The Contents of About Mathematics


Preface

Introduction

Chapter 1. Getting Started (24 programs)
1. Calculator logistics
2. Calculator history
3. Some characteristics of your calculator

  • Accuracy
  • Speed
4. Order and Disorder
  • Disorder
  • Order
  • A middle ground: Fractals
Chapter 2. Algorithms, Functions and Equations (16 programs)
1. An unexpected algorithm
2. An old school algorithm
  • How the square root algorithm works
3. Functions
  • Functions and equations
4. Solving equations
  • Logistics
5. Graphing implicit functions
6. Programs replace complex computations
  • Determining a linear equation passing through two points
  • Determining a quadratic equation passing through three points
  • Finding the center and radius of a circle given three points
7. Solving systems of equations
  • Trial and error
  • Feedback
  • Matrices

Chapter 3. Dimensional Analysis (4 programs)
1. Using dimensions in converting units
2. Using dimensions in calculations
  • The Drake equation
  • Fermi problems
  • The Gimli Glider
3. Measurement precision and scientific notation

Chapter 4. Money Matters (23 programs)
1. Interesting interest
  • Greedy gangsters
  • Formulas and programs
2. Amortization
3. A detour through series sums
  • Summing an arithmetic series
  • Summing a geometric series
4. Deriving the amortization formula
5. Savings
6. Credit cards
7. At the casino
  • Roulette strategies
8. Beating the stock market
  • Using the Kelly criterion
  • Portfolio rebalancing
  • Math works for cheats as well
Chapter 5. The Science of Secrecy (20 programs)
1. Secret communication
  • Binary, octal and hexadecimal numeration
  • Some additional coding examples
2. Substitution ciphers
3. Edgar Allen Poe: an early cryptologist
4. The Vigeniere cipher
5. Modulo arithmetic
6. The RSA code
  • Setting up the RSA coding procedure
  • Sending an RSA coded message
  • Receiving an RSA coded message
7. Why RSA coding is such a breakthrough

Chapter 6. Calculus The Smallest Pebble on the Beach (6 programs)
1. Calculus: a perfect title
2. Slope and velocity
3. The derivative
  • Is that all there is to the differential calculus?
4. The integral calculus
5. The definite integral and area
6. The remarkable connection
7. The indefinite integral
8. Tying up loose ends
  • Numerical differentiation
  • Numerical integration

Chapter 7. Mathematicsl Models (17 programs)
1. Random number functions
2. Do we really need calculator-chosen random numbers?
3. Probability
4. Expectation
5. The St. Petersburg paradox
6. Monte Carlo simulations
7. Dart throwing for π
8. A project

Chapter 8. Revisiting Arithmetic (19 programs)
1. Counting
2. Addition
  • Adding more than ten digits
  • Speedy addition
3. Subtraction
  • An aside about binary representation
4. Multiplication
  • Duplation and mediation
5. Division

Chapter 9. Powers, logarithms and exponential change (30 programs)
1. Integer Powers
2. Square Root
3. Rational and real powers
4. Logarithms
  • How logs were used from 1620 to 1980
5. Exponents and logarithms
6. Exponential Change
  • Newton's Law of Cooling
  • Banking
  • Doubling time and its consequences
  • Population growth and Thomas Malthus
7. Logistic growth

Chapter 10. Alternative Geometries (13 programs)
1. Triangle area formulas
2. Euclid's
Elements
3. Problems with Euclid
4. Geometry on the Earth
5. Where am I?
  • Local time vs. standard time
  • Determining latitude and logitude
6. Distance and direction on a sphere
7. The other alternative to Postulate 5

Answers

Teaching
About Mathematics