units, and hundreds of units, and we should divide the number into periods of three figures each by commas, the first period would be the period of units; the second, the period of thousands; the third of millions; the fourth of billions, &c. The right hand figure in each period expresses units or ones of the denomination of that period, while the second figure expresses tens, and the third, or left hand figure, expresses hundreds of that denomination. Let the pupils now answer the following questions. 1. What is the name of the first period at the left of the point? of the second period? of the third? of the fourth? of the fifth of the sixth? of the seventh? of the eighth ? 2. What is the name of the period occupying the first, second and third places at the left of the point? 3. Occupying the fourth, fifth, and sixth places? 4. Occupying the seventh, eighth, and ninth? 5. Occupying the tenth, eleventh, and twelfth ? 6. Occupying the thirteenth, fourteenth, and fifteenth ? 7. Occupying the sixteenth, seventeenth, and eighteenth? 8. Occupying the nineteenth, twentieth, and twenty-first? 9. Occupying the twenty-second, twenty-third, and twentyfourth? Which place from the point is occupied by 10. The millions? the quintillions? the units? 11. The billions? the quadrillions? the trillions? 12. The thousands? the sextillions? How many places are there between 13. The point and the thousands' period? 14. The point and the billions' period? 15. The point and the sextillions' period? 16. The point and the trillions' period? 17. The point and the millions' period? 18. The point and the quintillions' period? 19. The point and the units' period? 20. The point and the quadrillions' period? In which period and in which place of the period would 21. The fourth figure from the point be? 22. The seventh? the tenth ? the thirteenth ? 23. The sixteenth? the nineteenth? the twenty-second? 24. The second? the fifth? the eighth ? 25. The eleventh? the fourteenth? the seventeenth ? 26. The twentieth? the twenty-third? the third ? 27. The twenty-first? the eighteenth? the twelfth? 28. The fifteenth? the sixth? the twenty-fourth? 29. What would be the denomination of a figure in each place named in the above examples, from number 21 to number 28 ? 30. Where must a figure be placed to represent trillions? millions? thousands? ten-millions? hundred-trillions? units? hundreds? ten-thousands? hundred-billions? quadrillions? tenbillions? hundreds of sextillions? C. The places are usually only marked by the figures occupying them, and hence, that we may determine the place of a figure so as to know the denomination it represents, it is necessary that every place between the figure and the point should be occupied by some one of the digits. The digit to be used in any intermediate place will depend on the number of units of the denomination of that place, which we wish to represent. If we wish to represent simply five hundred, we write the character 5 in the third place from the point, and 0 in the first and second places. The zeros show that the 5 belongs in the hundreds' place, and that there are no units or tens in the number, except those contained in the five hundred. If we wish to represent five hundred and twenty-seven, we place 5 in the third place as before, but the first is occupied by 7 and the second by 2, showing that the 5 represents 5 hundreds, and that there are two tens and seven units in the number besides. How will you write 4 so that it shall represent 1. Four hundreds? four ten-thousands? 2. Four millions? four hundred-millions? How will you write 7 so that it shall represent 3. Seven thousands? seven hundred-thousands? 4. Seven ten-millions? seven billions? How will you write the figures 5 and 9 so that they shall respectively represent 5. Ten-thousands and thousands? 6. Hundred-thousands and ten-thousands? 7. Millions and hundred-thousands? How will you write the figures 8, 3, and 4, so that they shall respectively represent 8. Hundred-millions, ten-millions, and millions? 9. Hundred-billions, ten-billions, and billions? 10. Millions, thousands, and units? 11. Ten-millions, ten-thousands, and units? - 15. Hundred-trillions, hundred-billions, and hundred-millions? D. In reading a number represented by figures, we ordinarily commence at the left hand and read each period as though its figures stood alone, giving afterward the name of the period. For instance, the number 42,947,315,111,300,531. would be read in the same way and would express the same value as if written 42 quadrillions, 947 trillions, 315 billions, 111 millions, 300 thousand, and 531. Let the pupil now read each of the following numbers. E. 1. Represent by figures five hundred and twenty-seven thousand, four hundred and eighty-nine. 2. Eight thousand four hundred and seven. 3. Eighty-five thousand. 4. Eighty-five thousand and one. 5. Eighty-five thousand and thirty-one. 6. Nine million, eight hundred and fifty-six thousand, seven hundred and twenty-one. 7. Twelve million, twelve thousand, and twelve. 8. Four billion, eight hundred seventy-six million, five hundred and four thousand, three hundred and one. 9. Four billion, eight hundred and four million, eight hundred and four thousand, eight hundred and four. 10. Thirty-seven million, eight hundred and fifty-nine thou sand. 11. Thirty-seven billion, eight hundred and fifty-nine million. 12. Thirty-seven billion, eight hundred and fifty-nine thousand. 13. Thirty-seven million, eight hundred and fifty-nine. 14. Forty billion, three hundred and forty million, four hundred and eighty-seven thousand, five hundred and nine. 15. Five billion, eight hundred and seventy-six thousand, seven hundred and forty-six. 16. Seventy-five trillion, eight hundred and seventy-six billion, four hundred and eighty-two million, four hundred and seventy-six thousand, three hundred and twenty-seven. 17. Four trillion, seven hundred and sixty-four billion, eight hundred and twenty-one million, six hundred and seventeen thousand, four hundred and fifty-one. 18. Seven hundred and twenty-five trillion, eight hundred and seventy-six billion, four hundred and three million, eight hundred and fifty thousand, and four hundred. 19. Three hundred and six trillion, eighteen billion, four hundred million, three thousand, four hundred and seventy-five. 20. Three trillion, three hundred and ninety-nine billion, three hundred and ninety-nine million, three hundred thousand, four hundred and three. 21. Eighty-seven trillion, five hundred and four billion, three hundred million, seven thousand, six hundred and seventy-five. F. Combinations of numbers, wherever placed, can be read as though they stood alone, if the name of the place of the right-hand figure be given after reading the figures. For instance: 347 always stands for, and may be read as, three hundred and forty-seven of the denomination of the place occupied by the 7. To illustrate this still further, we have written opposite each of the following numbers the value expressed by 347 in that number. Let the pupil give the value expressed by the 409 in each of the following numbers, and also the value expressed by the 27: 5. 40,927. 7. 274,090. 9. 8,172,764,096,134. 6. 568,340,927. 8. 40,927,534. 10. 27,409. G. What is the greatest number of tens that can be taken— 1. From 1768? from 7,128,904 ? 2. From 306,702,537? from 20,000? 3. From 8,576,423,721? from 100,637,052,980,112? 4. What is the greatest number of hundreds that can be taken from each of the above? of thousands? of billions*? of tens of millions? of hundreds of thousands? of tens of billions? 5. How will you express the value of each of the above in terms of tens and units? Ans. 1768 176 tens + 8 units. 7,128,904 = &c. How will you express the value of each of the above in terms of — 6. Hundreds and units? billions and units? 7. Tens of billions and units? thousands and units? H. Since, as we have seen, the figure in any place expresses ten times the value it would express if written one place further towards the right, it follows of necessity that it expresses only one-tenth of the value it would express if written one place further towards the left. To give more full illustrations of this than we have yet done, we will compare the numbers 42 and 420. In the first number the figure 2 represents 2 units, while in the second it represents 2 tens, or ten times 2 units. In the first number the 4 represents 4 tens, while in the second it represents 4 hundreds, or ten times 4 tens. Therefore, each figure found in both numbers represents ten times the value in the second that it represents in the first, or one-tenth the value in the first that it represents in the second, and as there is no figure except zero in one that is not in the other, the value of the second number must be 10 times that of the first, and the value of the first one-tenth that of the second. * If any number is less than a billion, as 1768, the answer may be, 1768 is less than a billion, and, therefore, no billions can be taken from it. |