The chapter on Factors has been made as complete as possible for an elementary text-book, with a view to shorten subsequent work. The easy method of resolving quadratic trinomials into factors, whether the coefficient of the square of the letter involved is unity or greater than unity, and the Factor Theorem, explained on page 102, will be found of very great service in abridging algebraic processes. Examples of short methods for finding the highest common factor of compound expressions are given on page 118; and examples of short methods for solving quadratic equations by resolving them into factors are given on pages 272 and 273. A five-place table of logarithms is placed at the end of the book instead of a four-place table. Five-place logarithms are in common use for practical calculations, and are required by most colleges and science schools for the solution of problems set in entrance examination papers. The exercises throughout the book are carefully graded. They are sufficiently varied and interesting, and are not so difficult as to discourage the learner, or so easy as to deprive him of the satisfaction of well-earned success. The author has spared no pains to make this a model. text-book in subject-matter and mechanical execution. The remarkable favor with which his other Algebras have been received is shown by the fact that nearly a million copies have already been sold, and the sale continues to increase from year to year. The author trusts that this new candi date for favor will have the same generous reception, and be found to meet fully the requirements of the recent advance in the science and method of teaching Elementary Algebra. The author is under obligations to many teachers for valuable suggestions, and he will be thankful for corrections or criticisms. EXETER, N. H., June, 1898. G. A. WENTWORTH. NOTICE TO TEACHERS. Pamphlets containing the answers will be furnished without charge to teachers for their classes, on application to GINN & COMPANY, Publishers. NEW SCHOOL ALGEBRA. CHAPTER I. DEFINITIONS AND NOTATION. Numbers and Number-Symbols. 1. Algebra. Algebra, like Arithmetic, treats of numbers. 2. Units. In counting separate objects or in measuring magnitudes, the standards by which we count or measure are called units. Thus, in counting the boys in a school, the unit is a boy; in selling eggs by the dozen, the unit is a dozen eggs; in selling bricks by the thousand, the unit is a thousand bricks; in expressing the measure of short distances, the unit is an inch, a foot, or a yard; in expressing the measure of long distances, the unit is a rod, or a mile. 3. Numbers. numbers. Repetitions of the unit are expressed by A single unit and groups of units formed by successive additions of a unit may be represented as follows: These representative groups are named one, two, three, four, five, six, seven, eight, nine, ten; and are known collectively under the general name of numbers. It is obvious that these representative groups will have the same meaning, whatever the units may be that are counted. 4. Quantities. A number of specified units of any kind. is called a quantity; as 4 pounds, 5 oranges. NOTE. Quan.ities are often called concrete numbers, the adjective concrete being transferred from the units counted to the numbers that count them; but a number signifies the times a unit is taken, whether the unit is expressed or understood, and is always abstract. Thus, 4 barrels of flour means 4 times 1 barrel of flour; and 10 cords of wood means 10 times 1 cord of wood. Instead of groups of 5. Number-Symbols in Arithmetic. straight marks, we use in Arithmetic the arbitrary symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, called Arabic numerals, for the num*bers one, two, three, four, five, six, seven, eight, nine. The next number, ten, is indicated by writing the figure 1 in a different position, so that it shall signify not one, but ten. This change of position is effected by introducing a new symbol, 0, called nought or zero, and signifying none. All succeeding numbers up to the number consisting of 10 tens are expressed by writing the figure for the number of tens they contain in the second place from the right, and the figure for the number of units besides in the first place. The hundreds of a number are written in the third place from the right. The thousands are written in the fourth place from the right; and so on. 6. Number-Symbols in Algebra. Algebra employs the letters of the alphabet in addition to the figures of Arithmetic to represent numbers. The letters of the alphabet are used as general symbols of numbers to which any particular values may be assigned. In any problem, however, a letter is understood to have the same value throughout the problem. 7. Terms Common to Arithmetic and Algebra. Terms common to Arithmetic and Algebra, as addition, sum, subtraction, minuend, subtrahend, difference, etc., have the same meaning in both; or an extended meaning in Algebra consistent with the sense attached to them in Arithmetic. |