Guide to Essential Math

To the Reader

Preface to Second Edition

Chapter 1. Mathematical Thinking

1.1 The NCAA March Madness Problem

1.2 Gauss and the Arithmetic Series

1.3 The Pythagorean Theorem

1.4 Torus Area and Volume

1.5 Einstein’s Velocity Addition Law

1.6 The Birthday Problem

1.7 Fibonacci Numbers and the Golden Ratio

1.8 in the Gaussian Integral

1.9 Function Equal to Its Derivative

1.10 Stirling’s Approximation for!

1.11 Potential and Kinetic Energies

1.12 Riemann Zeta Function and Prime Numbers

1.13 How to Solve It

1.14 A Note on Mathematical Rigor

Chapter 2. Numbers

2.1 Integers

2.2 Primes

2.3 Divisibility

2.4 Rational Numbers

2.5 Exponential Notation

2.6 Powers of 10

2.7 Binary Number System

2.8 Infinity

Chapter 3. Algebra

3.1 Symbolic Variables

3.2 Legal and Illegal Algebraic Manipulations

3.3 Factor-Label Method

3.4 Powers and Roots

3.5 Logarithms

3.6 The Quadratic Formula

3.7 Imagining i

3.8 Factorials, Permutations and Combinations

3.9 The Binomial Theorem

3.10 e is for Euler

Chapter 4. Trigonometry

4.1 What Use is Trigonometry?

4.2 Geometry of Triangles

4.3 The Pythagorean Theorem

4.4 in the Sky

4.5 Sine and Cosine

4.6 Tangent and Secant

4.7 Trigonometry in the Complex Plane

4.8 de Moivre’s Theorem

4.9 Euler’s Theorem

4.10 Hyperbolic Functions

Chapter 5. Analytic Geometry

5.1 Functions and Graphs

5.2 Linear Functions

5.3 Conic Sections

5.4 Conic Sections in Polar Coordinates

Chapter 6. Calculus

6.1 A Little Road Trip

6.2 A Speedboat Ride

6.3 Differential and Integral Calculus

6.4 Basic Formulas of Di