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College Algebra
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AMS Chelsea Publishing
1961; 631 pp; hardcover
Volume: 354
Reprint/Revision History:
reprinted 2005
ISBN-10: 0-8218-3863-6
ISBN-13: 978-0-8218-3863-1
List Price: US$76 Member Price: US$68.40
Order Code: CHEL/354.H

At the beginning of the twentieth century, college algebra was taught differently than it is nowadays. There are many topics that are now part of calculus or analysis classes. Other topics are covered only in abstract form in a modern algebra class on field theory. Fine's College Algebra offers the reader a chance to learn the origins of a variety of topics taught in today's curriculum, while also learning valuable techniques that, in some cases, are almost forgotten.

In the early 1900s, methods were often emphasized, rather than abstract principles. In this book, Fine includes detailed discussions of techniques of solving quadratic and cubic equations, as well as some discussion of fourth-order equations. There are also detailed treatments of partial fractions, the method of undetermined coefficients, and synthetic division.

The book is ostensibly an algebra book; however, it covers many topics that are found throughout today's curriculum:

• calculus and analysis: infinite series, partial fractions, undetermined coefficients, properties of continuous functions,
• number theory: continued fractions,
• probability: basic results in probability.

Though the book is structured as a textbook, modern mathematicians will find it a delight to dip into. There are many gems that have been overlooked by today's emphasis on abstraction and generality. By revisiting familiar topics, such as continued fractions or solutions of polynomial equations, modern readers will enrich their knowledge of fundamental areas of mathematics, while gaining concrete methods for working with their modern incarnations. The book is suitable for undergraduates, graduate students, and researchers interested in algebra.

Reviews

"The author has arranged a great variety of classical, elementary material in a very original manner, which every college student or grammar-school master can still considerably profit from, even so in these days."

-- Zentralblatt MATH

From a review of the previous edition:

"This book contains more than would seem possible from the title ... the author demonstrates that he is taking pains to bring scientific rigor into accord with pedagogical considerations."

-- translation of Jahrbuch Database review cited in Zbl. Reviews

Numbers
• The natural numbers-counting, addition, and multiplication
• Subtraction and the negative
• Division and fractions
• Irrational numbers
• The imaginary and complex numbers
Algebra
• Preliminary considerations
• The fundamental operations
• Simple equations in one unknown letter
• Systems of simultaneous simple equations
• The division transformation
• Factors of rational integral expressions
• Highest common factor and lowest common multiple
• Rational fractions
• Symmetric functions
• The binomial theorem
• Evolution
• Irrational functions. Radicals and fractional exponents
• A discussion of the quadratic equation. Maxima and minima
• Equations of higher degree which can be solved by means of quadratics
• Simultaneous equations which can be solved by means of quadratics
• Inequalities
• Indeterminate equations of the first degree
• Ratio and proportion. Variation
• Arithmetical progression
• Geometrical progression
• Harmonical progression
• Method of differences. Arithmetical progressions of higher orders. Interpolation
• Logarithms
• Permutations and combinations
• The multinomial theorem
• Probability
• Mathematical induction
• Theory of equations
• The general cubic and biquadratic equations
• Determinants and elimination
• Convergence of infinite series
• Operations with infinite series
• The binomial, exponential, and logarithmic series
• Recurring series
• Infinite products
• Continued fractions
• Properties of continuous functions