Algebra in Ancient and Modern Times
About this Title
V. S. Varadarajan, University of California, Los Angeles, Los Angeles, CA
Publication: Mathematical World
Publication Year 1998: Volume 12
ISBNs: 978-0-8218-0989-1 (print); 978-1-4704-2478-7 (online)
MathSciNet review: MR1621032
MSC: Primary 01A05; Secondary 11-03, 12-03
From the reviews:
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.
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 quaternions. It is geared toward undergraduates who have no background in calculus.
V. S. Varadarajan is a professor of mathematics at the University of California, Los Angeles.
Undergraduate mathematics majors, graduate students, research mathematicians and historians interested in the history of mathematics.
Table of Contents
Some history of early mathematics
- 1. Eucild–Archimedes–Diophantus
- 2. Pythagoras and the Pythagorean triplets
- 3. Āryabhaṭa–Brahmagupta–Bhāskara
- 4. Irrational numbers: construction and approximation
- 5. Arabic mathematics
- 6. Beginnings of algebra in Europe
- 7. The cubic and biquadratic equations
Solution of the cubic and biquadratic equations
Some themes from modern algebra
- 10. Numbers, algebra, and the real world
- 11. Complex numbers
- 12. Fundamental theorem of algebra
- 13. Equations of degree greater than four
- 14. General number systems and the axiomatic treatment of algebra