CONTENTS. IV. To reduce a compound fraction to a simple one.... V. To multiply a whole number by a fraction.. VI. To divide a whole number by a fraction VII. To multiply a fraction by a whole number. VIII. To divide a fraction by a whole number. XI. To reduce fractions to the same denominator.....36 XVI. Multiplication of whole Numbers and Decimal Parts. 88 ARITHMETIC. ARITHMETIC teaches to compute numbers by means of certain signs and characters. The word Arithmetic is derived from the Greek arithmos, which means number. The fundamental rules of Arithmetic are Notation, Enumeration, Addition, Subtraction, Multiplication, and Division. The scholar should become thoroughly acquainted with these rules before leaving them. I. Notation. Notation is the method of expressing numbers by characters or figures. Different kinds of notation have been employed by different nations. The Greeks and Romans employed the letters of their alphabets to express numbers. The Arabic method of notation employs these ten characters,-1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, O cipher. It will be necessary to consider, in this work, only the Roman and Arabic methods of expressing numbers. 1. ROMAN METHOD OF NOTATION. In this method, seven letters are used to express numbers, viz. I, V, X, L, C, D, M; I denoting 1; V, 5; X, 10; L, 50; C, 100; D, 500; and M, 1000. I are sometimes used for D, and CIO for M. 2 TABLE. ....10 C ....100 M........1000 X.. ......10000 ..... II...2 XX....20 CC.....200 MM or II.2000 XX ..20000 ..30000 .4000 XXXX....40000 .......5000 I3333 ...50000 V...5 L ...50 D ......500 V ..6000 C. ...7000 M ..8000 MM ...9000l 2. ARABIC METHOD OF NOTATION. .....100000 ..1000000 ....2000000 In this method of notation, the following characters are employed, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These characters possess two values, primitive and local. The primitive value of a figure is that which it has when compared with another of the same denomination. Thus, 1, 2, 3, 4, being units, 2 has twice the value of 1; 3 has three times the value of 1, &c. If 1, 2, 3, 4, are considered any other denomination than units, but are all of the same kind, then 2 has twice the value of 1; 3 has three times the value of 1, as before. The local value of a figure is that which it has, when compared with another figure of a different denomination. Thus, 1, 1, 1, taken as different denominations, have three different values, according to their locality. The local value of figures increases from the right hand to the left, in a ten-fold proportion. If we wish to express the number of days in a year, in Arabic characters, thus, 365, the first right-hand figure, 5, represents five days; the next left-hand figure, 6, represents sixty days; and the figure 3 represents three hundred days. If we wish to express in figures, one dollar, twenty-five cents, and five mills, thus, 1,255, the right-hand figure 5 represents five mills; the next left-hand figure, 5, represents five cents; the next left-hand figure, 2, represents twenty cents; and the figure 1, one hundred cents, or one dollar. On the other hand, the local value of figures diminishes in a ten-fold proportion, from the left hand to the right. II. Enumeration. Enumeration teaches to express in language the value of figures, by ascertaining the local value of each figure in the given sum. The principles of Enumeration may be best learned from the following Should the learner find it necessary to employ a larger number of figures than are given in the preceding table, he can proceed with trillions, as he did with billions, making use of quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, &c. III. Addition. 1. If John has one marble in his hand, and four in his pocket, how many has he? 2. If William find 4 eggs in one nest, and 5 in another, how many will he find? 3. If a man sell 5 bushels of apples to one man, and 9 to another, how many bushels will he sell? |