Ans. In the number 217,426,389, taken as an exam ple, there are, 3. You have learned, in Lessons XIV. and XV., that units in the second place are ten times larger than in the first; a hundred times in the third, and a thousand times in the fourth; that is, that units increase in value ten times for every place they advance to the left; does the same scale of increase continue through larger numbers? Ans. The same principle of numeration is carried through numbers of any size; namely: any figure is ten times as much as the same figure on its right. 4. What name is given to this system of numeration? It is called the DECIMAL SYSTEM, from the Latin word decem, which means ten, because it numerates by the scale of tens, or decimal scale. 5. How would you write a number composed of several periods? In order to write any number, beginning at the left, the periods are written successively as all other numbers of three figures, taking care to occupy by Os all the places for which there are no significant figures. N.B. It is a good practice to divide the periods by commas. EXAMPLES. 1. Forty-five millions, six hundred and three thousand. nine hundred and twenty-five, would be written Millions. Thousands. Units. 2. Two hundred and one millions, five hundred thousand, seven hundred and thirty, N.B. Beginners will do well to prepare the operation by putting a dot in the place of each figure, thus, Milions. Thousands. Units. ... and then setting down each figure in its proper place over the dots, and substituting Os for the remaining dots. Thus, for instance, three hundred and four millions, five thousand and ten, would be written first In this way the pupil will be sure to make no mistake. EXERCISES. 1. Write: Six thousand and twenty-five. 2. Twelve thousand six hundred and fifty. 3. Seventy thousand and seventy. 4. Ninety-five thousand and ninety. 5. Two hundred and twenty thousand, four hundred and fiftyfive. 6. Four hundred and five thousand, six hundred and thirty seven. 7. Nine hundred and twenty millions, six thousand three hundred and seven. 8. Four hundred and twenty-one thousand and two hundred units. 9. Ninety one millions, twenty-six thousand and five. 10. One hundred and eight millions, two hundred and six thousand, five hundred. 11. Eighty millions, eight hundred thousand and eight units. 12. Six hundred millions, five thousand units. 13. Four billions, four millions, four thousand and four. 14. Nine hundred millions and nine units. 15. Sixty-six billions, six hundred and sixty millions, sixty-six thousand and sixty-six. LESSON XVIII. ADDITION. 1. What is the sign of addition, and how is it read? The sign of addition is placed between two numbers: it reads plus. 2. What is the sign of equality? The sign of equality is =, and reads is or are equal to, as the case may be. 3. Give examples of both signs. 5 apples +7 apples 12 apples. Read: 5 apples plus 7 apples are equal to 12 apples. Again, 6 +8: 14, or 14 = 6 +8. In the first case read: 6 plus 8 are equal to 14. 4. What things or units can be added together? You can add only things or units of the same denomination (that is, which have the same name). You could not, for instance, add 2 apples and 3 eggs and call the sum 5, either apples or eggs. Nor can you add units with tens, or with hundreds; but you may add units with units, tens with tens, hundreds with hundreds, &c. 5. How do you perform addition in writing? Ans. The numbers to be added are written under each other, so that units may be under units, tens under tens, hundreds under hundreds, &c., and a line is drawn beneath the whole: then, beginning at the right, each column is added separately, either upwards or downwards, as may be preferred. Let it be proposed, for example, to add 113 + 340 + 426: Arrange them as directed; and then, beginning with the units, say, 3 and 6 are 9, which set down under the column of units, as shown here. Then 1 and 4 are 5 and 2 are 7; write 7 under the column of tens. OPERATION. ∞ hundreds 113 340 426 879 Finally, 1 and 3 are 4 and 4 are 8; set down 8 under the hundreds; and the whole sum or amount is found to be 879. 6. In the preceding addition you will observe that the sum of each column does not exceed 9; how would you proceed if the sum of any of them were greater than 9? Any number over 9, contains units and tens; the units should be set down under the column that has been added, and the tens carried to the next column to be added with it. 7. Explain this by an example. Let it be proposed, for example,* to add 278189 +297 +185. The addition of the column of units amounts to 29, which is (Lesson XIII.) 2 tens and 9 units; set down the 9 under its column and carry the 2 forward to the column of tens, with which it must be added. OPERATION. 32 278 189 297 185 949 Now add the tens together, including the 2 tens carried, which make in all 34 tens; that is 3 hundreds and 4 tens; set down the 4 under the tens and carry the 3 to the column of hundreds, to be added in with them. Finally, the sum of the column of hundreds together with the 3 carried is 9; and the whole amount 949. N.B. It is always a good practice, especially for beginners, to write, as shown here, the numbers carried in smaller characters at the top of the column with which they are to be added. 8. Take as a third example, 986 +24+6+ 859+461. Answer. The numbers having been set under each other according to the rule, the column of units is added and gives 26; 6 is set down and 2 carried. The column of tens with the 2 carried gives 20, that is two hundreds and no tens; OPERATION. 22 986 24 6 859 461 2306 * It will be well for the pupil to explain on another example than that of the book, in this as well as in other cases. therefore 0 is set down to occupy the place of the absent tens (XIV. 6); and the 2 hundreds are carried to their proper column, with which they make 23 hundreds, which are written at once; and the total amount is 2,306; that is twenty-three hundred and six, or, otherwise, two thousand three hundred and six, as you may choose to read it. 9. What is then the complete RULE FOR ADDITION? I. Write down the numbers to be added under each other, units under units, tens under tens, hundreds under hundreds, &c.; and draw a line underneath the whole. II. Then, begin at the right hand and add successively either downwards or upwards, the figures of each column, and set down the amount under the column, if it do not exceed 9; but, if it do, set down the right hand figure, and carry the other part to the next left hand column. III. When you come to the last column, set down under it the whole of its sum. 10. May it not happen that the number to be carried is composed of more than one figure? Yes; this occurs frequently in long additions. 11. How do you carry in that case? Exactly in the same way: the whole number carried is added at once to the next left hand column. 12. As an example add together the following: Ans. In this example the sum of the column of units is 109; 9 is set down, and 10 carried, as shown at the top of the second column. Then we say, 10 and 8 are 18, and 9 are 27, &c., and find for the column of tens the sum 112; we set down 2 and carry 11. Now, 11 and 7 are 18, and 2 are 20, &c, and the sum of the column of hundreds is 43, which is set down, making the whole amount 4,329. OPERATION. 11 10 788 299 198 99 178 89 769 188 98 189 76 4,329 |