Read the following numbers: 13. 5006; 14. 36741; 400560; 13061; 49000; 100010. For convenience in reading large numbers, we may point them off, by commas, into periods of three figures each, counting from the right hand or unit figure. This pointing enables us to read the hundreds, tens, and units in each period with facility as seen in the following 28. Figures occupying different places in a number, as units, tens, hundreds, etc., are said to express different orders of units. 29. In numerating, or expressing numbers verbally, the various orders of units have the following names : Write and read the following numbers : 1. One unit of the third order, two of the second, five of the first. Ans. 125; read, one hundred twenty-five. 2. Two units of the 5th order, four of the 4th, five of the 2d, six of the 1st. Ans. 24056; read, twenty-four thousand fifty-six. 3. Seven units of the 4th order, five of the 3rd, three of the 2d, eight of the 1st. 4. Two units of the 7th order, nine of the 6th, four of the 3d, one of the 1st, seven of the 2d. 5. Three units of the 6th order, four of the 2d. 6. Nine units of the 8th order, six of the 7th, three of the 5th, seven of the 4th, nine of the 1st. 7. Four units of the 10th order, six of the 8th, four of the 7th, two of the 6th, one of the 3d, five of the 2d. 8. Eight units of the 12th order, four of the 11th, six of the 10th, nine of the 7th, three of the 6th, five of the 5th, two of the 3d, eight of the 1st. 30. Since the number expressed by any figure depends upon the place it occupies, it follows that figures have two values, Simple and Local. 31. The Simple Value of a figure is its value when taken alone; thus, 4, 7, 2. 32. The Local Value of a figure is its value when used with another figure or figures in the same number. Thus, in 325, the local value of the 3 is 300, of the 2 is 20, and of the 5 is 5 units. When a figure occupies units' place, its simple and local values are the same. 33. The leading principles upon which the Arabic notation is founded are embraced in the following GENERAL LAWS. I. All numbers are expressed by applying the ten figures to the different orders of units. II. The different orders of units increase from right to left, and decrease from left to right, in a tenfold ratio. III. Every removal of a figure one place to the left, increases its local value tenfold; and every removal of a figure one place to the right, diminishes its local value tenfold. From this analysis of the principles of Notation and Numeration, we derive the following rules : RULE FOR NOTATION. I. Beginning at the left hand, write the figures belonging to the highest period. II. Write the hundreds, tens, and units, of each successive period in their order, placing a cipher wherever an order of units is omitted. RULE FOR NUMERATION. I. Separate the number into periods of three figures each, commencing at the right hand. II. Beginning at the left hand, read each period separately, and give the name to each period, except the last, or period of units. 34. Until the pupil can write numbers readily, it may be well for him to write several periods of ciphers, point them off, over each period write its name, thus, and then write the given numbers underneath, in their appropriate places. EXERCISES IN NOTATION AND NUMERATION. Express the following numbers by figures : 1. Four hundred thirty-six. 2. Seven thousand one hundred sixty-four. 4. Fourteen thousand two hundred eighty. 5. One hundred seventy-six thousand. 7. Ninety-five million. 8. Four hundred eighty-three million eight hundred sixteen thousand one hundred forty-nine. 9. Nine hundred thousand ninety. 10. Ten million ten thousand ten hundred ten. Point off, numerate, and read the following numbers: 23. Write seven million thirty-six. 24. Write five hundred sixty-three thousand four. 25. Write one million ninety-six thousand. 26. A certain number contains 3 units of the seventh order, 6 of the fifth, 4 of the fourth, 1 of the third, 5 of the second, and 2 of the first; what is the number? 27. What orders of units are contained in the number 2.90648 ? ADDITION, 35. Addition is the process of uniting several numbers of the same kind into one equivalent number. 36. The Sum or Amount is the result obtained. 6 and 2 are 6 and 3 are 9 6 and 4 are 10 6 and 5 are 11 6 and 6 are 12 6 and 7 are 13 6 and 8 are 14 6 and 9 are 15 6 and 10 are 16 6 and 11 are 17 6 and 12 are 18 10 and 1 are 11 10 and 2 are 12 10 and 3 are 13 10 and 4 are 14 10 and 5 are 15 10 and 6 are 16 10 and 7 are 17 10 and 8 are 18 10 and 9 are 19 10 and 10 are 20 10 and 11 are 21 10 and 12 are 22 8 9 and 1 are 10 9 and 2 are 11 9 and 3 are 12 9 and 4 are 13 8 and 5 are 13 8 and 6 are 14 11 and 4 are 15 8 and 7 are 15 8 and 8 are 16 8 and 9 are 17 8 and 10 are 18 8 and 11 are 19 8 and 12 are 20 12 and 1 are 13 12 and 2 are 14 12 and 3 are 15 12 and 4 are 16 12 and 5 are 17 12 and 6 are 18 9 and 9 and 9 are 18 9 and 10 are 19 9 and 11 are 20 9 and 12 are 21 13 and 1 are 14 13 and 2 are 15 13 and 3 are 16 13 and 4 are 17 13 and 5 are 18 13 and 6 are 19 13 and 7 are 20 13 and 8 are 21 13 and 9 are 22 13 and 10 are 23 13 and 11 are 24 13 and 12 are 25 8 are 17 |
