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Birkhäuser

Exploring, Investigating and Discovering in Mathematics

  • Textbook
  • © 2004

Overview

Authors:
  1. Vasile Berinde

    1. Department of Mathematics and Computer Science, North University of Baia Mare, Baia Mare, Romania

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  • Uses competition-like problem solving as a platform for training typical inventive skills
  • Aims to open the way from the problem solving process to the very beginning of a research attitude
  • Assumes only a basic knowledge of elementary mathematics
  • Provides solutions to over 100 problems and includes hints or clear ideas of solving for over 150 further problems and exercises

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Table of contents (25 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. Chase problems

    • Vasile Berinde
    Pages 1-13

  3. Sequences of Integers Simultaneously Prime

    • Vasile Berinde
    Pages 15-25

  4. A Geometric Construction Using Ruler and Compass

    • Vasile Berinde
    Pages 27-33

  5. Solving a Class of Nonlinear Systems

    • Vasile Berinde
    Pages 35-45

  6. A Class of Homogenous Inequalities

    • Vasile Berinde
    Pages 47-52

  7. The First Decimal of Some Irrational Numbers

    • Vasile Berinde
    Pages 53-59

  8. Some Elementary Problems Regarding the Polynomial Approximation of Continuous Functions

    • Vasile Berinde
    Pages 61-68

  9. On an Interesting Divisibility Problem

    • Vasile Berinde
    Pages 69-73

  10. Determinants with Alternate Entries

    • Vasile Berinde
    Pages 75-82

  11. Solving Some Cyclic Systems

    • Vasile Berinde
    Pages 83-93

  12. On a Property of Recurrent Affine Sequences

    • Vasile Berinde
    Pages 95-106

  13. Binomial Characterizations of Arithmetic Progressions

    • Vasile Berinde
    Pages 107-114

  14. Using Duality in Studying Homographic Recurrences

    • Vasile Berinde
    Pages 115-128

  15. Exponential Equations Having Exactly Two Solutions

    • Vasile Berinde
    Pages 129-135

  16. A Class of Functional Equations

    • Vasile Berinde
    Pages 137-147

  17. An Extension of the Leibniz-Newton Formula

    • Vasile Berinde
    Pages 149-156

  18. A Measurement Problem

    • Vasile Berinde
    Pages 157-160

  19. A Class of Discontinuous Functions Admitting Primitives

    • Vasile Berinde
    Pages 161-174

  20. On Two Classes of Inequalities

    • Vasile Berinde
    Pages 175-181
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Keywords

About this book

We present here the English version of the Romanian first edition (V. Berinde: Ex­ plorare, 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 diacrit­ ical marks. Our purpose is to provide an introduction to creative problem solving tech­ niques 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 teach­ ers, 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 cre­ ating 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.

Reviews

"In the spirit of George Polya's classic treatise How to Solve It: A New Aspect of Mathematical Method, this book is Vasile Berinde's attempt to "algorithmetize" the creative process involved in problem-solving.... The author's stated goals are twofold. The most obvious aim is to provide methods for solving some difficult problems. But on a deeper level, he is trying to grow new research mathematicians by planting the seeds that will develop into the skills they will need to do original research.... Each of the 24 chapters begins with a source or starting problem and its solution. Then the fun begins! Remarks about the "essence" of the problem and its solution suggest new directions to explore, and these investigations lead to generalizations and the formulation of related problems, sometimes building a "factory" of new problems.

From the first page I eagerly grabbed my pencil and a stack of scratch paper, and set to work as I happily read along.... I recommend the book for all lovers of mathematics, but especially students and teachers who participate in mathematics contests and practice problem solving."

—MAA Online

Authors and Affiliations

  • Department of Mathematics and Computer Science, North University of Baia Mare, Baia Mare, Romania

    Vasile Berinde

Bibliographic Information

  • Book Title: Exploring, Investigating and Discovering in Mathematics

  • Authors: Vasile Berinde

  • DOI: https://doi.org/10.1007/978-3-0348-7889-0

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 2004

  • Softcover ISBN: 978-3-7643-7019-0Published: 17 December 2003

  • eBook ISBN: 978-3-0348-7889-0Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: XIX, 246

  • Number of Illustrations: 2 b/w illustrations

  • Topics: Algebra, Analysis, Mathematics, general, Geometry

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