College AlgebraAt 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. |
Contents
| 24 | |
PAGE | 44 |
PART SECONDALGEBRA | 79 |
THE FUNDAMENTAL OPERATIONS | 93 |
SIMPLE EQUATIONS IN ONE UNKNOWN LETTER | 110 |
SYSTEMS OF SIMULTANEOUS SIMPLE EQUATIONS | 127 |
THE DIVISION TRANSFORMATION | 155 |
FACTORS OF RATIONAL INTEGRAL EXPRESSIONS | 176 |
INDETERMINATE EQUATIONS OF THE FIRST DEGREE | 342 |
RATIO AND PROPORTION VARIATION | 347 |
ARITHMETICAL PROGRESSION | 354 |
GEOMETRICAL PROGRESSION | 357 |
HARMONICAL PROGRESSION | 362 |
SIONS OF HIGHER ORDERS INTERPOLATION | 364 |
LOGARITHMS | 374 |
PERMUTATIONS AND COMBINATIONS | 393 |
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE | 196 |
RATIONAL FRACTIONS | 213 |
THE BINOMIAL THEOREM | 252 |
EVOLUTION | 260 |
IRRATIONAL FUNCTIONS RADICALS | 271 |
QUADRATIC EQUATIONS | 298 |
DISCUSSION OF THE QUADRATIC EQUATION | 304 |
SIMULTANEOUS EQUATIONS WHICH CAN BE SOLVED BY MEANS OF QUADRATICS | 317 |
INEQUALITIES | 340 |
THE MULTINOMIAL THEOREM | 408 |
PROBABILITY | 409 |
MATHEMATICAL INDUCTION | 424 |
THEORY OF EQUATIONS XXX CUBIC AND BIQUADRATIC EQUATIONS | 425 |
DETERMINANTS AND ELIMINATION XXXII CONVERGENCE OF INFINITE SERIES | 520 |
OPERATIONS WITH INFINITE SERIES | 539 |
THE BINOMIAL EXPONENTIAL AND LOGARITHMIC SERIES | 553 |
RECURRING SERIES | 560 |
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Common terms and phrases
a₁ ab² algebraic approach arithmetical progression arranged ascending powers ax² b₁ binomial binomial theorem called coefficients common factor complex numbers convergent corresponding definition denominator denote digits divide divisor equal exactly divisible exponent expression factor x formula given equation graph Hence highest common factor identity imaginary infinite integers integral function irrational number leading term less limit logarithms lower degree method multiply negative nth root obtain pair of equations partial fractions polynomial positive number prime factors proper fraction prove quadratic quotient R₁ radical radicand rational number real numbers reduced remainder result root of f(x sequence simple equations Simplify solution Solve square root Substituting subtract symmetric functions theorem tion transformed u₁ unknown letters values vanishes variable x²y x²y²