## The Pythagorean Theorem: The Story of Its Power and BeautyAlthough we all remember the Pythagorean Theorem from our school days, not until you read this book will you find out about the marvelous treasures this most famous mathematical concept holds. In an easily understood manner, the author entertains us with the wonders surrounding this theorem. This is the sort of treatment that will help popularize mathematics!-Charlotte K. Frank, PhD, SVP, research and development, McGraw-Hill Education, The McGraw-Hill CompaniesUsing the familiar Pythagorean theorem as the main theme the authors show the power and beauty of mathematics as we would have perhaps wished to have seen it when we were first introduced to this ubiquitous theorem in our school days. This book is a must read for anyone with even a small interest in mathematics.-Daniel Jaye, principal, Bergen County Academies, Hackensack, NJThe first time I have enjoyed anything about mathematics.-Bob Simon, 60 Minutes CorrespondentNot only is this book a very valuable resource for mathematics teachers, but it is also a book that can convince the general public that there is genuine beauty in mathematics. Perhaps this book will help bring 'converts' to mathematics!-Dr. Anton Dobart, director general, Austrian Ministry for Education, Art and CultureIt is often overheard in academic environments that 'math is'fun!' This little book on the Pythagorean theorem is surely proof enough, especially since, like the theorem, the fun is on almost every page.-Leon M. Lederman, Nobel laureateThe Pythagorean theorem may be the best-known equation in mathematics. Its origins reach back to the beginnings of civilization, and today every student continues to study it. What most nonmathematicians don't understand or appreciate is why this simply stated theorem has fascinated countless generations.In this entertaining and informative book, veteran math educator Alfred S. Posamentier makes the importance of the Pythagorean theorem delightfully clear.He begins with a brief history of Pythagoras and the early use of his theorem by the ancient Egyptians, Babylonians, Indians, and Chinese, who used it intuitively long before Pythagoras's name was attached to it.He then shows the many ingenious ways in which the theorem has been proved visually using highly imaginative diagrams. Some of these go back to ancient mathematicians; others are comparatively recent proofs, including one by the twentieth president of the United States, James A. Garfield.After demonstrating some curious applications of the theorem, Posamentier then explores the Pythagorean triples, pointing out the many hidden surprises of the three numbers that can represent the sides of the right triangle (e.g, 3, 4, 5 and 5, 12, 13). And many will truly amaze the reader.He then turns to the Pythagorean means (the arithmetic, geometric, and harmonic means). By comparing their magnitudes in a variety of ways, he gives the reader a true appreciation for these mathematical concepts.The final two chapters view the Pythagorean theorem from an artistic point of view-namely, how Pythagoras's work manifests itself in music and how the Pythagorean theorem can influence fractals.Posamentier's lucid presentation and gift for conveying the significance of this key equation to those with little math background will inform, entertain, and inspire the reader, once again demonstrating the power and beauty of mathematics!Alfred S. Posamentier, Ph.D. (New York, NY), is dean of the School of Education and professor of mathematics education at The City College of the City University of New York. He has published more than 40 books in the area of mathematics and mathematics education, including The Fabulous Fibonacci Numbers, Pi: A Biography of the World's Most Mysterious Number, and Math Charmers: Tantalizing Tidbits for the Mind. |

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### Contents

Acknowledgments | 9 |

Proving the Pythagorean Theorem | 37 |

Applications of the Pythagorean | 77 |

Pythagorean Triples | 123 |

The Pythagorean Means | 169 |

Tuning the Soul Pythagoras | 185 |

The Pythagorean Theorem | 211 |

Final Thoughts | 235 |

About the Mathematics Work | 237 |

Some Selected Proofs | 253 |

List of Primitive Pythagorean | 273 |

303 | |

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### Common terms and phrases

ABCD algebraic altitude apply the Pythagorean area of square area of triangle arithmetic mean BCPR complex number consecutive numbers consider consonance construction Demonstration diagonal diezeugmenon equations established Euclidean formula famous theorem Fibonacci numbers fourth fractal geometric mean Greek hammer harmonic mean hypote hypotenuse inradius integer intervals Jos de Mey legs magic square mathematician mathematics meson n2 Primitive natural numbers notice octave odd numbers original triangle pattern perfect square perimeter pitch Posamentier Primitive 38 Primitive 46 Primitive 50 Primitive m n a=m Primitive or multiple primitive Pythagorean triple produce proof proportion prove the Pythagorean Pythago Pythagoras Pythagoras’s Pythagorean means Pythagorean Theorem Pythagorean tree Pythagorean triangle quadrilateral ratio rean Theorem rean triples rectangle relationship right triangle ABC semiperimeter sequence shown in figure side lengths simply square ACNM Stage strings superparticular ratio tetrachord tetraktys trapezoid triangular number whole tone