hard to translate the book into English. I am grateful to Tony Gardiner from the University of Birmingham who commented on early sections.

September 2003 Baia Mare Vasile Berinde

Contents

Preface

Introduction

1 Chase problems 1.1 Source problem . . . . . . . . . . . . . . . . . . . . . . 1.2 A Close Examination of the Problem and its Solution 1.3 First Direction of Investigation . . . 1.4 Exploring the Investigation Results. 1.5 Bibliographical Comments . 1.6 References...............

2 Sequences of Integers Simultaneously Prime 2.1 Source Problems . . . . . . . . . . . . . 2.2 The Solution Algorithm ........ . 2.3 The Particularity of Problems 2b and 2h . 2.4 Conclusions......... 2.5 Bibliographical Comments . 2.6 References ......... .

3 A Geometric Construction Using Ruler and Compass 3.1 Source Problem .................. . 3.2 A Close Examination of the Problem and its Solution 3.3 Other Directions . . . . . . . . . . . 3.4 Consequences. Yet More Directions . 3.5 Exploring the Suggestions . 3.6 Bibliographical Comments . 3.7 References..........

4 Solving a Class of Nonlinear Systems 4.1 Source Problem . . . . . . . 4.2 Solving the Source Problem 4.3 Other Directions . . . . . .

vii

xv

1 1 3 3 5

12 13

15 15 17 22 23 24 24

27 27 28 29 31 32 32 32

35 35 35 41

x

4.4 Bibliographical Comments . 4.5 References ......... .

Contents

45 45

5 A Class of Homogenous Inequalities 47 47 47 48 52 52

5.1 Source Problems ....... . 5.2 Solving the Source Problems ............ . 5.3 A Creative Approach to Problem 5 and Problem 5' . 5.4 Bibliographical Comments . 5.5 References ................. .

6 The First Decimal of Some Irrational Numbers 53 6.1 Source Problem . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 Solving the Source Problem . . . . . . . . . . . . . . . . . . 53 6.3 A Close Examination of the Solution. Exploring Around It 55 6.4 Other Directions of Investigation 58 6.5 Bibliographical Comments . . . . . . . . . . . . 59

7 Polynomial Approximation of Continuous Functions 61 7.1 Source Problem. . . . . . . . . 61 7.2 Exploring the Source Problem. 63 7.3 Bibliographical Comments . 68 7.4 References............ 68

8 On an Interesting Divisibility Problem 69 8.1 Source Problem. . . . . . . 69 8.2 Other Related Problems . . 72 8.3 Bibliographical Comments . 73 8.4 References.......... 73

9 Determinants with Alternate Entries 75 9.1 Source Problem. . . . . . . . . . . . . . . . . . . . . . 75 9.2 A Close Examination of the Problem and its Solution 77 9.3 Other Directions of Investigation 79 9.4 Bibliographical Comments . 81

10 Solving Some Cyclic Systems 10.1 Source Problem ..... 10.2 Solving the Source Problem . . . . . . . . . . . . . . . . . 10.3 A Unitary Method for Solving a Class of Cyclic Systems. 10.4 Other Exploratory Directions 10.5 Bibliographical Comments . 10.6 References . . . . . . . . . . .

83 83 84 86 91 92 92

Contents

11 On a Property of Recurrent Affine Sequences 11.1 Source Problem . . . . . . . . . . . . . . . 11.2 Solution of the Source Problem ..... . 11.3 A Close Examination of the Problem and its Solution 11.4 Recurrences with Many Successive Terms 11.5 Sub-Convex Recurrent Sequences 11.6 The Case of k Successive Terms. 11. 7 Bibliographical Comments . 11.8 References ............ .

12 Binomial Characterizations of Arithmetic Progressions 12.1 Source Problem ...... . 12.2 Bibliographical Comments . 12.3 References . . . . . . . . . .

13 Using Duality in Studying Homographic Recurrences 13.1 Introduction. . . . . . . . . . . . . . . . . . . . . 13.2 Source Problem . . . . . . . . . . . . . . . . . . . 13.3 A Close Examination of the Solution. First Investigations 13.4 Dual Homographical Sequences . . . . . . . . . . 13.5 The Characteristic Equation Has Complex Roots 13.6 Homographic Functional Equations . 13.7 Bibliographical Comments . 13.8 References . . . . . . . . . . . . . . .

14 Exponential Equations Having Exactly Two Solutions 14.1 Source Problem .................. . 14.2 Exploring the Solvability of the General Equation. 14.3 Other Investigation Directions. 14.4 Bibliographical Comments . 14.5 References . . . . . . . . . .

15 A Class of Functional Equations 15.1 Source Problem ...... . 15.2 A Close Examination of the Problem and its Solution . 15.3 Other Functional Equations. Non-Binomial Coefficients 15.4 Other Exploratory Directions. Functional Inequalities 15.5 Bibliographical Comments . 15.6 References . . . . . . . . . . . . . . . . . .

16 An Extension of the Leibniz-Newton Formula 16.1 Source Problem .............. . 16.2 A Close Examination of the Problem and its Solution 16.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .

xi

95 95 95 96 98

100 101 105 105

107 107 113 114

115 115 116 118 120 125 127 128 128

129 129 131 133 135 135

137 137 138 142 145 147 147

149 149 151 154

xii

16.4 Bibliographical Comments. 16.5 References . . . . . .

Contents

155 155

17 A Measurement Problem 157 157 160 160

17.1 Source Problem. Some Exploratory Directions . 17.2 Bibliographical Comments . 17.3 References .................... .

18 A Class of Discontinuous Functions Admitting Primitives 18.1 Source Problem . . . . . . . . . . . . . . . . . . . . .

161 161

18.2 Exploring the Class of Discontinuous Functions Admitting Primitives164 18.3 A Characterization of Functions Admitting Primitives 168 18.4 Weakening the Assumptions . . . . . . . . . . 170 18.5 The Function Has More Discontinuity Points 171 18.6 Bibliographical Comments . 173 18.7 References . . . . . . . . . . . . . . . . . . . . 173

19 On Two Classes of Inequalities 175 19.1 First Source Problem: Telescopic Sums. . . . . . . . . . . . 175 19.2 Second Source Problem: Sequences of the Harmonic Series . 177 19.3 Bibliographical Comments . 181 19.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

20 Another Problem of Geometric Construction 183 20.1 Source Problem. Some Exploratory Directions. 183 20.2 Bibliographical Comments . 187 20.3 References. . . . . . . . . . . . . . . . . . . . . 188

21 How Can We Discover New Problems by Means of the Computer 189 21.1 Introduction. . . . . . . . . . . . . . 189 21.2 First Source Problem. . . . . . . . . . . . . . . . . . . . . 189 21.3 Extending the First Source Problem . . . . . . . . . . . . 190 21.4 Another Type of Problem Obtained Using the Computer. 193 21.5 The Creative Approach .. 194 21.6 Third Source Problem . . . 197 21.7 Bibliographical Comments. 199 21.8 References. . . . . . . . . . 199

22 On the Convergence of Some Sequences of Real Numbers 201 22.1 Source Problem. . . . . . . . . . . . . . . . . . . . . 201 22.2 A Close Examination of the Problem and its Solution 203 22.3 A Necessary and Sufficient Criterion for Convergence. 208 22.4 The Rate of Convergence of Sequences of Real Numbers 210 22.5 Bibliographical Comments . 216 22.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Contents

23 An Application of the Integral Mean 23.1 Source Problem . . . . . . . . 23.2 The Integral Mean at a Point 23.3 Bibliographical Comments . 23.4 References . . . . . . . . . . .

24 Difference and Differential Equations 24.1 Difference Equations of First Order. 24.2 Equations of Order Greater than 1 24.3 Bibliographical Comments . 24.4 References . . . . . . . . . . . . . .

Addendum Basic Principles Regarding Creativity in Solving Problems. Conclusions References. . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

219 219 222 229 229

231 231 236 241 242

243 243 245 245

Introduction

Many years have passed since the idea of writing this book took precise shape, and another five or six years were needed to write it, thus the entire process of "making" it covers almost fifteen years. In spite of this, the book is not a very elab­orate one. My first intention was to publish it in 1995, the year when the journal Cazeta Matematica celebrated a century of existence. But, because of overwhelm­ing administrative duties at that time (duties that consumed considerable time and intellectual energy), I had to interrupt my work for many years. I only began to work again at this book in 2000, The World Mathematics Year. I do not con­sider that the book is finished now as it was initially planned, but this is the best written version to date.

How did the idea of writing this book evolve?

It was simply a corollary of my activity as a problem proposer for mathe­matics journals and contests. Now I have to mention that I only sporadically dealt with this part of elementary mathematics, as an immense nostalgia for the times of my training as a problem solver of Cazeta Matematica. An important role in favoring elementary mathematics was also played by Lucrarile Seminarului de Creativitate Matematica, a journal that was founded in 1991 exactly with the aim of emphasizing the creative approach in practicing mathematics.

What Does the Book Deal with?

As the title suggests, the book is intended to be an introduction to mathe­matical research by means of some elementary themes. I have been asked many times, by pupils, students and high school teachers, if there is anything left to be discovered in mathematics, which they took for a completely formed field, with no other development possibilities.

Well, this book is a wide and insistent affirmative answer to this question. Its aim is to offer a concrete and as exact as possible description of the act of discovery in mathematics.

xvi Introduction

Using examples which I consider appropriate and that were also close to my own interests, I tried to show the nature of a creative approach to any mathematics problem. Thus I have tried to capture the attitude, the "mind-set", of a researcher contemplating a problem, whether that problem has been chosent a priori or has arisen in the context of a related question. This seems to me the natural behavior of any researcher, whatever his/her field of activity might be.

When facing a problem (in mathematics, technology, arts, etc.) one has es­sentially the same attitude as assumed in any creative process. First of all one must know the problem thoroughly, then probe it from all possible angles, looking for its essential features, its deepest meanings, and the reasons for its behavior. Then one must develop conclusions from which to push the evaluation and investigation of these aspects as far as possible towards not only a solution but a connection with the wider theme.

The steps of this creative approach are inherent in the very title of the book: "Exploring, Investigating and Discovering in Mathematics". We first explore the inner structure, the "deposit of precious metals" of a certain problem, then, once the "mother lode" has been found, we investigate all, or as many as possible, suggested directions; if our creative approach has been well conceived, at the end of these efforts the result, the "precious metal" , will certainly appear - and that is the discovery.

This same creative approach can be applied to higher mathematics or to any field of scientific research. To demonstrate this point without decreasing the accessibility of the book, some problems in Sections 23 and 24 go beyond the elementary level.

To Whom is This Book Addressed?

This book was written so that it can be read (at least in some of its parts) by anyone with an interest in and some study of mathematics. But of course it is mainly addressed to those who practise mathematics as a scholarly subject: students, teachers and researchers.

As already mentioned, it is first of all a guide for those who are starting their scientific research and need to develop a precise working method. Reading and thinking about one or another of the themes in the book might well lead a reader's first steps in mathematical investigation, showing how a problem is born and gives birth to other problems - demonstrating how one can always push the level of his/her knowledge, no matter how small and limited the original research theme might have been.

The 24 themes included in this book touch upon almost all elementary math­ematics, starting with arithmetic and continuing through algebra, geometry, cal­culus and even applied mathematics. They reflect not only the authors personal tastes and professional interests but his commitment to elementary mathematics - a tribute to his training as a problem solver with the outstanding monthly Gazeta Matematica.

Introduction xvii

What is the Structure of the Book?

The book comprises 24 chapters and an Addendum that tries to synthesize the main principles that have guided me in developing my own problem solving method. Each chapter deals with a specific "theme". Each theme is treated with a number of variations, similar to the techniques of musical composition, or perhaps to the action of a spider, weaving a web of reflections in an infinite set of mirrors. Starting with a known source, a chain of variations, or of reflections, creates a new image, often at least as valuable as the one that was the starting point. In science as in art, the creative principles are essentially the same.

In its turn, each theme (chapter) has a more or less fixed structure, depending on the material and on the imagination and inspiration of the investigator. The following arrangement of the material can be found in almost every theme:

1. The text of the source problem (the starting problem)

2. The solution (or solutions) of the source problem, highlighting the creative principles that this solution (solutions) are based on:

(a) algorithmicity;

(b) generality (of the method);

( c ) generalization (of the data and requirements);

3. Remarks regarding the essence of:

(a) the problem;

(b) the given solution;

and the outline of the first exploratory directions;

4. The investigation of the directions outlined in 3, considering:

(a) the generality principle;

(b) the generalization principle;

5. Finalizing the explorations, systemizing the conclusions of all effectively fi­nalized investigations and outlining the results by:

(a) reformulating the problem;

(b) formulating new problems (which are already solved);

(c) possibly building a "factory" of new problems;

(d) setting up a list of related problems (which the reader is invited to solve);

(e) suggesting other possible investigation directions;

xviii Introduction

6. Bibliographical comments about the source problem, similar or related prob­lems, books or articles where the general theory, on which the problem is based, is grounded, and suggested further reading.

We do not pretend that our references are complete. On the contrary, we restricted them to the Romanian mathematics journals to illustrate how cer­tain themes developed exclusively within our own mathematics literature: obvious treatments of some of the themes can be found in the international literature as well.

How original is the book? What are its merits?

The author found a great joy of the spirit in writing this book, the sensation of expressing the very core of his ideas about mathematics and research work. As no research is exclusively the domain of an individual, it is easy to see how original such a book can be. We claim originality for our book mainly through the author's conception of solving problems, a conception that is illustrated through some themes that have been of interest to him for varying periods of time. There are other publications dealing in a similar manner with elementary mathematics problems (the best known of which is G. Polya's book Discovery in Mathematics), still we were guided here by our own method, presented in the Addendum. How original and useful it is, is left to the reader to decide.

The goal of originality also motivated us to avoid the use of some common terminology, e.g., "the heuristic method", etc. Instead we preferred to use, when possible, concepts that are much closer to the creative act in mathematics as we see it, and that could be summarized as follows: never when we finish writing down the solution of a mathematics problem should we be tempted to state that the problem is now completely solved. No way! "Exhaustive" is a term that every mathemati­cian should avoid. As there is always something new to be said related to a solved problem, new approach angles, new investigation and exploratory directions can appear at any moment.

There is still to be said that a lot of the results and "discoveries" of this book belong equally to other mathematicians, and strictly limiting or deciding upon the priority of one or another was not the intention of the author. Still, in most the cases the presentation itself will suggest the evolution of ideas on that theme.

And finally I wrote this book hoping that it would express one of my funda­mental beliefs regarding the human spirit.

I strongly believe in the existence of some latent creative resources in each of us - sparks of our creator and reflections of the divine act of creation. I would even dare to say that these latent creative resources pre-exist in the human being. An adequate methodological framework, and especially the tools specific to creation, are necessary to awaken them. No matter ifhe is an artist, a poet, a mathematician, an engineer, a physicist, a chemist, a tailor, a cook or a peasant, the one who seeks the way to creation, learns how to use these tools and finds the secret of creation will attain spiritual self-fulfillment.

Introduction xix

Succeeding in a profession, whatever that profession might be, depends mainly on finding the tools necessary for revealing this creative part of ourselves. Blessed are those who have come to this realization, obtained the appropriate tools, learned their functions, and take the time to use them intensely for creation!

Vasile Berinde

Donetsk, Ukraine 29th of May - 1st of June 2001