## The Pythagorean Theorem: The Story of Its Power and Beauty"The first time I have enjoyed anything about mathematics."Bob Simon 60 Minutes correspondent "It 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 laureate "Although 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 Senior vice president of research and development, McGraw-Hill Education, McGraw-Hill Companies "Not 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 Culture "Using the familiar Pythagorean Theorem as the main theme, the author shows 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, NJ The 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. How did the Pythagorean Theorem evolve? Why has this geometric relationship fascinated countless generations? Who was this colorful and controversial man called Pythagoras? Are there applications for the theorem outside the realm of mathematics? In this entertaining and informative book, veteran math educator Alfred S. Posamentier makes the importance of the Pythagorean Theorem delightfully clear. Posamentier begins with a brief history of Pythagoras himself 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. Following this introduction to the topic, he shows the many ingenious ways in which the theorem has been proved visually by 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 a right triangle (e.g., 3, 4, 5 and 5, 12, 13). The relationships among the numbers of a Pythagorean triple will truly amaze the reader. Posamentier next turns to "Pythagorean means" (the arithmetic, geometric, and harmonic means). Outlining Pythagoras's contributions to the methods used for measuring and comparing quantities in a variety of ways gives the reader a true appreciation for these valuable mathematical concepts. Finally, the last two chapters take a some what different approach to the topic and view the Pythagorean Theorem from an artistic point of view. The author shows how Pythagoras's work manifests itself in music and how the Pythagorean Theorem has influenced fractals, including the founding of a new class of fractals called "Pythagorean trees." 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. |

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### Contents

Acknowledgments | 9 |

Proving the Pythagorean Theorem | 37 |

Applications of the Pythagorean | 77 |

Copyright | |

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### Common terms and phrases

AABC ABCD algebraic altitude apply the Pythagorean area of square area of triangle arithmetic mean BCPR complex number consecutive numbers consider consonance construction Curiosity Demonstration diagonal diezeugmenon equal equations established Euclidean formula famous theorem Fibonacci numbers fourth fractal geometric mean Greek hammer harmonic mean Heron's formula hypaton hyperbolaion hypotenuse inradius integer intervals legs magic square mathematician mathematics meson natural numbers notice octave odd numbers original triangle pattern perfect square perimeter pitch Primitive 38 Primitive 46 Primitive 47 Primitive 50 Primitive or multiple primitive Pythagorean triple produce proof proportion prove the Pythagorean Ptolemy's Theorem Pythago Pythagoras Pythagoras's Pythagorean means Pythagorean Theorem Pythagorean tree Pythagorean triangle quadrilateral ratio rean Theorem rean triples rectangle rectangular solid relationship right triangle ABC segment semiperimeter sequence shown in figure side lengths simply square ACNM strings tetrachord tetraktys trapezoid triangular numbers whole tone