Expanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions

221 Pages·2023·12.688 MB·English

by  Boris Pritsker

Expanding Mathematical
Toolbox: Interweaving Topics,
Problems, and Solutions
This book offers several topics from different mathematical disciplines and shows how
closely they are related. The purpose of the book is to direct the attention of readers who
have an interest in and talent for mathematics to engaging and thought-provoking prob-
lems that should help them change their ways of thinking, entice further exploration, and
possibly lead to independent research and projects in mathematics. In spite of the many
challenging problems, most solutions require no more than a basic knowledge covered in
a high-school math curriculum.
To shed new light on a deeper appreciation for mathematical relationships, the problems
are selected to demonstrate techniques involving a variety of mathematical ideas. Included
are some interesting applications of trigonometry, vector algebra and Cartesian coordi-
nate system techniques, in addition to geometrical constructions and inversion in solv-
ing mechanical engineering problems and in studying models explaining non-Euclidean
geometries.
Expanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions is primarily
directed at secondary school teachers and college professors. The book will be useful in
teaching mathematical reasoning because it emphasizes how to teach students to think
creatively and strategically and how to make connections between math disciplines. The
text also can be used as a resource for preparing for mathematics Olympiads. In addi-
tion, it is aimed at all readers who want to study mathematics, gain a deeper understand-
ing and enhance their problem-solving abilities. Readers will find fresh ideas and topics
offering unexpected insights, new skills to expand their horizons in math studies, and an
appreciation for the beauty of mathematics.

Expanding Mathematical
Toolbox: Interweaving Topics,
Problems, and Solutions
Boris Pritsker
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Boris Pritsker
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ISBN: 9781032417387 (hbk)
ISBN: 9781032417356 (pbk)
ISBN: 9781003359500 (ebk)
DOI: 10.1201/9781003359500
Typeset in Palatino
by KnowledgeWorks Global Ltd.
To the memory of my beloved parents.

Contents
Preface ..............................................................................................................................................ix
About the Authors ......................................................................................................................xiii
1. Beauty in Mathematics ..........................................................................................................1
2. Euclidean Constructions .....................................................................................................19
3. Inversion and Its Applications ...........................................................................................33
4. Using Geometry for Algebra. Classic Mean Averages’ Geometrical
Interpretations.......................................................................................................................49
5. Using Algebra for Geometry ..............................................................................................69
6. Trigonometrical Explorations ............................................................................................89
7. Euclidean Vectors ...............................................................................................................113
8. Cartesian Coordinates in Problem Solving ..................................................................131
9. Inequalities Wonderland ..................................................................................................147
10. Guess and Check Game.....................................................................................................169
Appendix .....................................................................................................................................185
References ...................................................................................................................................203
Index .............................................................................................................................................205
vii

Preface
Mathematics isn’t a palm tree, with a single long straight trunk covered with scratchy formulas.
It’s a banyan tree, with many interconnected trunks and branches - a banyan tree that has grown
to the size of a forest, inviting us to climb and explore.
William P. Thurston
While studying mathematics in middle and then in secondary school, students face arith-
metic, algebra, geometry, trigonometry, pre-calculus, and statistics and probability as
separate disciplines. A stereotypical view has been to “divide” mathematics into catego-
ries that are, by implication, close to disjoint. Don’t we apply pure algebraic techniques
for solving quadratic or cubic equations? What about solving construction problems in
geometry; don’t we rely on the geometric properties of figures and use already proven
geometric theorems? Moreover, it tends to be that synthetic solutions are usually valued
more than algebraic or trigonometric. Certainly, this is true. Many algebraic and geometric
problems are efficiently solved without referring to other math disciplines. Some schol-
ars even consider, for instance, trigonometry as the killer of geometrical beauty, meaning
that tedious trigonometric modifications sometimes overshadow pure elegant geometri-
cal ways of problem-solving. In fact, it all depends on the problem one is solving, and
in most cases, it is up to you to decide what methods and techniques to apply to get the
most elegant and short solution. But it is an erroneous view that mathematical branches
are not associated with each other and each one exists in its own universe under its own
laws. One cannot succeed in mathematical studies without relating math disciplines to
ix