36. From this it is seen that ten units of a lower order make one unit of the next higher order; the system of notation is therefore called the Decimal System, from the Latin, decem, ten. WRITTEN EXERCISES. 1. Write and read two units of thousands ? In 5 thousands? In the second order, and four of the 60 tens? In 3 millions ? first. 9. What is the result of chang. 2. Write and read four units of ing a figure one place to the left the third order, and six of the Two places? Three places ? gecond. 10. What is the result of chang3. Write and read nine units of ing a figure one place to the right? the third order, and three of the Two places? Three places ? Girst. 11. What is the result of plac4. Write and read three units ing one naught to the right of one of the fifth order, six of the third, or more figures? Two naughts ? and seven of the first. Three naughts? 5. Write and read five units of 12. What place is occupied by the sixth order, four of the third, tens figure? Hundreds figure? and eight of the second. Ten-thousands ? Millions? Bil6. How many units in one ten? | lions? In 2 tens? In 3 tens? In 4 hun 13. What is the denomination dreds ? of a figure in third place? In fifth 7. How many tens in 1 hun- place? In fourth place? In sev. dred ? In two hundreds ? 3 hun- enth place? In ninth place? In dreds ? 2 thousands ? sixth place? In eighth place? 8. How many hundreds in 3 37. The table which has been given enables us to read a numerical expression consisting of twenty-one figures; the periods which follow them in order are as follows: Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tertio-decillions, Quarto-decillions, Quinto-decillions, Sexto-decillions, Septo-decillions, Octo-decillions, Nono-decillions, Vigi'lions. With these, and those already given, we can write and read a numerical expression consisting of sixty-six places. NOTES.-1. The first of the nine Arabic characters are called digits, from the Latin word digitus, a finger, owing to the fact that the ancients reck. onel by counting the fingers. They are also called significant figures, because they always indicate a definite number of units. The character 0, called zero, cipher, or naught, always indicates an absence of units. 2. The Arabic Notation is named from the Arabs, who introduced it into Europe by their con quest of Spain during the 11th century. The Arabs obtained it from the Hindoos, by whom it was probably invented more than 2000 years ago. 3. There are three theories for the origin of the Arabic characters, tor wbich see Brooks's Philosophy of Arithmetie. THE DECIMAL SCALE. 38. Since figures express different orders of units in dif ferent places, they are said to have two kinds of values ; a simple value and a local value. 39. The Simple Value of a figure is its value when it stands alone or in units place. The Local Value of a figure s its value when in some other place than units. 40. The value of a figure is increased tenfold for every place it is moved from right to left; and is decreased ten. fold for every place it is moved from left to right. This law is called the scale of the system. 41. Since the value of terms decreases from left to right at a tenfold rate, if we fix the place of units by a point (.), we may extend the decimal scale to the right of units. 42. The first place on the right of the point will be onetenth of units or tenths, the second place one-tenth of tenths, or hundredths, the third place, thousandths, etc. 43. Such terms are called decimals, and the point is called the decimal point. The expression 48.37 is read 48 units, 3 tenths, and 7 hundredths, or 48 and 37 hundredths. 44. The Money of the United States is expressed by the decimal system. The dollar is the unit, and is indicated by the symbol $. The first place at the right of the decimal point is called dimes; the second place, cents, etc. 45. Dimes and Cents are usually read as a number of cents. Thus, $4.65 is read 4 dollars and 65 cents; ang $72.48 is read 72 dollars and 48 cents. The Decimal system of numeration had its origin in the practice, common to all nations, of counting by groups of tens. EXAMPLES FOR PRACTICE. Read the following: 1. 12.5 5. 46.375 9. $125.45 2. 36.25 6. 89.625 10. $437.05 47.75 7. $16.25 11. $548.475 4. 86.50 18. $85.35 12. $768.605 Write the following: 1. Fourteen and five tenths. 5. Sixty dollars and seven cents. 2. Eighty-four and twenty-five 6. Eighty-seven dollars and hundredths. twenty-five cents. 3. Two hundred and three, and 7. Four hundred and fifty dol. sixty-seven hundredths. lars, and fifty cents. 4. Seven hundred and ninety 8. Eight hundred and sixty. six, and eight hundred seventy-four dollars, thirty-seven cents five thousandths. and five mills. NOTE.—1t is recommended that only advanced or finishing classes study the English Method of Nugeration. ENGLISH METHOD OF NUMERATION. 46. The method of numeration by dividing numbers into periods of three figures each, is called the French Method. 47. The English Method uses periods of six figures each, calling the first period units, the second millions, the third billions, the fourth trillions, etc. 48. The places in each period are units, tens, hundreds, thousands, tens of thousands, hundreds of thousands. The method is represented in the following table : NOTE.-The French Method is used throughout the United States, France, etc., and being much more convenient, is very likely to supersede the other method in England. EXAMPLES FOR PRACTICE. 1. Write by the English method, one million; one billion; one trillion; one quadrillion. ROMAN NOTATION. 49. The Roman Method of Notation employs seven letters of the Roman alphabet. Thus, I represents one ; V, five ; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand. 50. To express other numbers these characters are combined according to the following principles : 1. Every time a letter is repeated its value is repeated. 2 When a letter is placed before one of greater value, the DIFFERENCE of their value is the number represented. 3. When a letter is placed after one of a greater value, the sum of their values is the number represented 4. A dush placed over an expression increases its value a thousand fold. Thus, VII denotes seven thousand. 51. These principles are exhibited in the following table, which the pupil will examine carefully : BOMAN TABLE. VII . . I One. XXX . Thirty. Two. XL Forty. Fifty. Four. LX Sixty. Five. LXX Seventy. Ninety. One hundred. Eight. Two hundred. Five hundred. Six hundred. Eleven. DCCCC Nine hundred One thousand. Fifteen. MM Two thousand XIX Nineteen. MCLX One thousand one hundred XX Twenty. MDCCCLXXVI, 1876 (and sixty 52. The Roman Method is named from the Romans, who invented and used it. It is now only employed to denote the chapters and sections of books, pages of preface and introduction, and in other places for prominence and distinction. WRITTEN EXERCISES. Express the following numbers by the Roman method : 1. Twenty-seven. 2. Seventy-seven. 3. Two hundred and one. 4. Six hundred and fifty-six. 5. One thousand seven hundred and seventy-six. 6. Four thousand seven hundred and fifty-seven, 7. 25007. 8. 206484. Read the following numbers: 1. LXXVII. 2. MX0. 3. MDCCCLXXVI. 4. MMMCCCXXXIII. 5. XVDCCXLIV. 6. clxxxviii. 7. xlix. 8. xcix. LUMBERMEN'S NOTATION. 53. Lumbermen in marking lumber employ a modification of the Roman Method of Notation. The first four char. acters are like the Roman. The others are as follows: Î ŞU ŞI All X X XI XII XIII X!!! 싶 12 13 14 25 16 17 18 19 20 21 22 24 25 26 28 29 30 40 NOTE.--For a full discussion of Arithmetical Language, see the author's Philosophy of Arithmetic, |