In the number Three hundred and forty-seven, for example, we find seven units of the first order, four of the second, and three of the third. In the number Five thousand, nine hundred and four, we find four units of the first order, none of the second, nine of the third, and five of the fourth. How many units of the first and second orders, respectively, do you find in the number Twenty-five? In the number Thirty-one? In the number Seventy-six ? In the number Nineteen ? In the number Ninety-nine ? How many units of distinct orders are found in the number Five hundred and thirty-four? In the number Nine hundred and fifty-one? In Three thousand seven hundred and sixteen? In Eight thousand one hundred and seventy? Scale of Numeration. 11. Ten of any lower order of units make one of the next higher order; or, one of a higher order makes ten of the next lower order. Thus ten units (of the first order) make one ten. How many Tens make One hundred ? How many Hundreds make One thousand? How many Hundreds make Two thousand? How many Hundreds make Three thousand? One million is how many Hundred thousand? One billion is how many Hundred millions? One trillion is how many Hundred billions? Numeration Table. § 12. The ascending orders of units are given from right to left, in the following Table : Billions, &c. Hund. of mill., Tens of mill., Millions, Tens of thous., Hundreds, Tens, Units, Recite the orders of units, ascending, from Units to Decillions. Recite them, descending, beginning with Hundreds. Beginning with Thousands. Beginning with Tens of thousands. Beginning with Hundreds of thousands. Beginning with Millions. Beginning with Billions. What are the relative values of these different orders of units (§ 11)? ΝΟΤΑΤΙΟΝ. § 13. NOTATION is the method of denoting numbers by numeral characters or figures. These Figures-sometimes called the digits of numbers-are 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, and the 0 zero or cipher, which denotes nothing. The figures from 1 to 9 inclusive, are significant; the 0 is insignificant. Tens, Hundreds, &c.-how denoted. $ 14. Tens, hundreds, &c., are denoted by two, three, &c., figures in a row, right and left, the first on the right being units,-the second, tens,-the third, hundreds, and so on, according to the ascending orders of units. (§ 12.) Thus 12, 1 ten and 2, that is, twelve; 123, 1 hundred, 2 tens, and 3; or one hundred and twenty-three. What would be the value of 4 in the first place on the right? In the second place? In the third place? In the fourth place? Simple and Local Values. § 15. The simple value of a figure is that which is expressed by the simple name of the figure-being its value when used alone, or in the units' place. Thus the simple value of 5 is five; and 5 has its simple value in 25, 2 tens and 5, or twenty-five. § 16. The local value of a figure is that which arises from its place in a row of figures-being its simple value increased tenfold for each place it is removed from units toward the left. In 25, 2 has the local value 2 tens, which is 10 times the simple 2; and in 125, 1 has the local value 1 hundred. 3 in the second place from the right would be how many times the simple 3? In the third place? In the fourth place?. In the fifth? Use of 0 Zero or Cipher. § 17. The 0 zero or cipher, having no value, is used to occupy vacant places in notation; and thus to assign a required local value to another figure, by removing it the proper distance from the units' place. Thus 10, 1 with a 0 occupying the units' place, denotes Ten. so, 100, 1 with two Os for tens and units, is One hundred. How may the figure 2 be made to denote 2 tens, or twenty? How may it be made to denote 2 hundred ? 2 thousand? 2 millions? How may 3 be made to denote 3 hundred? 3 thousand? 30 thou3 hundred millions? RULES OF ARITHMETIC. $ 18. A Rule, in Arithmetic, is a prescribed mode of performing an Arithmetical operation. RULE I. § 19. To Numerate or read a row of Figures. Call the successive figures units, tens, hundreds, &c., from right to left (§ 12), and then read them according to their respective names or values from left to right. To read the number EXAMPLE. 70436 0521 Calling the figures, one after another, units, tens, hundreds, &c., from right to left, we find the last figure 7 to be hundreds of millions. Then reading from left to right, we say, Seven hundred and four millions, Three hundred and sixty thousand, Five hundred and twenty one. EXERCISES. Read each of the following numbers,-from the top to the bottom of each column. RULE II. § 20. To write in Figures any given Number. Place proper figures from left to right, to denote the number in each of the descending orders of units, from the highest in the given number down to simple units; observing to fill each vacant place with a 0. (§ 17.) EXAMPLE. To write in figures the number Three millions, twenty-five thousand, and thirty. The descending orders of units in this number, are 3 millions, 2 tens of thous., 5 thousands, and 3 tens. Hence we write it, 3025030; in which the vacant places of hundreds of thousands, hundreds, and units, are filled with Os. EXERCISES. Write in figures each of the following numbers:— 1. One hundred. 2. Two hundred and one. 10. One thousand. 11. Two thousand and nine. 17. Four thousand, five hun- 18. Ten thousand. 21. Fifty-four thousand, one hundred and twenty-three. 22. Eighty-seven thousand, five hundred and seventy-eight. 23. Seventy-one thousand, two hundred and one. 24. Forty thousand, three hundred and two. 25. One hundred thousand. thousand, one hundred. sand, two hundred and three. 28. Seven hundred and thirteen thous., four hundred and fifty. 29. Nine hundred and ninety nine thousand, and one. 30. Eight hundred thousand, and seven hundred. 19. Twelve thousand and ten. 20. Twenty thousand and nine. [sand, one hundred and nine. 31. Nine hundred and one thou 32. One million. 43. One hundred millions. 33. Five millions, five hundred 44. Three hundred millions, one hundred thousand. thousand. 34. Nineteen millions, two hun- 45. Five hundred and thirtydred and forty-seven thous. four millions, nine hundred. 35. Thirty millions, one hun- 46. Six hundred and nine mildred and fifty thousand, seven hundred. lions, fifty thousand, one hundred and twenty-five. 36. Seventy-five millions, eight 47. Nine hundred and seven hundred and sixty-four thou sand, nine hund. and twelve. teen millions, five hundred thous.,four hundred and sixty. ty thousand, three hundred and four. 37. Eleven millions, seven hun- 48. Five hundred millions, sixdred and fourteen thousand. 38. Twenty-nine millions, four hundred and one thousand, 49. two hundred and ten. 39. Thirty millions, nine hundred and twenty thousand. Seven hundred and ten millions, one hundred thousand, five hundred and ninety-one. 40. Seven millions, eighty-five 50. Three hundred and one milthousand, six hundred and forty-nine. 41. Eighty-five millions, eighty- lions, seven hundred and ten. 51. Eight hundred and six millions, nine hundred and nineteen thousand, one hundred. 52. Nine hundred and ninetynine millions, nine hundred. and ninety-nine thousand, nine hund. and ninety-nine. French and English Numeration. § 21. In the French system of Numeration, which prevails in continental Europe, and in America, a thousand millions make one billion, a thousand billions make one trillion, and so on (§ 9 and § 12). In the English system, a million millions make one billion, a million billions make one trillion, and so on. Hence, in this system, after hundreds of millions, the ascending order of units are, thousands of millions, tens of thousands of millions, hundreds of thousands of millions, billions; and in like manner after hundreds of billions, &c. For example, the number 3 840 930 670 820, in the French system, is 3 trillions, 840 billions, 930 millions, 670 thousand, eight hundred and twenty. In the English, it is 3 billions, 840930 millions, 670820. |