Items related to Advanced Course in Algebra
Synopsis
In preparing the present work, the author has endeavored to meet the needs of Colleges and Scientific Schools of the highest rank. The development of the subject follows in the main the author's College Algebra; but numerous improvements have been introduced. Attention is especially invited to the following : 1. The development of the fundamental laws of Algebra for the positive and negative integer, the positive and negative fraction, and zero, in Chaps. I and II. In the above treatment, the author has followed to a certain extent The Number System of Algebra, by Professor H. B. Fine ; who has very courteously permitted this use of his treatise. 2. The development of the principles of equivalence of equations, and systems of equations, both linear and of higher degrees; see §§ 116-123, 182, 233-6, 396, 442, 470, 477, and 478. 3. The prominence given to graphical representation. In Chap. XIV, the student learns how to obtain the graphs of linear equations with two unknown numbers, and of linear expressions with one unknown number. He also learns how to represent graphically the solution of a system of two linear equations, involving two unknown numbers, and sees how inde- terminate and inconsistent systems are represented graphically. The graphical representation of quadratic expressions, with one unknown number, is taken up in § 465 ; and, in § 467, the graphical representation of equal and imaginary roots. The principles are further developed for simultaneous quad- ratics, in §§ 482 and 483; and for expressions of any degree, with one unknown number, in §§ 744 and 745. At the end of Chap. XVIII, the student is taught the graphi- cal representation of the fundamental laws of Algebra for pure imaginary and complex numbers. In Chap. XXXVII, the graphical representation is given of Derivatives (§ 751), of Multiple Boots (§ 755), of Sturm's Theorem (§ 762), and of a Discontinuous Function (§ 766). 4. In Chap. VII, there are given the Remainder and Factor Theorems, and the principles of Symmetry. 5. In Chap VIII will be found every method of factoring which can be done advantageously by inspection, including factoring of symmetrical expressions. In this chapter is also given Solution of Equations by Factoring (§ 182). 6. In the earlier portions of Chap. XI, the pupil is shown that additional solutions are introduced by multiplying a fractional equation by an expression which is not the L.C.M. of the given denominators ; and is shown how such additional solutions are discovered. 7. In §§ 264 and 265, the student is taught how to find the values of expressions taking the indeterminate forms ^j — > Ox 00, and oo — oo. 8. All work coming under the head of the Binomial Theo- rem for positive integral exponents is taken up in the chapter on Involution. 9. In developing the principles of Evolution, all roots are restricted to their principal values. 10. In the examples of § 398, the pupil is taught to reject all solutions which do not satisfy the given equation, when the roots have their principal values. 11. The development of the theory of the Irrational Num- ber, and its graphical representation (§§ 399-406). 12. The development of the fundamental laws of Algebra for Pure Imaginary and Complex Numbers (Chap. XVIII). 13. The use of the general form ax^ + 6a; + c = 0, in the theory of quadratic equations (§§ 454-6). 14. The discussion of the maxima and minima values of quadratic expressions (§ 461). 15. The chapter on Convergency and Divergency of Series (Chap. XXVI). 16. In Chap. XXVIII is given Euler's proof of the Binomial Theorem, for any Eational Exponent. 17. The solution of logarithmic equations (§ 604). 18. The proof of the formula for the number of permuta- tions of n different things, taken r at a time (§ 624).
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