5 single units; the 3 occupying the second place represents 3 tens, or 3 units each ten times the size or value of a unit of the first place; and the 2 occupying the third place represents 2 hundreds, or 2 units each one hundred times the size or value of a unit of the first place; the value expressed by any figure being always made tenfold by each removal of it one place to the left. 31. The cipher becomes significant when connected with other figures, by filling a place that otherwise would be vacant; as in 10 (ten) where it occupies the vacant place of units, and in 102 (one hundred and two) where it fills the vacant place of tens. 32. The simple value of a unit is the value expressed by a figure standing alone; or, in a collection, when standing in the right-hand place. Thus 2 alone, or in 32 (thirty-two), expresses a simple value of two single units or ones. 33. The local value of a unit is the value expressed by a figure when it is used in combination with another figure or figures, and depends upon the place the figure occupies. Thus, in 44 (forty-four), the 4 in the first place expresses the local value of 4 units, and the 4 in the second place, the local value of 4 tens, or forty. 34. The successive places occupied by figures are often called orders. Thus a figure in the first or units' place is called a figure of the first order, or of the order of units; a figure in the second place is a figure of the second order, or of the order of tens; in the third place, of the third order, or of the order of hundreds; and so on, each figure next to the left belonging to a distinct order, the unit of which is tenfold the size or value of a unit of the order at the right. EXERCISES. 1. Write three units of the first order. 2. Write five units of the first order. 3. Write eight units of the second order, with seven of the first. 4. Write two units of the third order, with none of the second, and one of the first. 5. Write seven units of the fourth order, with two of the third, none of the second, and none of the first. 6. Write one unit of the fifth order, with none of the four lower orders. 7. Write six units of the sixth order, five of the fifth, four of the fourth, three of the third, one of the second, and two of the first. 8. Write one unit of the eighth order, with none of the seven lower orders. 9. Write nine units of the ninth order, with six of each of the eight lower orders. 10. Write two units of the twelfth order, with none of the eleventh, none of the tenth, one of the ninth, five of the eighth, nine of the seventh, none of the sixth, none of the fifth, none of the fourth, three of the third, none of the second, and three of the first. 11. Write three units of the fifteenth order, with none of the fourteenth, none of the thirteenth, none of the twelfth, one of the eleventh, seven of the tenth, five of the ninth, one of the eighth, none of the seventh, none of the sixth, five of the fifth, three of the fourth, two of the third, two of the second, and seven of the first. 12. Write four units of the twenty-fifth order, with three of the twenty-fourth, two of the twenty-third, none of the twentysecond, none of the twenty-first, none of the twentieth, none of the nineteenth, none of the eighteenth, none of the seventeenth, five of the sixteenth, and none of the fifteen lower orders. NUMERATION. 35. NUMERATION is the process of reading numbers when expressed by figures. 36. There are two methods of numeration; the French and the English. FRENCH NUMERATION. 37. The French method of numeration is that in general use on the continent of Europe and in the United States. Beginning at the right, figures occupying more than three places being separated into as many groups as possible of three figures each, called periods, it gives a distinct name to each period. FRENCH NUMERATION TABLE. Hundreds of Sextillions. Hundreds of Quintillions. Hundreds of Quadrillions. Tens of Quadrillions. Hundreds of Trillions. ∞ Tens of Trillions. 2 3. 7 8 9, 1 2 3, 4 5 6, 7 8 9, 1 2 3, 4 5 6, 7 8 9, 8th Period. 7th Period. 6th Period. 5th Period. 4th Period. 3d Period. 2d Period. 1st Period. lions. Quintil. Quadril. Billions. The value of the number represented in the table is, seven hundred eighty-nine sextillions, one hundred twenty-three quintillions, four hundred fifty-six quadrillions, seven hundred eighty-nine trillions, one hundred twenty-three billions, four hundred fifty-six millions, seven hundred eighty-nine thousands, one hundred twenty-three. 38. The unit of the first period, or right-hand group, is 1; of the second, 1 thousand; of the third, 1 million; of the fourth, 1 billion; of the fifth, 1 trillion; of the sixth, 1 quadrillion; of the seventh, 1 quintillion; of the eighth, 1 sextillion, etc. The periods above sextillions, in their order, are, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions, etc. Before NOTE. The idea of number is the latest and most difficult to form. the mind can arrive at such an abstract conception, it must be familiar with that process of classification by which we successively ascend from individuals to species, from species to genera, from genera to orders. The savage is lost in his attempts at numeration, and significantly expresses his inability to proceed by holding up his expanded fingers or pointing to the hair of his head. It is, indeed, difficult for any mind to form an adequate idea of the larger numbers. To count a million, at the rate of one in a second, would require upward of twenty-three days of twelve hours each. A billion is equal to a million a thousand times repeated, or a number so great, as to exceed all the seconds of time that would elapse in thirty-two years. 1 39. To read numbers represented by figures according to the French method; Begin at the right hand, and point off the figures into as many periods as possible of three places each. Then, commencing at the left hand, read the figures of each period, giving the name of each period excepting that of units. EXERCISES. Read orally, or write in words, the numbers represented by the following figures, according to the French method: 24. 407144140070060700007101800808 40. To write numbers in figures according to the French method; Begin at the left hand, and write in each successive order the figure belonging to it. If any intervening order would otherwise be vacant, fill the place by a cipher. EXERCISES. Represent by figures, and read, the following numbers, according to the French method: 1. Twenty-nine. 2. Four hundred and seven. 3. Twenty-three thousand and seven. 4. Five millions and twenty-seven. 5. Seven millions, two hundred five thousand and five. 6. Two billions, two hundred seven millions, six hundred four thousand and nine. 7. One hundred five billions, nine hundred nine millions, three hundred eight thousand two hundred and one. 8. Nine quintillions, eight billions and forty-six. 9. Fifteen quintillions, thirty-one millions and seventeen. 10. Five hundred seven septillions, two hundred three trillions, fifty-seven millions and eighteen. 11. Nine nonillions, forty-seven trillions, seven billions, two millions, three hundred ninety-two. 12. Fifteen duodecillions, ten trillions, one hundred twentyseven billions, twenty-six millions, three hundred twenty thousand four hundred twenty-six. ENGLISH NUMERATION. 41. The English method of numeration is that generally used in Great Britain, and in the British Provinces. It divides numbers into periods of six figures, and gives a distinct name to each. The value of the figures in the above table, expressed in words according to the English method, is, Three hundred ninety-eight thousand, eight hundred thirty-two trillions; five hundred sixty-three thousand, eight hundred seventy-one billions; three hundred fifty-one thousand, six hundred fifteen millions; one hundred twenty-three thousand five hundred sixty-one. |