of thousands, or by writing it at the left hand of four ciphers; thus, 10000; and one hundred thousand is expressed by removing the unit, 1, still one place further to the left, or by writing it at the left hand of five ciphers; thus, 100000. We can express thousands, tens of thousands, and hundreds of thousands in one number, in the same manner as we express units, tens, and hundreds in one number. To express five hundred twenty-one thousand eight hundred three, we write 5 in the sixth place, counting from units, 2 in the fifth place, 1 in the fourth place, 8 in the third place, 0 in the second place, (because there are no tens,) and 3 in the place of units; thus, The greatest number that can be expressed by five figures is 99999; and by six figures, 999999. EXAMPLES FOR PRACTICE. Write the following numbers in figures: 1. Twenty thousand. 2. Forty-seven thousand. 3. Eighteen thousand one hundred. 4. Twelve thousand three hundred fifty. 5. Thirty-nine thousand five hundred twenty-two. 7. Eleven thousand twenty-four. 8. Forty thousand ten. 9. Sixty thousand six hundred. 10. Two hundred twenty thousand. 11. One hundred fifty-six thousand. 12. Eight hundred forty thousand three hundred. Greatest number expressed by five figures? Six figures? 13. Five hundred one thousand nine hundred sixty-four. 14. One hundred thousand one hundred. 15. Three hundred thirteen thousand three hundred thir teen. 16. Seven hundred eighteen thousand four. 17. One hundred thousand ten. 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. 30. Next above hundreds of thousands we have, successively, units, tens, and hundreds of millions, and then follow units, tens, and hundreds of each higher name, as seen in the period. period. period. period. period. period. period. period. period. How may figures be pointed off? Next period above millions, what? period. One million, how expressed? Give the name of each successive NOTE. This is called the French method of pointing off the peri ods, and is the one in general use in this country. 31. Figures occupying different places in a number, as units, tens, hundreds, &c., are said to express different orders of units. and so on. Thus, 452 contains 4 units of the third order, 5 units of the second order, and 2 units of the first order. 1,030,600 contains 1 unit of the seventh order, (millions,) 3 units of the fifth order, (tens of thousands,) and 6 units of the third order, (hundreds.) EXAMPLES FOR PRACTICE. Write and read the following numbers : 1. One unit of the third order, four of the second. 2. Three units of the fifth order, two of the third, one of the first. 3. Eight units of the fourth order, five of the second. 4. Two units of the seventh order, nine of the sixth, four of the third, one of the second, seven of the first. 5. Three units of the sixth order, four of the second. 6. Nine units of the eighth order, six of the seventh, three of the fifth, seven of the fourth, nine of the first. 7. Four units of the tenth order, six of the eighth, four of the seventh, two of the sixth, one of the third, five of the second. 8. Eight units of the twelfth order, four of the eleventh, six of the tenth, nine of the seventh, three of the sixth, five of the fifth, two of the third, eight of the first. Units of different orders are what? 32. From the foregoing explanations and illustrations, we derive several important principles, which we will now pre sent. 1st. Figures have two values, Simple and Local. The Simple Value of a figure is its value when taken alone; thus, 2, 5, 8. The Local Value of a figure is its value when used with another figure or figures in the same number; thus, in 842 the simple values of the several figures are 8, 4, and 2; but the local value of the 8 is 800; of the 4 is 4 tens, or 40; and of the 2 is 2 units. NOTE. When a figure occupies units' place, its simple and local values are the same. 2d. A digit or figure, if used in the second place, expresses tens ; in the third place, hundreds; in the fourth place, thousands; and so on. 3d. As 10 units make 1 ten, 10 tens 1 hundred, 10 hundreds 1 thousand, and 10 units of any order, or in any place, make one unit of the next higher order, or in the next place at the left, we readily see that the Arabic method of notation is based upon the following TWO GENERAL LAWS. I. The different orders of units increase from right to left, and decrease from left to right, in a tenfold ratio. II. 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. Thus, 6 is 6 units. 60 is 10 times 6 units. 600 is 10 times 6 tens. 6000 is 10 times 6 hundreds. 60000 is 10 times 6 thousands. First principle derived? What is the simple value of a figure? Local Second principle? Third? First law of Arabic notation? Second? 4th. The local value of a figure depends upon its place from units of the first order, not upon the value of the figures at the right of it. Thus, in 425 and 400, the value of the 4 is the same in both numbers, being 4 units of the third order, or 4 hundred. NOTE. Care should be taken not to mistake the local value of a figure for the value of the whole number. For, although the value of the 4 (hundreds) is the same in the two numbers, 425 and 400, the value of the whole of the first number is greater than that of the second. 5th. Every period contains three figures, (units, tens, and hundreds,) except the left hand period, which sometimes contains only one or two figures, (units, or units and tens.) 33. As we have now analyzed all the principles upon which the writing and reading of whole numbers depend, we will present these principles in the form of 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, Trillions, Billions, Millions, Thousands, Units. 000, 000, 000, 000, 000 Fourth principle? What caution is given? Fifth principle? Rule for notation? Numeration? |