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Index
Cover Title Page Copyright Page CONTENTS INTRODUCTION CHAPTER 1: ARITHMETIC NOVELTIES
When Is a Number Divisible by 3 or 9? When Is a Number Divisible by 11? Divisibility by Prime Numbers Squaring Numbers Quickly Squares and Sums Using Squares to Multiply Arbitrary Numbers An Alternative Method for Extracting a Square Root Sensible Number Comparisons Euclidean Algorithm to Find the GCD Sums of Positive Integers Sums of Odd Positive Integers The Realm of Nonterminating Decimals Atoms in the Universe of Numbers Fun with Number Relationships Friendly Numbers Palindromic Numbers Prime Numbers Infinite Primes The Neglected Triangular Numbers Perfect Numbers Making Mistaken Generalizations The Fibonacci Numbers
CHAPTER 2: ALGEBRAIC EXPLANATIONS OF ACCEPTED CONCEPTS
Simple Algebra Helps Logical Reasoning Division by Zero Irrationality of the Square Root of 2 Bisection Method to Approximate Square Roots Continued Fractions of Square Roots Fermat’s Method of Factoring Comparing Means Diophantine Equations Falling Squares Descartes’s Rule of Signs Horner’s Method to Evaluate Polynomials Generating Pythagorean Triples The Frobenius Problem
CHAPTER 3: GEOMETRIC CURIOSITIES
Parallelograms and Triangles Using a Grid to Calculate Areas The Center of a Quadrilateral Beyond the Formula for the Area of a Triangle Heronian Triangles A New Formula for the Area of Isosceles Triangles Pick’s Theorem When Intersecting Lines Meet a Circle Origins of Trigonometry Sines of Small Angles An Unconventional View of the Sine Surprising Proofs of the Pythagorean Theorem Beyond the Pythagorean Theorem—Part I Beyond the Pythagorean Theorem—Part II Beyond the Pythagorean Theorem—Part III The Pythagorean Theorem Extended to Three Dimensions Polyhedra: Sides, Faces, and Vertices Lunes and the Right Triangle Concurrency Similarity and the Golden Ratio A Relation between Points and Circles Constructions with Compasses Alone The Sphere and the Cylinder Regular Polygons and Stars Platonic Solids and Star Polyhedra
CHAPTER 4: PROBABILITY APPLIED TO EVERYDAY EXPERIENCES
How the Theory of Probability Began Benford’s Law The Birthday Phenomenon The Monty Hall Problem Bertrand’s Box The False Positive Paradox Pascal’s Triangle Random Walks The Poker Wild-Card Paradox
CHAPTER 5: COMMON SENSE FROM A MATHEMATICAL PERSPECTIVE
The Origins of Some Mathematics Symbols The Counterintuitive A Surprising Solution Don’t “Wine” over This Problem—A Problem-Solving Approach Organized Thinking Successive Percentages Rule of 72 A Mathematical Conjecture Unexpected Patterns An Infinity Conundrum The Concept of Infinity Counting the Uncountable Mathematics on a Bicycle The Parabola: A Remarkable Curve
ACKNOWLEDGMENTS APPENDIX NOTES INDEX
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