То secure an intelligent solution of arithmetical problems, students should become familiar with the generalizations of 52, 53, 54, and 55, relating to integers; of 113, 114, and 116, relating to common and decimal fractions; and of 213 and 214, relating to percentage and its applications. Students should be required to exhibit the application of these principles to particular problems as they are successively solved, until the general law of increase and decrease of numbers has been thoroughly mastered. Topical rules and special analyses are soon forgotten or inextricably blended; but the constant iteration and reiteration of a few general principles, applicable to all classes of problems, are likely to Leave a clear and permanent impression. By thus generalizing many perplexing arbitrary rules into a few imple common-sense rules, many independent bases of reasoning into one common basis, it is possible for business colleges in six months to impart to their students a more thorough and enduring knowledge of arithmetic than can be taught in twice that time under the old system of presentation. ARITHMETIC. INTRODUCTION. 1. Arithmetic treats of numbers. 2. A Number is an expression of a quantity by means of characters or figures. 3. The Unit of a number is one of the things which it expresses. Thus, one inch is the unit of nine inches, one mile of sixty miles. REMARK 1.-Numbers are classified as concrete when they name the kind of units which they represent, as five gallons, three pounds, six books; and as abstract when the kind of units is not named, as five, three, six. REM. 2.-Numbers are said to be like when they express the same kind of units, as two years and nine years; and unlike when they express different kinds of units, as three miles and four quarts. REM. 3.-Numbers are further distinguished as integers when they express whole units, and as fractions when they express parts of a whole unit. NUMERATION AND NOTATION. 4. Numeration is the art of reading numbers when expressed by figures. 5. Notation is the art of writing numbers. REM. The method of expressing numbers by figures is called Arabic Notation: and by letters, Roman Notation. 6. The Arabic method embraces the following ten figures: 0 23 4 5 6 7 8 9 Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. Each of the above figures, when standing alone, represents the number written under it. Consequently, nine is the greatest number that can be expressed by one figure; and numbers greater than nine are expressed by the artifice of increasing the local value of each figure tenfold by placing one figure to the right of it; a hundredfold by placing two figures to the right of it; a thousandfold by placing three figures to the right of it; etc. REM. The distance of a figure from the right of a number is called its order; thus, the first figure at the right is the first, or units' order; the second figure, the second, or tens' order; the third figure, the third, or hundreds' order; the fourth figure, the fourth, or thousands' order, etc.; ten units of any order equaling one unit of the next order on the left. Hence, the representative value of a figure increases tenfold for each order that it is moved to the left, and decreases tenfold for each order that it is moved to the right. For the names and position of the several orders, see DECIMAL NOTATION Table, 121. ILLUSTRATIVE EXERCISE. 7. Read 7862. EXPLANATION.-7862 is read seven thousand eight hundred and sixty-two; the ¡ocal value of 7 being thousand because it has three figures at its right, the local value of 8 being hundred because it has two figures at its right, the local value of 6 being tens because it has one figure at its right, and the local value of 2 being its simple value (two) because it has no figure at its right. Read, or express by written words, the following numbers: 8. With numbers greater than hundreds, the further artifice is employed of considering them in periods or sets of three figures each. The first set of three figures at the right is named the units' period; |