Essentials of Mathematical Thinking

Steven G. Krantz

• Publisher: Chapman and Hall/CRC
• Year: 2017
• ISBN:
  9781351642941
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Essentials of Mathematical Thinking addresses the growing need to better comprehend mathematics today. Increasingly, our world is driven by mathematics in all aspects of life. The book is an excellent introduction to the world of mathematics for students not majoring in mathematical studies.

The author has written this book in an enticing, rich manner that will engage students and introduce new paradigms of thought. Careful readers will develop critical thinking skills which will help them compete in today's world.

The book explains:

• 
  
  

• What goes behind a Google search algorithm

• 
  
  

• How to calculate the odds in a lottery

• 
  
  

• The value of Big Data

• 
  
  

• How the nefarious Ponzi scheme operates
  

Instructors will treasure the book for its ability to make the field of mathematics more accessible and alluring with relevant topics and helpful graphics. The author also encourages readers to see the beauty of mathematics and how it relates to their lives in meaningful ways.

Table of Contents

Preface

1 First Thoughts

1.1 What Is Mathematical Thinking?

1.2 How Does Mathematics Differ from Other Disciplines?

1.3 A Sample Problem

2 Diverse Mathematical Thoughts

2.1 A Fraction of the Time

2.2 How to Swindle on the Stock Market

2.3 The Bible Code

2.4 Winning on a Game Show

2.5 Cutting the Cake

2.6 A Lesson in Map Coloring

2.6.1 Analysis

2.6.2 Modern Developments

2.6.3 Denouement

2.7 The Complexity of Songs

2.8 Bertrand's Paradox

3 Strategy

3.1 It’s All in the Balance

3.2 See and Say

3.3 The Ponzi Scheme

3.4 Ham Sandwich Theorems

4 Focus

4.1 The Erdo˝s Number

4.2 Time Out

4.3 Days of the Week

5 Science

5.1 A Belt for the Earth

5.2 Your Next Breath

5.3 A Hairy Question

5.4 The Motions of the Planets

5.5 How Big Is Big Data?

6 Counting

6.1 Funny Numbers

6.2 The Pigeon Flew the Coop

6.3 Conditional Probability

6.4 Benford’s Law

6.5 Puzzling Birthdays

7 Games

7.1 How to Count

7.2 How to Beat the Lottery

7.3 The Eudaemonic Pie

7.4 A Dicey Bet

7.5 The Game of Life

7.6 The Tower of Hanoi

8 Geometry

8.1 Thoughts of Pythagoras

8.2 Symmetry

8.3 Buffon’s Needle Problem

8.4 Euler’s Formula

8.5 Sphere Packing

8.6 The Platonic Solids

8.7 Heron’s Problem

8.8 A Little Geometric Reasoning

9 Practical Matters

9.1 Strangers on a Plane

9.2 You’ve Got My Vote

9.2.1 The Plurality System

9.2.2 The Hare System

9.2.3 The Borda Count

9.2.4 Cumulative Voting

9.2.5 Approval Voting

9.2.6 Conclusions

9.3 Take Your Pill

9.4 Geometric Analysis and Facial Structure

9.4.1 Geometry and Facial Structure

9.4.2 Conformal Mapping

9.4.3 Wavelets and Filters

9.4.4 Summary Remarks

9.5 Beware the Raven

9.6 The Prisoner’s Dilemma

9.7 The Eyes Have It

9.8 A Sure Bet

9.9 Hilbert’s Hotel Infinity

10 Breaking the Code

10.1 Alan Turing and Cryptography

10.1.1 Background on Alan Turing

10.1.2 The Turing Machine

10.1.3 What Is Cryptography?

10.1.4 Encryption by Way of Affine Transformations

10.1.5 Digraph Transformations

10.2 RSA Encryption

10.2.1 Basics and Background

10.2.2 Preparation for RSA

10.2.3 Modular Arithmetic

10.2.4 Relatively Prime Integers

10.2.5 The RSA System Enunciated

10.2.6 The RSA Encryption System Explicated

11 Discrete Problems

11.1 Far-Reaching Dominoes

11.2 Surreal Life

11.3 A Problem with Marriage

11.4 Euler's Bridges

11.5 Scheduling Sporting Events

12 Advanced Ideas

12.1 Searching on Google

12.1.1 The Mathematics of a Google Search

12.1.2 The Directed Web Graph

12.1.3 Passage to the Web HyperLink Matrix

12.1.4 A Fix for Dangling Nodes

12.1.5 The Ultimate Google Matrix

12.2 A Needle Problem of Kakeya

12.3 Euclidean and Non‐Euclidean Geometry

12.4 Archimedes and the Calculation of the

12.4.1 The Genius of Archimedes

12.4.2 Archimedes’s Calculation of the Area of a Circle

13 Concluding Remarks

13.1 The Final Word

References

Index

Preface

It is becoming increasingly clear that mathematical thinking is essential to understanding the world around us. Police tell us that most crimes these days are committed with electronic devices (usually computers). Many if not most medical procedures are now robotically assisted. The carburation system of a modern automobile is all electronic. Geometric visualization is one of the most important aspects of modern technology.

And it is plain that the thinking behind all of the technology described in the last paragraph is mathematical. The Google search engine is based on a mathematical algorithm. Not just logic, but analysis and combinatorics and geometry and many other ideas from the mathematical sciences are critical parts of our world.

A college education consists of learning different modes of discourse. Every modern student should thus have some exposure to mathematical discourse. Whether or not the student has any intention of working in the mathematical sciences, he/she will need these modes of thought to survive in the modern marketplace.

And learning mathematical thought does not have be dry, desultory, and boring. It can be fun and exciting and challenging (in a productive fashion). It can relate to things that are meaningful to students—such as the paradigm behind a Google search, or the means by which a music CD is encoded, or the RSA encryption algorithm. In a course on mathematical thinking the student will obviously learn some mathematical skills, but he/she will also gain critical thinking skil ls. These are valuable life tools.

This book may be thought of as a book for nonmathematicians taking a mathematics course that is a breadth requirement for an undergraduate degree. Thus it has minimal mathematical prerequisites. It has very few equations. It has lots of pictures and lots of explanation. It is driven by examples. But those examples will lead students down new paths, and acquaint them with new paradigms of thought. They will stimulate and provoke and encourage. At the end of this journey, the student should feel that ...

Table of contents

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