Orient BlackSwan

About the BookAbout the AuthorTable of Contents

Learning Mathematics through Modelling and Simulation: An Investigative Approach, offers the readers a hands-on experience of discovering the beautiful aspects of mathematics through the use of technology. Three powerful tools falling into the categories of spreadsheet programs (Microsoft Excel), dynamic geometry software (GeoGebra) and computer algebra systems (Mathematica), are introduced. They are applied in the book to enable the readers to independently explore, visualise, conjecture, reason and solve problems.

The book highlights the role of mathematical software in enabling explorations leading to a deeper insight into the problems. Technology is used to generate phenomena and acquire data from which questions naturally arise and create a demand for appropriate theory.

Readers are introduced to the applications of Calculus, Probability, Number Theory, Linear Algebra and Discrete Mathematics. Each chapter includes a set of exercises, called Investigations, that prompt analytical thinking, encourage exploration of a concept, or test the readers’ ability to apply the concept to a realistic problem situation.

High school students, undergraduates and any reader keen to learn interesting mathematics with the help of technology will find this book both useful and enjoyable.

Jonaki Ghosh obtained her Ph.D in Applied Mathematics from the Jamia Milia Islamia University. Presently, she is Associate Professor in Lady Shri Ram College for Women, where she teaches courses related to mathematics education. She is keenly interested in the use of technology for mathematics instruction, professional development of mathematics teachers and popularisation of mathematics.

Amber Habib obtained his Ph.D from the University of California, Berkeley, and is currently Professor at the Shiv Nadar Institution of Eminence, Delhi NCR. He is engaged in making mathematics education more fulfilling through projects, special topics and its myriad links with other disciplines.

Geetha Venkataraman obtained her Ph.D from the University of Oxford. She worked at St Stephen’s College as Associate Professor and is currently Professor at Dr. B. R. Ambedkar University Delhi. She is a group theorist, interested in issues related to mathematics education, women in mathematics and leadership in academia.

Preface
Acknowledgements
1 Monte Carlo Methods: Estimating Π
1.1 Random Numbers
1.2 The Square–Circle Method for Estimating Π
1.3 The Buffon’s Needle Problem
1.4 Chebyshev’s Inequality and Accuracy of Estimates
2 Exploring Probability through Simulation
2.1 The Monty Hall Problem: A Historical Journey
2.2 Spreadsheet Simulation of the Monty Hall Problem
2.3 The Monty Hall Problem and Conditional Probability
2.4 The Game of Craps
2.5 Spreadsheet Simulation of the Game of Craps
3 Graph Theory: Seven Bridges Problem and the Tower of Hanoi
3.1 A Puzzle with Seven Bridges
3.2 The Konigsberg Bridges Problem and Graphs
3.3 Graphs, Walks, Trails and Paths
3.4 Eulerian and Semi-Eulerian Graphs
3.5 The Tower of Hanoi
3.6 Hanoi Graphs
4 The Mathematics of Ciphers: Public Key Encryption and RSA
4.1 Secret Messages
4.2 Encryption Schema
4.3 Modular Arithmetic and the Euler Phi Function
4.4 Some Important Results
4.5 The RSA Algorithm
4.6 Diffie–Hellman Key Exchange
5 Exploring Fractals
5.1 The Sierpinski Triangle
5.2 The Koch Snowflake
5.3 Space Filling Trees
5.4 Fractal Dimension
6 The Newton–Raphson Method and Fractals
6.1 The Newton–Raphson Method for Real Functions
6.2 The Newton–Raphson Method for Complex Functions
6.3 Cyclic and Chaotic Orbits
6.4 Polynomial Iteration
6.5 The Mandelbrot Set
7 Experiments in Symmetry: Frieze and Wallpaper Patterns
7.1 The Ubiquitous Nature of Symmetry
7.2 Symmetries of a Plane
7.3 Strip or Frieze Patterns
7.4 Classifying Strip Patterns
7.5 Wallpaper Patterns
8 Applications of Matrices
8.1 Ranking Players
8.2 Genes and Inheritance
8.3 Age-structured Population Growth
A Introduction to Software
A.1 Excel
A.2 Mathematica
A.3 GeoGebra
B Investigations: Hints and Results
Bibliography
Image Credits
Subject Index
Index of Names