Now numbers are divided, as we have already learned, into groups of units, tens, hundreds, tenths, hundredths, etc.; and when numbers are to be added, the parts into which they are divided may be added in any order we please, provided they are all counted; hence we may first add the units of one order, then the units of another order, and so on. 29. In order to add numbers, they should first be arranged so that their decimal points are in a vertical column. This will ensure that all the tenths shall be in the same vertical column, and so for the hundredths, etc.; and so also for the units, tens, hundreds, etc. arrangement is for convenience only. This The following examples will show how this principle enables us readily to find the sum of any given numbers. Ex. 1. Add 235.7 and 524.2. Since we wish to add the tenths by themselves, the units by themselves, etc., we write the numbers so that the decimal points are in a vertical column; thus, 235.7 Now 2 tenths and 7 tenths make 9 tenths, 4 units and 5 units make 9 units, 2 tens and 3 tens make 5 tens, and 5 hundreds and 2 hundreds make 7 hundreds. The required sum is generally placed just under the numbers to be added and separated from them by a horizontal line; thus, 235.7 524.2 759.9 Ex. 2. Add 548.6, 789, and 197.8. Write the numbers as in Ex. 1; thus, 548.6 789. 197.8 unit and 4 Now 8 tenths and 6 tenths make 14 tenths, that is, tenths. The 4 tenths can be put in the column for tenths, but the 1 unit must be counted with the other units. We then have 1 unit, 7 units, 9 units, and 8 units, which make 25 units; that is, 2 tens and 5 units. The 5 is put in the column for units, but the 2 tens are 'carried' (as it is called) and added with the other tens. So we proceed until all the columns are added. NOTE. Since ten units of any order make one unit of the next higher order, the figures in any column may be added without specifying the kind of units they represent; that is, without calling them tens, or hundreds, or thousands, etc., as the case may be. 25; Also, we should never use as many words as in the above explanations, but should say (see ex. 2) only 8, 14; 1 (carried), 8, 17, 2 (carried), 11, 19, 23; 2 (carried), 3, 10, 15. Of course the 4, 5, 3, and 15 are the figures to be written. In all cases the sums of numbers should be more prominent than the numbers themselves. 30. To detect mistakes in addition it is well to add each line of figures twice, once from bottom to top and once from top to bottom. An error is much more likely to be detected in this way than by simply repeating the addition in the same order, for the same mistake is very likely to be made again. Pupils should not be allowed to add more than one column at a time. EXAMPLES IV. Written Exercises. 1. Add 3104, 297, 5649, and 989. Find the sum of 2. 21.63, 5.24, 170.63, 27.59, 17. 3. 301.7, 30.17, 3.017, .3017, .03017. 4. 319, 562, 1230, 857, 4908, and 9087. 5. 235, 796, 804, 987, 359, and 856. 6. 170.2, 3.605, 17.35, 15.609, .0086. с 7. .0037, 21.85, 169.4, 17.9375, .90087. 8. 4.1372, 41.372, 4137.2, .41372, 41372. Add the numbers in each column and in each row of In the next three examples do not change the positions of the numbers. 18. Find 30.1 +297 + 35.16 +1079 + 8.017 +10.053. 19. Find 93084 + 15614 +3801.76 + 536174 + 123456 +40.404. 20. Find 218904 + 37.215.199 + 582163 + 397157 +81.4297.9163. 21. Add six hundred ninety-five, one thousand seventyfour, eleven thousand four hundred eighty-nine, and fiftyfour thousand three hundred seventy. 22. Add three million four hundred seventeen thousand thirty-five, nine hundred forty-six thousand seven hundred, fifteen million fifteen thousand fifteen, and sixty million sixteen hundred twenty-four. 23. Add six million five hundred nine thousand seven hundred six and twelve thousand four hundred thirtytwo hundred-thousandths, three hundred ninety thousand and four hundred twelve thousandths, eighteen million forty and six ten-thousandths. 31. Thus far we have studied numbers without reference to objects. When numbers are used without reference to any particular units, they are called Abstract Numbers. Two and five are abstract numbers. 9. Express in words 111111111, 1203405, 2314100, 504.0314, 20050060, and 30300074. 10. Express in words the numbers 3012004, 1101.11011, 201201201, 1000040305101, and 604102000300004. 11. Write in figures the numbers fifty-eight, eightyfive, two hundred eleven, three thousand twelve, six thousand forty, and nine thousand three hundred. Write in figures the following numbers: 12. Twelve and three-tenths. 13. Three hundred four and nine-tenths. 14. Twenty-five, three-tenths, and four hundredths. 15. Four, six-tenths, and seven-thousandths. 16. One million, four-tenths, and three-millionths. 17. Write in figures the numbers eleven hundred eleven, fourteen hundred sixty, twelve hundred thousand sixteen hundred, six million twelve hundred sixteen, and eleven billion eleven hundred eleven. 18. Write in figures twenty million twenty thousand, seventeen million fifty thousand nineteen, one hundred four million six hundred two thousand eleven, and six thousand three hundred seven million two thousand fifty six. 18. The ordinary system of notation was introduced. into Europe by the Arabians, and is still called the Arabic system of Notation although it is now known that the Arabians derived their knowledge from the Hindoos. ROMAN NUMERALS. 19. Besides the Arabic system of notation some use is still made of the cumbrous system employed by the Romans. |