# College Algebra

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*Henry Burchard Fine*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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 & Endorsements

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

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

#### Table of Contents

# Table of Contents

## College Algebra

- Cover Cover11
- Title page i2
- Preface iii4
- Contents vii8
- The natural numbers–counting, addition, and multiplication 110
- Subtraction and the negative 1625
- Division and fractions 2736
- Irrational numbers 3948
- The imaginary and complex numbers 7079
- Preliminary considerations 7988
- The fundamental operations 93102
- Simple equations in one unknown letter 110119
- Systems of simultaneous simple equations 127136
- The division transformation 155164
- Factors of rational integral expressions 176185
- Highest common factor and lowest common multiple 196205
- Rational fractions 213222
- Symmetric functions 245254
- The binomial theorem 252261
- Evolution 260269
- Irrational functions. Radicals and fractional exponents 271280
- Quadratic equations 298307
- A discussion of the quadratic equation. Maxima and minima 304313
- Equations of higher degree which can be solved by means of quadratics 309318
- Simultaneous equations which can be solved by means of quadratics 317326
- Inequalities 340349
- Indeterminate equations of the first degree 342351
- Ratio and proportion. Variation 347356
- Arithmetical progression 354363
- Geometrical progression 357366
- Harmonical progression 362371
- Method of differences. Arithmetical progressions of higher orders. Interpolation 364373
- Logarithms 374383
- Permutations and combinations 393402
- The multinomial theorem 408417
- Probability 409418
- Mathematical induction 424433
- Theory of equations 425434
- The general cubic and biquadratic equations 483492
- Determinants and elimination 492501
- Convergence of infinite series 520529
- Operations with infinite series 539548
- The binomial, exponential, and logarithmic series 553562
- Recurring series 560569
- Infinite products 564573
- Continued fractions 566575
- Properties of continuous functions 577586
- Answers 591600
- Index 627636
- Back Cover Back Cover1642