The Pythagorean Theorem

A 4,000-Year History

Author: Eli Maor

Publisher: Princeton University Press

ISBN: 0691148236

Category: Mathematics

Page: 280

View: 2803

The author presents a complex history of the Pythagorean Theorem, examining the earliest evidence of knowledge of the theorem to Einstein's theory of relativity.
Posted in Mathematics

Beautiful Geometry

Author: Eli Maor,Eugen Jost

Publisher: Princeton University Press

ISBN: 1400848334

Category: Mathematics

Page: 208

View: 8722

If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important branches of mathematics.
Posted in Mathematics

STEMathematics: Exercises in Applied Computation and Modeling (Volume 1)

Author: Elliott Ostler


ISBN: 0996674101


Page: 390

View: 4610

STEMathematics is an instructional resource designed primarily for secondary level mathematics teachers and students interested in discovering how mathematics describes (and is applied to) our natural world. This resource provides both the historical elements and the technical aspects of various topics in mathematics that provide instructional context in the sciences, technology, and engineering, (STEM) disciplines. The purpose of STEMathematics is to help teachers become more personally interested in the topics they teach and to gain a broader perspective of how mathematics can be integrated with other subject disciplines.
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Einstein Gravity in a Nutshell

Author: A. Zee

Publisher: Princeton University Press

ISBN: 1400847451

Category: Science

Page: 888

View: 9351

This unique textbook provides an accessible introduction to Einstein's general theory of relativity, a subject of breathtaking beauty and supreme importance in physics. With his trademark blend of wit and incisiveness, A. Zee guides readers from the fundamentals of Newtonian mechanics to the most exciting frontiers of research today, including de Sitter and anti-de Sitter spacetimes, Kaluza-Klein theory, and brane worlds. Unlike other books on Einstein gravity, this book emphasizes the action principle and group theory as guides in constructing physical theories. Zee treats various topics in a spiral style that is easy on beginners, and includes anecdotes from the history of physics that will appeal to students and experts alike. He takes a friendly approach to the required mathematics, yet does not shy away from more advanced mathematical topics such as differential forms. The extensive discussion of black holes includes rotating and extremal black holes and Hawking radiation. The ideal textbook for undergraduate and graduate students, Einstein Gravity in a Nutshell also provides an essential resource for professional physicists and is accessible to anyone familiar with classical mechanics and electromagnetism. It features numerous exercises as well as detailed appendices covering a multitude of topics not readily found elsewhere. Provides an accessible introduction to Einstein's general theory of relativity Guides readers from Newtonian mechanics to the frontiers of modern research Emphasizes symmetry and the Einstein-Hilbert action Covers topics not found in standard textbooks on Einstein gravity Includes interesting historical asides Features numerous exercises and detailed appendices Ideal for students, physicists, and scientifically minded lay readers Solutions manual (available only to teachers)
Posted in Science

Music by the Numbers

From Pythagoras to Schoenberg

Author: Eli Maor

Publisher: Princeton University Press

ISBN: 1400889898

Category: Mathematics

Page: 176

View: 5217

How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music explicitly based on mathematical principles. Yet Eli Maor argues that music has influenced math at least as much as math has influenced music. Starting with Pythagoras, proceeding through the work of Schoenberg, and ending with contemporary string theory, Music by the Numbers tells a fascinating story of composers, scientists, inventors, and eccentrics who played a role in the age-old relationship between music, mathematics, and the sciences, especially physics and astronomy. Music by the Numbers explores key moments in this history, particularly how problems originating in music have inspired mathematicians for centuries. Perhaps the most famous of these problems is the vibrating string, which pitted some of the greatest mathematicians of the eighteenth century against each other in a debate that lasted more than fifty years and that eventually led to the development of post-calculus mathematics. Other highlights in the book include a comparison between meter in music and metric in geometry, complete with examples of rhythmic patterns from Bach to Stravinsky, and an exploration of a suggestive twentieth-century development: the nearly simultaneous emergence of Einstein's theory of relativity and Schoenberg's twelve-tone system. Weaving these compelling historical episodes with Maor's personal reflections as a mathematician and lover of classical music, Music by the Numbers will delight anyone who loves mathematics and music.
Posted in Mathematics

"e:" The Story of a Number

The Story of a Number

Author: Eli Maor

Publisher: Princeton University Press

ISBN: 1400832349

Category: Mathematics

Page: 248

View: 3338

The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of e to mathematics and illuminates a golden era in the age of science.
Posted in Mathematics

Calculus in 3D: Geometry, Vectors, and Multivariate Calculus

Author: Zbigniew Nitecki

Publisher: American Mathematical Soc.

ISBN: 1470443600

Category: Calculus

Page: 405

View: 8371

Calculus in 3D is an accessible, well-written textbook for an honors course in multivariable calculus for mathematically strong first- or second-year university students. The treatment given here carefully balances theoretical rigor, the development of student facility in the procedures and algorithms, and inculcating intuition into underlying geometric principles. The focus throughout is on two or three dimensions. All of the standard multivariable material is thoroughly covered, including vector calculus treated through both vector fields and differential forms. There are rich collections of problems ranging from the routine through the theoretical to deep, challenging problems suitable for in-depth projects. Linear algebra is developed as needed. Unusual features include a rigorous formulation of cross products and determinants as oriented area, an in-depth treatment of conics harking back to the classical Greek ideas, and a more extensive than usual exploration and use of parametrized curves and surfaces. Zbigniew Nitecki is Professor of Mathematics at Tufts University and a leading authority on smooth dynamical systems. He is the author of Differentiable Dynamics, MIT Press; Differential Equations, A First Course (with M. Guterman), Saunders; Differential Equations with Linear Algebra (with M. Guterman), Saunders; and Calculus Deconstructed, AMS.
Posted in Calculus

Trigonometric Delights

Author: Eli Maor

Publisher: Princeton University Press

ISBN: 0691158207

Category: Mathematics

Page: 236

View: 929

Trigonometry has always been an underappreciated branch of mathematics. It has a reputation as a dry and difficult subject, a glorified form of geometry complicated by tedious computation. In this book, Eli Maor draws on his remarkable talents as a guide to the world of numbers to dispel that view. Rejecting the usual arid descriptions of sine, cosine, and their trigonometric relatives, he brings the subject to life in a compelling blend of history, biography, and mathematics. He presents both a survey of the main elements of trigonometry and a unique account of its vital contribution to science and social development. Woven together in a tapestry of entertaining stories, scientific curiosities, and educational insights, the book more than lives up to the title Trigonometric Delights. Maor, whose previous books have demystified the concept of infinity and the unusual number "e," begins by examining the "proto-trigonometry" of the Egyptian pyramid builders. He shows how Greek astronomers developed the first true trigonometry. He traces the slow emergence of modern, analytical trigonometry, recounting its colorful origins in Renaissance Europe's quest for more accurate artillery, more precise clocks, and more pleasing musical instruments. Along the way, we see trigonometry at work in, for example, the struggle of the famous mapmaker Gerardus Mercator to represent the curved earth on a flat sheet of paper; we see how M. C. Escher used geometric progressions in his art; and we learn how the toy Spirograph uses epicycles and hypocycles. Maor also sketches the lives of some of the intriguing figures who have shaped four thousand years of trigonometric history. We meet, for instance, the Renaissance scholar Regiomontanus, who is rumored to have been poisoned for insulting a colleague, and Maria Agnesi, an eighteenth-century Italian genius who gave up mathematics to work with the poor--but not before she investigated a special curve that, due to mistranslation, bears the unfortunate name "the witch of Agnesi." The book is richly illustrated, including rare prints from the author's own collection. Trigonometric Delights will change forever our view of a once dreaded subject.
Posted in Mathematics


Eulers Konstante, Primzahlstrände und die Riemannsche Vermutung

Author: Julian Havil

Publisher: Springer-Verlag

ISBN: 3540484965

Category: Mathematics

Page: 302

View: 3367

Jeder kennt p = 3,14159..., viele kennen e = 2,71828..., einige i. Und dann? Die "viertwichtigste" Konstante ist die Eulersche Zahl g = 0,5772156... - benannt nach dem genialen Leonhard Euler (1707-1783). Bis heute ist unbekannt, ob g eine rationale Zahl ist. Das Buch lotet die "obskure" Konstante aus. Die Reise beginnt mit Logarithmen und der harmonischen Reihe. Es folgen Zeta-Funktionen und Eulers wunderbare Identität, Bernoulli-Zahlen, Madelungsche Konstanten, Fettfinger in Wörterbüchern, elende mathematische Würmer und Jeeps in der Wüste. Besser kann man nicht über Mathematik schreiben. Was Julian Havil dazu zu sagen hat, ist spektakulär.
Posted in Mathematics

To Infinity and Beyond

A Cultural History of the Infinite - New Edition

Author: Eli Maor

Publisher: Princeton University Press

ISBN: 0691178119

Category: Mathematics

Page: 304

View: 490

To Infinity and Beyond explores the idea of infinity in mathematics and art. Eli Maor examines the role of infinity, as well as its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher; from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes the mathematician's fascination with infinity--a fascination mingled with puzzlement.
Posted in Mathematics

Was ist Mathematik?

Author: Richard Courant,Herbert Robbins

Publisher: Springer-Verlag

ISBN: 3662000539

Category: Mathematics

Page: N.A

View: 7951

47 brauchen nur den Nenner n so groß zu wählen, daß das Intervall [0, IJn] kleiner wird als das fragliche Intervall [A, B], dann muß mindestens einer der Brüche m/n innerhalb des Intervalls liegen. Also kann es kein noch so kleines Intervall auf der Achse geben, das von rationalen Punkten frei wäre. Es folgt weiterhin, daß es in jedem Intervall unendlich viele rationale Punkte geben muß; denn wenn es nur eine endliche Anzahl gäbe, so könnte das Intervall zwischen zwei beliebigen benachbarten Punkten keine rationalen Punkte enthalten, was, wie wir eben sahen, unmöglich ist. § 2. Inkommensurable Strecken, irrationale Zahlen und der Grenzwertbegriff 1. Einleitung Vergleicht man zwei Strecken a und b hinsichtlich ihrer Größe, so kann es vor kommen, daß a in b genau r-mal enthalten ist, wobei r eine ganze Zahl darstellt. In diesem Fall können wir das Maß der Strecke b durch das von a ausdrücken, indem wir sagen, daß die Länge von b das r-fache der Länge von a ist.
Posted in Mathematics

Die Geschichte der Null

Author: Robert Kaplan

Publisher: N.A

ISBN: 9783492239189

Category: Null

Page: 247

View: 3560

Posted in Null

The Exact Sciences in Antiquity

Author: Otto Neugebauer

Publisher: Courier Corporation

ISBN: 9780486223322

Category: Science

Page: 240

View: 4817

Based on a series of lectures delivered at Cornell University in the fall of 1949, and since revised, this is the standard non-technical coverage of Egyptian and Babylonian mathematics and astronomy, and their transmission to the Hellenistic world. Entirely modern in its data and conclusions, it reveals the surprising sophistication of certain areas of early science, particularly Babylonian mathematics. After a discussion of the number systems used in the ancient Near East (contrasting the Egyptian method of additive computations with unit fractions and Babylonian place values), Dr. Neugebauer covers Babylonian tables for numerical computation, approximations of the square root of 2 (with implications that the Pythagorean Theorem was known more than a thousand years before Pythagoras), Pythagorean numbers, quadratic equations with two unknowns, special cases of logarithms and various other algebraic and geometric cases. Babylonian strength in algebraic and numerical work reveals a level of mathematical development in many aspects comparable to the mathematics of the early Renaissance in Europe. This is in contrast to the relatively primitive Egyptian mathematics. In the realm of astronomy, too, Dr. Neugebauer describes an unexpected sophistication, which is interpreted less as the result of millennia of observations (as used to be the interpretation) than as a competent mathematical apparatus. The transmission of this early science and its further development in Hellenistic times is also described. An Appendix discusses certain aspects of Greek astronomy and the indebtedness of the Copernican system to Ptolemaic and Islamic methods. Dr. Neugebauer has long enjoyed an international reputation as one of the foremost workers in the area of premodern science. Many of his discoveries have revolutionized earlier understandings. In this volume he presents a non-technical survey, with much material unique on this level, which can be read with great profit by all interested in the history of science or history of culture. 14 plates. 52 figures.
Posted in Science


die Tugend des Vergessens in digitalen Zeiten

Author: Viktor Mayer-Schönberger

Publisher: N.A

ISBN: 9783940432902

Category: Computer storage devices

Page: 264

View: 5467

GOOGLE VERGISST NIE! Weshalb es lebenswichtig ist zu vergessen Das digitale Zeitalter ist eines der perfekten Erinnerung: niemals zuvor haben wir so viele Informationen sammeln können wie heute. Immer größer und billiger werdende Speichermedien, neue Methoden zur Erschließung von Informationen und der Zugriff auf Daten aus aller Welt über das Netz machen dies möglich. Welche Folgen hat diese Entwicklung? Wird nur noch das als wahr gelten, was die Form digitaler Daten annimmt? Oder werden wir für das, was dauert, für Verträge und sprachliche Kunstwerke, beim Papier bleiben? Ist das Vergessen nicht ein exzellenter, evolutionär bewährter Mechanismus zur Gewichtung von Informationen? Beeinträchtigt das permanente Erinnern nicht unser Urteils- und Entscheidungsvermögen und damit unsere Entwicklungsfähigkeit? Sind aus ihrem Kontext gerissene Informationen nicht anfällig für Manipulationen? Mayer-Schönberger öffnet uns die Augen für die Gefahren der ewigen digitalen Erinnerung und zeigt die wichtige Rolle auf, die das Vergessen in unserer Geschichte gespielt hat. Er plädiert für eine genial einfache Lösung: Dateien aller Art mit einem Verfallsdatum auszustatten, damit das Gedächtnis der Menschheit nicht unter der Datenflut zusammenbricht.
Posted in Computer storage devices

Mathematische Keilschrift-Texte

Mathematical Cuneiform Texts

Author: Otto Neugebauer

Publisher: Springer-Verlag

ISBN: 3662327945

Category: Mathematics

Page: 516

View: 5178

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.
Posted in Mathematics

Mathematik im mittelalterlichen Islam

Author: J. L. Berggren

Publisher: Springer-Verlag

ISBN: 9783540766889

Category: Mathematics

Page: 200

View: 5705

Die Mathematik im mittelalterlichen Islam hatte großen Einfluss auf die allgemeine Entwicklung des Faches. Der Autor beschreibt diese Periode der Geschichte der Mathematik und bezieht sich dabei auf die arabischsprachigen Quellen. Zu den behandelten Themen gehören Dezimalrechnen, Geometrie, ebene und sphärische Trigonometrie, Algebra sowie die Approximation von Wurzeln von Gleichungen. Das Buch wendet sich an Mathematikhistoriker und -studenten, aber auch an alle Interessierten mit Mathematikkenntnissen der weiterführenden Schule.
Posted in Mathematics

Ziel und Struktur der physikalischen Theorien

Author: Pierre Duhem,Lothar Schäfer,E Mach

Publisher: N.A

ISBN: 3787332146

Category: Philosophy

Page: 374

View: 2296

Pierre Duhem (1861-1916) gehörte zu jenen Wissenschaftlern, die im ausgehenden 19. Jahrhundert an der Umbildung der Physik im großen Stil arbeiteten und damit an der Vorbereitung der wissenschaftlichen Revolution beteiligt waren, die durch Planck und Einstein herbeigeführt wurde. Duhems klassisches Werk der modernen Wissenschaftstheorie hat auf die Entwicklung des logischen Empirismus nachhaltigen Einfluß ausgeübt. Das von Duhem beigezogene reichhaltige Material und seine konzisen Fallstudien stellen eine Fundgrube für jeden dar, der sich ernsthaft mit Wissenschaftstheorie beschäftigt.
Posted in Philosophy

Warum ist E = mc2?

Einsteins berühmte Formel verständlich erklärt

Author: Brian Cox,Jeff Forshaw

Publisher: Kosmos

ISBN: 3440152065

Category: Science

Page: 256

View: 3132

E = mc2 ist die berühmteste Formel der Welt. Mit ihr brachte Einstein es auf den Punkt: Energie und Masse sind zwei Seiten derselben Medaille und die Lichtgeschwindigkeit c ist ihr Wechselkurs. Doch warum besteht dieses so einfache Verhältnis? Wie ist Albert Einstein zu diesem Schluss gekommen? Und welche Folgen für das Verständnis des Universums ergeben sich daraus? Brian Cox, Professor für Physik und in England durch seine Sendungen auf BBC sehr bekannt, hat sich zusammen mit seinem Kollegen Jeff Forshaw, Professor für theoretische Physik, die scheinbar einfache Einstein-Gleichung vorgenommen, um sie mit viel Energie ausführlich und verständlich zu erklären.
Posted in Science