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Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed. Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath. By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.
Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Pythagorean theorem. --- Pythagoras' theorem --- Pythagorean proposition --- Theorem, Pythagorean --- Geometry, Plane
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The purpose of the conference “On Pythagoreanism”, held in Brasilia in 2011, was to bring together leading scholars from all over the world to define the status quaestionis for the ever-increasing interest and research on Pythagoreanism in the 21st century. The papers included in this volume exemplify the variety of topics and approaches now being used to understand the polyhedral image of one of the most fascinating and long-lasting intellectual phenomena in Western history. Cornelli’s paper opens the volume by charting the course of Pythagorean studies over the past two centuries. The remaining contributions range chronologically from Pythagoras and the early Pythagoreans of the archaic period (6th-5th centuries BCE) through the classical, hellenistic and late antique periods, to the eighteenth century. Thematically they treat the connections of Pythagoreanism with Orphism and religion, with mathematics, metaphysics and epistemology and with politics and the Pythagorean way of life.
Philosophy, Ancient --- Pythagoras and Pythagorean school --- Pythagorean theorem --- Pythagoras' theorem --- Pythagorean proposition --- Theorem, Pythagorean --- Geometry, Plane --- Plato --- Pythagoras --- Platon --- Aflāṭūn --- Aplaton --- Bolatu --- Platonas --- Platone --- Po-la-tʻu --- Pʻŭllatʻo --- Pʻŭllatʻon --- Pʻuratʻon --- Πλάτων --- אפלטון --- פלאטא --- פלאטאן --- פלאטו --- أفلاطون --- 柏拉圖 --- 플라톤 --- Платон --- プラトン --- Pitágora --- Pitagora di Samo --- Pitágoras --- Pitágoras de Samos --- Pythagore --- Πυθαγόρας --- فيثاغورس --- Presocratics. --- Pythagoras. --- Pythagoreanism.
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Was Plato a Pythagorean? Plato's students and earliest critics thought so, but later scholars have been more skeptical. This book reconsiders this question by arguing that a specific type of Pythagorean philosophy, called 'mathematical' Pythagoreanism played a profound role in Plato's philosophy.
Philosophy, Ancient --- Mathematics, Ancient. --- Pythagorean theorem. --- Pythagoras and Pythagorean school. --- Philosophie ancienne --- Mathématiques anciennes --- Pythagore, Théorème de --- Pythagorisme --- Plato. --- Pythagoriciens --- Platon, --- Et les mathématiques --- Mathématiques anciennes --- Pythagore, Théorème de --- Mathematics, Ancient --- Pythagoras and Pythagorean school --- Pythagorean theorem --- Ancient mathematics --- Pythagoras' theorem --- Pythagorean proposition --- Theorem, Pythagorean --- Geometry, Plane --- Aflāṭūn --- Aplaton --- Bolatu --- Platonas --- Platone --- Po-la-tʻu --- Pʻŭllatʻo --- Pʻŭllatʻon --- Pʻuratʻon --- Πλάτων --- אפלטון --- פלאטא --- פלאטאן --- פלאטו --- أفلاطون --- 柏拉圖 --- 플라톤 --- Plato --- Platon --- Platoon --- Théorème de Pythagore --- Théorème de Pythagore. --- Pythagoriciens. --- Платон --- プラトン --- Et les mathématiques.
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Archytas of Tarentum is one of the three most important philosophers in the Pythagorean tradition, a prominent mathematician, who gave the first solution to the famous problem of doubling the cube, an important music theorist, and the leader of a powerful Greek city-state. He is famous for sending a trireme to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC. This 2005 study was the first extensive enquiry into Archytas' work in any language. It contains original texts, English translations and a commentary for all the fragments of his writings and for all testimonia concerning his life and work. In addition there are introductory essays on Archytas' life and writings, his philosophy, and the question of authenticity. Carl A. Huffman presents an interpretation of Archytas' significance both for the Pythagorean tradition and also for fourth-century Greek thought, including the philosophies of Plato and Aristotle.
Ancient mathematics --- Ancient philosophy --- Antieke filosofie --- Antieke wiskunde --- Filosofie [Antieke ] --- Filosofie [Griekse ] --- Filosofie [Romeinse ] --- Filosofie van de Oudheid --- Greek philosophy --- Griekse filosofie --- Mathematics [Ancient ] --- Mathématiques anciennes --- Mathématiques de l'antiquité --- Philosophie ancienne --- Philosophie antique --- Philosophie de l'Antiquité --- Philosophie grecque --- Philosophie romaine --- Philosophy [Ancient ] --- Philosophy [Greek ] --- Philosophy [Roman ] --- Pythagoras' theorem --- Pythagorean proposition --- Pythagorean theorem --- Roman philosophy --- Romeinse filosofie --- Stelling van Pythagoras --- Theorem Pythagorean --- Théorème de Pythagore --- Wiskunde [Antieke ] --- Wiskunde van de oudheid --- Archytas, --- Archytas of Tarentum --- Mathematicians --- Greece --- Biography --- Scientists --- Ps.-Archita --- Ps.-Archytas --- Pseudo Archita --- Pseudo-Archytas --- Tarentum, Archytas of --- Archytas de Tarente --- Arts and Humanities --- History --- Archytas, - of Tarentum.
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By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and examples of their practical applications. Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins, detailed algorithm descriptions, and many figures and worked examples. The book may appeal, in whole or in part, to mathematicians, computer scientists, and engineers.
Mathematics. --- Computer graphics. --- Computer-aided engineering. --- Algebra. --- Computer mathematics. --- Geometry. --- Computational intelligence. --- Computer-Aided Engineering (CAD, CAE) and Design. --- Computational Mathematics and Numerical Analysis. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Computational Intelligence. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Mathematics --- Euclid's Elements --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematical analysis --- CAE --- Engineering --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Engineering graphics --- Image processing --- Math --- Science --- Data processing --- Digital techniques --- Pythagorean-hodograph curves. --- Hodograph equations. --- Pythagorean theorem. --- Geometry, Analytic. --- Curves --- Computer-aided design. --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Shapes --- Analytical geometry --- Geometry, Algebraic --- Algebra --- Pythagoras' theorem --- Pythagorean proposition --- Theorem, Pythagorean --- Geometry, Plane --- Equations, Hodograph --- Differential equations, Partial --- Fluid mechanics --- PH curves --- Graphic methods --- Computer aided design. --- Computer science --- Computer vision. --- Engineering. --- Construction --- Industrial arts --- Technology --- Machine vision --- Vision, Computer --- Pattern recognition systems --- CAD (Computer-aided design) --- Computer-assisted design --- Computer-aided engineering --- Design --- Optical data processing. --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Optical equipment
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