554 PREFACE THE Advanced Course of the High School Algebra contains a review of all topics treated in the Elementary Course, together with such additional topics as are required to make it amply sufficient to meet the entrance requirements of any college or technical school. Its development is based upon. the following important considerations: 1. The pupil has had a one year's course in algebra, involving constant application of its elementary processes to the solution of concrete problems. This has invested the processes themselves with an interest which now makes them a proper object of study for their own sake. 2. The pupil has, moreover, developed in intellectual maturity and is, therefore, able to comprehend processes of reasoning with abstract numbers which were entirely beyond his reach in the first year's course. This is particularly true if, in the meantime, he has learned to reason with the more concrete forms of geometry. In consequence of these considerations, the treatment throughout is from a more mature point of view than in the Elementary Course. The principles of algebra are given in the form of theorems the proofs of which are based upon a definite set of axioms. As in the Elementary Course, the important principles are used at once in the solution of concrete and interesting problems, which, however, are here adapted to the pupil's greater maturity and experience. But relatively greater space and emphasis are given to the manipulation of standard algebraic M306421 forms, such as the student is likely to meet in later work in mathematics and physics, and especially such as were too complicated for the Elementary Course. The division of the High School Algebra into two distinct courses has made it possible to give in the Advanced Course a more thorough treatment of the elements of algebra than could be given if the book were designed for first-year classes. It has thus become possible to lay emphasis upon the pedagogic importance of viewing each subject a second time in a manner more profound than is possible on a first view. Attention is specifically called to the following points: The clear and simple treatment of equivalent equations in Chapter III. The discussion by formula, as well as by graph, of inconsistent and dependent systems of linear equations, pages 40 to 44. The unusually complete treatment of factoring and the clear and simple exposition of the general process of finding the Highest Common Factor, in Chapter V. The careful discrimination in stating and applying the theorems on powers and roots in Chapter VI. The unique treatment of quadratic equations in Chapter VII, giving a lucid exposition in concrete and graphical form of distinct, coincident, and imaginary roots. The concise treatment of radical expressions in Chapter X, and especially an innovation much needed in this connection the rich collection of problems, in the solution of which radicals are applied. CHICAGO AND BOSTON, April, 1908. H. E. SLAUGHT. |