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This book was written as a textbook to be used in the standard American university and college courses devoted to the theory of equations. As such it is elementary in character and, with few exceptions, contains only material ordinarily included in texts of this kind. But the presentation is made so explicit that the book can be studied by students without a teacher’s help.
Everything that is stated in the text is presented with full development, and nowhere is reference made to results that are beyond the scope of this book. This accounts for the fact that the book, though containing chiefly the same matters as other currently used texts, surpasses them in size. A few topics that might be omitted without harm are marked by black stars. Numerous problems are added after each significant section. For the most part they are simple exercises, but the more difficult among them are marked by asterisks.
In four chapters the exposition differs considerably from custom. In the chapter on complex numbers the superficial approach so common in many books is replaced by a simple and yet thorough presentation of the theory of complex numbers. The author’s experience shows that students, almost without exception, follow this presentation without difficulty.
In the chapter on separation of roots the author gives a very efficient method for separating of real roots, much superior in practice to that based on Sturm’s Theorem. He believes that no other book mentions this method, which he invented many years ago and has been teaching to his students for a number of years.
In the chapter on numerical computation of roots Horner’s method is presented in the original form, including the process of contraction, which unfortunately has disappeared from American texts. Also, a thorough examination is made of the error caused by contraction.
Determinants are introduced not by formal definition, as is usual, but by their characteristic properties, following Weierstrass. The advantage of this is apparent, for example, in the proof of the theorem of multiplica¬tion of determinants. Some elementary notions about the algebra of matrices are also developed in this chapter.
Certain matters because of their intrinsic difficulty are referred to the appendixes. Appendix I deals with the fundamental theorem of algebra.
The author chose as the most intuitive proof, and therefore most suitable for beginners, the fourth proof by Gauss.
Appendix II gives the proof of a theorem of Vincent on which is based the method of separation of roots mentioned above.
Appendixes III and IV were added on the advice of Professor S. P. Timoshenko as likely to interest engineering students. Appendix III is devoted to a simple derivation of criteria for an equation to have all roots with negative real part. Appendix IV deals with iterative solution of the frequency equation.
Appendix V gives an explanation of Graeffe’s method for computing roots and is of particular value in the calculation of the imaginary roots of an equation.
Preface.
Complex Numbers.
Polynomials in One Variable.
Algebraic Equations and Their Roots.
Limits of Roots. Rational Roots.
Cubic and Biquadratic Equations.
Separation of Roots.
The Theorem of Sturm.
Approximate Evaluation of Roots.
Determinants and Matrices.
Solution of Linear Equations by Determinants. Some Applications of Determinants to Geometry.
Symmetric Functions.
Elimination.
Appendixes
The Fundamental Theorem of Algebra.
On the Theorem of Vincent.
On Equations Whose Roots Have Negative Real Part.
Iterative Solution of the Frequency Equation.
Graeffe’s Method.
Answers.
Index