« ΠροηγούμενηΣυνέχεια »
the units only of each rank are set down, and the tens carried and added in as units to the figures of the next rank to the left, agreeably to the third principle of arithmetic, p. 117. The operation of adding may be commenced in any of the ranks when the long process is used [let the pupil try this, by commencing separately with each of the four ranks], but in the shortened process, it is necessary to commence with the first rank on the right, because tens of each rank are to be added in as units of the next rank to the left.
Proof. Suggestive Questions. — If columns of figures be correctly added downwards as well as upwards, will their sums be the same, or will they be different ? Will they mutually prove each other, then? If one or more columns of figures be added together, and then all the lines in the column or columns be again added except the lower one,* what will be the difference between these two sums? If the second sum be now added to the line that was cut off, will this third sum be the same as that of the first, if the first has been added correctly? Here, then, are two methods, either of which will prove whether the work has been performed properly. If numbers that are placed vertically may be proved by adding them downwards as well as upwards, how may numbers added horizontally be proved on the same principle ?
7. Arrange vertically and add the following numbers: 1st. 246+3582+72+9873+855+2144+3792+53.
2d. 4321+2153+3946+2604+4098+24452+13246+ 6944+8175+4924+5678.
3d. 75630+76042+4942+3294+6757+4275+8641+ 1975+4132+360979063+7429.
4th. 21539+172+184 +64577+73722+35392+9077+ 1814+6137+1691 +6105+26+3284.
5th. 1619 7161013612795471+63300+14713+832+ 9468+3215+403+123+8678+9136+7924+8706+321.
6th. 152-13+326-78+49-237+1736-46+5897+13068+ 9:415+725+842-19+335-86+973437+8642+54:32.
7th. 7172.12+553-14+241•177+877-35+927'13+5679'12+684 24+68+539+28+135+9232+465-12+8472+ 8579.
* It is usual to omit the upper line in this method of proof, but such a plan is more liable to error, since the figures to be added will recur nearly in the same order. Should the teacher prefer to have the upper line cut off, the other figures should be added downwards
8th. 3157.13+9711.82+76131'31+285432+1646-78+ 532-964+72 341+555666+8404-26+9373.28+13576246+ 8891.372.
9th. 14132+37-48+96-23+45-67+89-13+579.24+1245+670-86+26:26+3333-34+975'342+87248+94:26+ 38.12.
10th. 1234+5678+9123+4567+8912+5790+2040+ 6735+9813+4276+1358+9342+88761+3456+72352+ 4638+1926.
11th. 3825+9638+1326 +5431+9425+3873+4284+ 7965+9123+4476+135879123+4782+91234+48245+ 5796-+-4312.
12th. 94:37+2823+496.01+182.94+529-87+651-32+ 825973+5687.74 +5035-69+679856+6073-87 +9087.26 +8709-56.
13th. 5687 23+8235 26 +1829 4+8708+1324+5639-24 +5824:37+8308-12+2358:09+9263-57+9123-24+3281.
14th. 917.23+287246+5387+1316:28+4827+325-16 +3243+9127+4832'44+9123-45+1055+6249+ 554246 +3264.
15th. 3408-26+1357-95+2186-37+9345-35+8421:38+ 1796*24+3875+9394-32+8218677+54134.28+3276.45+ 9137-42 +376.
8. Add the following numbers horizontally, and then find the amount of their respective sums vertically. 37628+49-56–32.06+56-28+72 54+6247 24:36+8.45+94.7+37'4+28.84+97.28+365 15+246 +375+942+87-43+215.64 •52 + 125 +25+375 +2.25 +1.125+3•64 28+572+3.75 +4.375+28-49-+57 32. 30+9.02+5076+28+94+3+72+8656 25+50+3.25+9-125+8626+9:47+5.4 32.04.+567+95-25+7.84+9'375+1-236 5.84+3-57+4.92+65+3715+6*25. 8.27+4328 +59625+06+32 5+5'42. 23-85+29.47+86-32+89:45+27.886 81321643975472 +38-43 +127-+55 24.375 +88-84+37.25 +17.94+13+85-92
14:57 +93-29+6*25+4375+19 25+16
[Two, three, or more exercises like the above may now be formed from the figures in Example 7, and these may again be used by taking the figures in a backward order, thus changing 32:45 to 54-23, and so forth. Addition should be practised till the pupil can run up a column correctly with the utmost ease and rapidity. It would be well if the classes should occasionally practise the addition of columns the whole length of the slate, until he has finished his course of arithmetic. No
operation is so often called for in practical business as the summation of numbers.
Specimen of different methods of adding Large Amounts.
1. Reading by three figures at once. First step. Nineteen and eighteen are thirty-seven, and seventeen are fifty-four, and eleven are sixty-five ; carry six to twenty-one are twenty-seven, and seventeen are forty-four, and sixteen are sixty, and nineteen are seventy-nine ; carry seven to eighteen are twenty-five, and twenty-four are forty-nine, and twelve are sicty-one, and eighteen are seventy-nine ; carry seven to fifteen are twenty-two, and nine are thirty-one, and twenty-four are fifty-five, and six are sixty-one. As soon as this step has been practised with various long columns of figures till it can be performed with ease and rapidity, the pupil may proceed to the Second STEP, which consists in omitting all the words in the first step except those in italics. The process of adding the above will then require only the following words : Nineteen, thirty-seven, fiftyfour, sixty-five; twenty-seven, forty-four, sixty, seventy-nine; twenty-five, forty-nine, sixty-one, seventy-nine; twenty-two, thirty-one, fifty-five, sixty-one.
2. Reading by four figures at once.—This method is the same as the last, except that four figures are read at the same time, in place of three.
3. Additional Abbreviations.-a. Never stop at ty; that is, if, in the summation of a column, you come to sixty, seventy, or
any other exact number of tens, let the eye catch the amount of two, three, or four figures more, while you are pronouncing the word mentally, so that you may take sixty-six, or sixty-nine, in place of simply sixty, &c. Thus, if in summing up a column you come to the number 50, and, while that number is passing through your mind, you see 4 and 5 as the next two figures above, then you just add 9 to your 50, without even mentally repeating the word fifty. Thus, in running up a column, the words forty, fifty, sixty, &c., never occur alone, but always in combination with one, two, or three, &c., additional numbers.
6. Select the tens as much as practicable; that is, if you have, say thirty-four, and see a six above in the column, even though it may not be adjoining, call your number forty; and, while mentally pronouncing that word, add in two or three more figures, so as, not even in such a case as this, to stop at ty. By careful practice of this method, an intelligent pupil will soon be able to read off six, eight, or even sometimes ten figures at once.
C. Let the eye glance up one column while you are writing the units of the preceding one.
d. Dispense with words altogether. That the mind can call up the idea of the sum of three, four, or more figures, without thinking of the names of the individual numbers, will be evident to any one who will give the experiment a fair trial. We open a book, and possibly the first word that meets the eye is one of many letters and syllables, such as incomprehensibility, and instantly the idea strikes the mind, without its taking cognizance of any of the nineteen letters or eight syllables. The mind seizes it as a whole, without special regard to its individual parts.
Such a power as this may be acquired with numeral characters as well as with letters, and the saving of time will be found to be beyond all calculation. Nor is this all. The mental discipline thus acquired will be of incalculable value in every other study.
Questions to be put by the teacher, before the pupil commences the next Section, and to be repeated from time to time till they are answered without hesitation.- What is addition? See p. 56. What is the result of addition called ? What is the sign of addition? What is it called ? See p. 57. What is the sign of equality ? In what order can figures be placed most conveniently for addition? Should they always be placed in this order ? See p. 124. Why? Where should we begin