PREFACE This book contains a brief review of the fundamental operations of algebra followed by a thorough presentation of the topics usually included in the work of the third half year. The material is so arranged that the choice of topics for review or advance study may be easily made. The book contains not only a large number of practical problems but also practical applications of graphs. Material for drill in the manipulation of exponents and radicals will be found in Chapter VII. Great pains have been taken to make the subject of logarithms accessible to beginners. The most difficult part in learning to compute with logarithms is the process of “interpolation.” Of the various arrangements of logarithmic tables which have been suggested, we have selected the one which renders interpolation easiest. The chapter on logarithms is introduced earlier than usual, on account of the great practical importance of the subject. If desired, it can be taken up later, after Chapter VI. The concept of a function does not receive isolated and abstract treatment; it is presented as a fundamental idea in proportion, variation, and graphics. Its connection with problems of everyday life is firmly established. The aims of the First Year Book have been kept in view in the preparation of this text. Emphasis is thrown upon clearness of exposition and the use of expressions which recall the axiomatic processes involved. Continued stress has been laid upon oral exercises. Several of the practical problems given in the text were suggested by the perusal of an English book, T. Percy Nunn's Exercises in Algebra (including Trigonometry), Part I, 1913. The authors have received help from several teachers. Special mention should be made of Mr. E. A. Cummings of the North Side High School in Denver, who has worked most of the exercises and offered valuable suggestions. FLORIAN CAJORI. TABLE OF CONTENTS I. ELEMENTARY DEFINITIONS AND OPERATIONS Order of Fundamental Operations Simultaneous Linear Equations . Fractional and Negative Exponents Fractions, their Multiplication and Division Highest Common Factor and Least Common Multiple Addition and Subtraction of Fractions II. MORE ADVANCED THEORY AND OPERATIONS Remainder Theorem and Factor Theorem Binomial Theorem (preliminary treatment) Systems of Linear Equations. Determinants III. PROPORTION, VARIATION, Function Different Modes of Variation . Problems in Proportion and Variationi Graphs Exhibiting Empirical Data V. QUADRATIC EQUATIONS AND THEIR PROPERTIES Relations between Roots and Coefficients Graph of the Quadratic Equation ax2 + bx+c=y VI. SYSTEMS OF EQUATIONS SOLVABLE BY QUADRATICS A System of Two Equations, One Linear A System of Two Equations, Both Quadratic |