Attention is called to the unusually large number of problems contained in this book as well as in the other three books of this series. It has been thought best to present the subjects of Longitude and Time, Standard Time, Square Root, Partial Payments, Foreign Exchange, and the Metric System in the body of the book rather than to reserve them for an appendix. This does not, of course, prevent the teacher from omitting these subjects if it seems desirable to do so, as the question of their omission is one that any capable teacher of grammar grades should be fully competent to decide. Par In the preparation of the manuscript, the authors desire to acknowledge the helpful assistance of many teachers who have kindly offered suggestions and criticisms of great value and supplied many of the problems that the book contains. ticular acknowledgment should be made of the valuable assistance of Miss A. J. Meadowcroft, Principal of the Burnham School, Haverhill, Mass., Miss Gertrude E. Bigelow, Supervisor of Practice, of the Boston Normal School, Mr. John D. Billings, Principal of the Webster School, Cambridge, Mass., Mr. James W. Applebee, Ingalls School, Lynn, Mass., and Mr. Thomas W. Davis, formerly Principal of the Howard School, Cambridge, Mass. FOURTH BOOK INTRODUCTION 1. TO THE PUPIL. - Arithmetic is the science that treats of numbers and their use. Every business transaction, no matter if it is but the purchase or sale of a paper of pins, a pound of nails, or a half yard of tape, requires the use of arithmetic. To be able to apply it is a great satisfaction and convenience. In order to use it successfully, you must study and practice it long enough and carefully enough to fix its facts and principles firmly in mind. 2. The occasion for the use of arithmetic is the measurement of quantities of things of various kinds. 3. A single thing, or a single quantity taken as a standard for measuring quantities of the same kind, is called a unit. A boy, a cent, a yard, a pailful, a dozen, are units. 4. To show how great any quantity is when compared with a unit of its kind, we use numbers. 5. A number that refers to some particular kind of object or unit of measure is called a concrete number. 10 bottles, 6 yards, 3 pens, 4 cents, $2, are concrete numbers. 6. A number that does not refer to any particular kind of object or unit of measure is called an abstract number. Forty-five, six hundred eleven, 347, are abstract numbers. 7. The knowledge of numbers includes a knowledge of their expression, combination, and relations. 8. The oral expression of numbers is called numeration. 9. The expression of numbers by written symbols — letters or figures is called notation. 10. A single unit is expressed by the word one or the symbol (figure) 1. One with one is expressed by the word two, or the figure 2. The simplest groups of numbers are expressed in words and in figures as follows: One, two, three, four, five, six, seven, eight, nine. 1, 2, 3, 4, 5, 6, 7, 8, 9. The numbers from one to nine are called digits. One with nine is ten; ten is expressed in figures thus, 10. The figure 0 is called zero or naught; it stands for no number. Groups of tens are expressed in words and figures as follows: Ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. 10, 20, 30, 40, 50, 60, 70, 80, 90. Ten tens are one hundred, expressed in figures thus, 100. Ten hundreds are one thousand, expressed in figures thus, 1000. 11. By taking tens of smaller groups, we form larger groups, hundreds, thousands, millions, and so on. This system of numbering by tens is called the decimal system of notation. 1 unit is called a unit of the first order; 1 ten is called a unit of the second order; 1 hundred a unit of the third order, etc. 12. Units with tens are expressed as follows : |