Learning mathematics through modelling and simulation: an investigative approach / by Jonaki Ghosh, Amber Habib and Geetha Venkataraman.

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.

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