Translated by Robert G. Burns
Translated by Robert G. Burns

Book DetailsAnneli Lax New Mathematical LibraryVolume: 47; 2017; 304 pp
Reprinted edition available: NML/52
Portal through Mathematics is a collection of puzzles and problems mostly on topics relating to secondary mathematics. The problems and topics are fresh and interesting and frequently surprising. One example: the puzzle that asks how much length must be added to a belt around the Earth's equator to raise it one foot has probably achieved old chestnut status. Ivanov, after explaining the surprising answer to this question, goes a step further and asks, if you grabbed that too long belt at some point and raised it as high as possible, how high would that be? The answer to that is more surprising than the classic puzzle's answer. The book is organized into 29 themes, each a topic from algebra, geometry or calculus and each launched from an opening puzzle or problem. There are excursions into number theory, solid geometry, physics and combinatorics. Always there is an emphasis on surprise and delight. And every theme begins at a level approachable with minimal background requirements. With well over 250 puzzles and problems, there is something here sure to appeal to everyone.
Portal through Mathematics will be useful for prospective secondary teachers of mathematics and may be used (as a supplementary resource) in university courses in algebra, geometry, calculus, and discrete mathematics. It can also be used for professional development for teachers looking for inspiration. However, the intended audience is much broader. Every fan of mathematics will find enjoyment in it.

Table of Contents

Cover

Half Title Page

Copyright

Title Page

Contributors

Anneli Lax New Mathematical Library

Contents

Foreword

Preface for anAmerican Readership

Author's Preface

Part I Surprising and Easy

1 Surprising right triangles

2 Surprisingly short solutions of geometric problems

3 A natural assertion with a surprising proof

4 Surprising answers

5 A surprising connection between three sequences

Part II Algebra, Calculus, and Geometry: problems

6 Five problems and a function

7 Five solutions of a routine problem

8 Equations of the form f(x, y) = g(x, y) and their generalizations

9 The generalized version of Viete's formula

10 Multiple roots of polynomials

11 Nonroutine applications of the derivative

12 Complex numbers, polynomials, and trigonometry

13 Complex numbers and geometry

14 Areas of triangles and quadrilaterals

15 Constructions in solid geometry

16 Inequalities

17 Diophantine equations

18 Combinatorial tales

19 Integrals

Part III Algebra, Calculus, and Geometry: theory (a little way beyond high school mathematics)

20 Functional equations of elementary functions

21 Sequences given by recurrence relations

22 The "golden ratio" or solving equations of the form f( x) = x

23 Convex functions: inequalities and approximations

24 Taylor's formula, Euler's formula, and a combinatorial problem

25 Derivatives of vectorfunctions

26 Polynomials and trigonometric relations

27 Areas and volumes as functions of coordinates

28 Values of trigonometric functions and sequences satisfying certain recurrence relation

29 Do there exist further "numbers" beyond complex numbers?

Solutions of the supplementary problems

Index


Additional Material

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 Book Details
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Reprinted edition available: NML/52
Portal through Mathematics is a collection of puzzles and problems mostly on topics relating to secondary mathematics. The problems and topics are fresh and interesting and frequently surprising. One example: the puzzle that asks how much length must be added to a belt around the Earth's equator to raise it one foot has probably achieved old chestnut status. Ivanov, after explaining the surprising answer to this question, goes a step further and asks, if you grabbed that too long belt at some point and raised it as high as possible, how high would that be? The answer to that is more surprising than the classic puzzle's answer. The book is organized into 29 themes, each a topic from algebra, geometry or calculus and each launched from an opening puzzle or problem. There are excursions into number theory, solid geometry, physics and combinatorics. Always there is an emphasis on surprise and delight. And every theme begins at a level approachable with minimal background requirements. With well over 250 puzzles and problems, there is something here sure to appeal to everyone.
Portal through Mathematics will be useful for prospective secondary teachers of mathematics and may be used (as a supplementary resource) in university courses in algebra, geometry, calculus, and discrete mathematics. It can also be used for professional development for teachers looking for inspiration. However, the intended audience is much broader. Every fan of mathematics will find enjoyment in it.

Cover

Half Title Page

Copyright

Title Page

Contributors

Anneli Lax New Mathematical Library

Contents

Foreword

Preface for anAmerican Readership

Author's Preface

Part I Surprising and Easy

1 Surprising right triangles

2 Surprisingly short solutions of geometric problems

3 A natural assertion with a surprising proof

4 Surprising answers

5 A surprising connection between three sequences

Part II Algebra, Calculus, and Geometry: problems

6 Five problems and a function

7 Five solutions of a routine problem

8 Equations of the form f(x, y) = g(x, y) and their generalizations

9 The generalized version of Viete's formula

10 Multiple roots of polynomials

11 Nonroutine applications of the derivative

12 Complex numbers, polynomials, and trigonometry

13 Complex numbers and geometry

14 Areas of triangles and quadrilaterals

15 Constructions in solid geometry

16 Inequalities

17 Diophantine equations

18 Combinatorial tales

19 Integrals

Part III Algebra, Calculus, and Geometry: theory (a little way beyond high school mathematics)

20 Functional equations of elementary functions

21 Sequences given by recurrence relations

22 The "golden ratio" or solving equations of the form f( x) = x

23 Convex functions: inequalities and approximations

24 Taylor's formula, Euler's formula, and a combinatorial problem

25 Derivatives of vectorfunctions

26 Polynomials and trigonometric relations

27 Areas and volumes as functions of coordinates

28 Values of trigonometric functions and sequences satisfying certain recurrence relation

29 Do there exist further "numbers" beyond complex numbers?

Solutions of the supplementary problems

Index