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Algebra in Ancient and Modern Times
V. S. Varadarajan, University of California, Los Angeles, CA
A co-publication of the AMS and Hindustan Book Agency.
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Mathematical World
1998; 142 pp; softcover
Volume: 12
ISBN-10: 0-8218-0989-X
ISBN-13: 978-0-8218-0989-1
List Price: US$29
Member Price: US$23.20
Order Code: MAWRLD/12
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This text offers a special account of Indian work in diophantine equations during the 6th through 12th centuries and Italian work on solutions of cubic and biquadratic equations from the 11th through 16th centuries. The volume traces the historical development of algebra and the theory of equations from ancient times to the beginning of modern algebra, outlining some modern themes such as the fundamental theorem of algebra, Clifford algebras, and quarternions. It is geared toward undergraduates who have no background in calculus.

For other wonderful titles written by this author see: Euler through Time: A New Look at Old Themes, Supersymmetry for Mathematicians: An Introduction, The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysis, and The Selected Works of V.S. Varadarajan.

This book is copublished with the Hindustan Book Agency (New Delhi) and is distributed worldwide, except in India, Sri Lanka, Bangladesh, Pakistan, and Nepal by the American Mathematical Society.

Readership

Undergraduate mathematics majors, graduate students, research mathematicians and historians interested in the history of mathematics.

Reviews

"The book was written for freshmen students who should learn algebra by its history. So the topics mentioned above are treated from a mathematical as well as a historical point of view. The material is presented in a way that students should see how ideas have emerged. In some cases a rough look forward to the modern development is given. Many sections are supplemented by notes and exercises, which contain a lot of mathematics as well as additional historical facts. The book is completed by a very short list of references and an index."

-- Zentralblatt MATH

"This is a fine book on two counts. First ... there is the singularly excellent treatment of the solution of biquadratic equations. Second, it paints a strong picture of mathematics as a very long sequence of accomplishments, each building on the ones before, in a way that beginning mathematicians can understand and appreciate it. It paints the picture in a concise and economical style, the style that mathematicians find elegant. I would particularly recommend Algebra in Ancient and Modern Times to strong high school students, to high school algebra teachers, to people who want a history of mathematics with a lot of mathematics in the history, and to anyone who needs to know how to find an analytic solution to a nasty fourth degree polynomial."

-- MAA Online

"Varadarajan spins a captivating tale, and the mathematics is first-rate. The book belongs on the shelf of any teacher of algebra ... The great treasure of this book is the discussion of the work of the great Hindu mathematicians Aryabhata (c.476-550), Brahmagupta (c.598-665), and Bhaskara (c.1114-1185). Teachers of mathematics history will be especially interested in Varadarajan's exposition of the remarkable cakravala, an algorithm for solving \(X^2 - NY^2= \pm 1\). The book contains many exercises that enhance and supplement the text and that also include historical information. Many of the exercises ask readers to apply the historical techniques. Some of the exercises are quite difficult and will challenge any student."

-- Mathematics Teacher

"Varadarajan gives us nice treatment of the work of Indian mathematicians on the so-called Pell equation as well as a very detailed yet teachable discussion of the standard story of the solution of cubic and quartic equations by del Ferro, Tartaglia, Cardano, and Ferrari in sixteenth-century Italy."

-- Mathematical Reviews

Table of Contents

Some history of early mathematics
  • Euclid-Archimedes-Diophantus
  • Pythagoras and the Pythagorean triplets
  • Āryabhaṭa-Brahmagupta-Bhāskara
  • Irrational numbers: construction and approximation
  • Arabic mathematics
  • Beginnings of algebra in Europe
  • The cubic and biquadratic equations
Solution of the cubic and biquadratic equations
  • Solution of the cubic equation
  • Solution of the biquadratic equation
Some themes from modern algebra
  • Numbers, algebra, and the real world
  • Complex numbers
  • Fundamental theorem of algebra
  • Equations of degree greater than four
  • General number systems and the axiomatic treatment of algebra
  • References
  • Chronology
  • Index
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