« ΠροηγούμενηΣυνέχεια »
9. A man paid debts, as follows: one of 427 dollars; another of 763 dollars; another of 654; another of 500; another of 325; and another of 3,250. How much did he pay ? Ans. 5,919 dollars.
10. In 1820, the following was the population of the several New-England States. How many in all ? Maine,
298,335 New Hampshire, 244,161 Vermont,
235,764 Massachusetts, 523,287 Rhode Island, 83,059
Connecticul, 275,248 Ans. 1,659,854. 11. How many in all the Middle States, containing severally,
Louisiana, 153,407 Ans. 3,879,379. 13. How many inhabitants in the following States and Territories ? Ohio,
66,586 Arkansas Ter. 14,273 Michigan Ter.
8,896 Dist. of Columbia, 33,039 Ans. 906,617. 14. Then, how many inhabitants were there in the whole United States, in 1820 ?
Ans. 9,625,734. The best method of ascertaining whether you have performed your addition correctly, is, to
LOOK OVER THE EXAMPLE CAREFULLY A SECOND TIME, ADDING THE COLUMNS IN A REVERSED ORDER ; THAT IS, DOWNWARDS, IF YOU FISRT ADDED THEM UPWARDS, AND UPWARDS, IF YOU FIRST ADDED THEM DOWNWARDS.
The ascertaining whether an operation in Arithmetic is correctly performed, is called proving the operation, and thr process by which this is done, is called PROOF.
EXAMPLES FOR PRACTICE. 1. A man owns five farms. The first is worth 16,750 dollars; the second, 12,875; the third, 4,387; the fourth, 9,321 ; and the fifth, 6,223 dollars. What is the value of all ?
2. A man possesses property to the amount of 11,764 dollars. Ilo draws a prize in a lottery of 20,000 ; he obtains by marriage, 8,650 ; his uncle dies and leaves him 14,825; in a speculation he gains 5,346; and by his regular trade, he acquires 7,432 dollars. How much has he in all ?
3. Add the following numbers : 1,827,554 ; 9,200,305 ; 723; 5; 68; 23; 537,452; 12,900,020 ; 15; 804.
4. Add 12; 37,853,000,503; 2; 91; 16,020,755; 81 ; 307 ; 205 ; 7,033; 15,000,333; 16,275,850,905,306.
5. Add 555,777,333,222,111; 888,222,666,444,999; 678,678,678, 678,678; 324,324,324,324,324 ; 123,123,123,123,123; 987,987,987, 987,987; 111,222,333,444,555.
6. Add 891,011,123; 12,765,892; 987,654; 321,234 ; 12,621 ; 13,789 ; 15,910; 64; 7,835,876 ; 2,867,955.
7. Add 272 ; 5,004; 10,000 ; 20,000,002 ; 59; 892,736,552 : 11,111,222 ; 901 ; 87; 5,600,222. 8. 9.
13. 123456789123456789123456789 100020003000400050006000789 22203330444055506660777000
8009003004000030085 34562871954327834639385 134614783689000023641
OBSERVATIONS ON ADDITION, FOR ADVANCED PUPILS.
§ IX. The rule directs, that units of the same order should be written under each other, in the same column, before the process of adding is commenced. But it is manifestly of little importance, what arrangement is adopted, if we only add together units of the same orders, and avoid adding those of different orders to each other. The arrangement prescribed by the rule, is adopted, because most convenient, since it brings as near together as possible, the numbers to be added to one another.
The rule directs to begin with the units' order. That this is not necessary, the following example will show,
Add 2357 2357 8468
Add, first the tens ; then the thousands; then 8468 9116
the units ; and then the hundreds. Then add 9116
these amounts, as they stand
It will be seen that the answer is the same, as 19941
when the numbers are added by the rule. The 2i example, however, shows that it would be impos8..
sible to carry, mentally, unless the addition were
to commence with the first order. Hence the 19941
advantage of following the rule. The learner will perceive, that, in order to add with ease, it is necessary to commit to memory an Addition table as far as 9; that is, a table containing the sums of all the possible combinations of two digits. For, thus far, a separate character is used for every number. Numbers greater than 9, are expressed by more than one figure, and the addition is made in each order with as much simplicity as in that of units. Now if we could not add the number ex. pressed by any single figure, all at once, we should be obliged to add unit after unit, until we arrived at the number of units, express. ed by the figure. This is the manner in which children always add. For this reason, they often count on their fingers, in order to ascertain when they have added the proper number of units. All uneducated persons do the same.
These remarks go to show the advantage of our scheme of notation. If we had an independent name and character for every num. bor, we should find it very difficult to perform examples, where large numbers were concerned, and we should either be obliged to commit to memory long and tedious addition tables, or we should be under the necessity of adding unit after unit, as children do. With our present scheme of notation, the largest numbers are managed as easily as units under nine. If the ratio of increase towards the left hand had been twelve fold, instead of ten fold, it would have been necessary to extend our addition tables as high as 11. 1f it had been thirty fold, it would have been necessary to continue them to 29, &c. Of course, we may extend them farther than necessity requires, if We choose. Our common tables extend to 12.
It only remains that we consider the mode of proof. No method can be de. vised, which is a perfect verification of the accuracy of the operation. By none of those, which have been proposed, will an error be necessarily detected, though there is a strong probability that it may: The most natural mode of proving an operation, is to reverse the
process, and go back, from the final result, to the starting point. Hence, the natural mode of proving Addition, ie by Subtraction. As the student, who peruses these observations, is supposed to be acquainted, at least, with the ground rules, we will explain this method, a little more fully: . If two numbers be added, we obtain a third equal to both together. If from this sum then, we take away, or subtract, one of the numbers added, the remainder ought to be the other. This affords us an easy proof, when but two numbers are added. But it usually happens, that several are to be united into one sum. It would be a tedious process, to subtract these, one by one, from the sum; and hence the proof by subtraction, is not used. We might, it is true, add the given numbers into two sums, and take one of these from the total. The remainder would be the other. But there are easier modes than this.
We will briefly mention all, that have been suggested. The first, and sim. plest, is, to add the numbers over again in the same manner as at first. But, if ihere was an error, in the first process, we shall be likely to fall into the same error again ; because all the numbers recur in the same order of succession.
It is better, therefore, to reverse the order, as directed in the last section. Yet, in this case, we cannot positively assert, that we have made no error; though, if the two results agree, it is very probable that the addition has been correctly performed.
A third method is, to divide the example into parts, and to find the amount of each part, separately. Then, add these several amounts, and, if the first result was correct, and this is correct likewise, the two will agree. Thus,
6,878,186 6,878,186 In this example, we made, by way of proof, four separate additions ; dividing the numbers added, into as many different portions. Adding the results, thus obtained, we have a total, which agrees with that first found. Whence we con. clude the operation right. This method of proof is the same, in principle, with that found in common Arithmetics; viz., cut off the upper, or lower line, add the other figures, and, to the result
, add the line cut off. It may be objected to this proof, 'as to the one first mentioned, that the recur. rence of the numbers in the same order, renders it liable to error. This objection, however, is not very strong, when the first row is cut off, since, though the others recur in the same order, each figure is added to a different number, from
that to which it was added in the first process. Thus, in the example above, if we begin at the bottom, 8 is added to 5, making 13; 4 to 13, making 17, &c. If* in the proof, the first row be cut off, 4, which was before added to 13, is added to 8, making 12; 6, which was before added to 17, is added to 12, making 18, and so on ; each figúre, in the proof, being added to a different number, from that to which it was added in the first process. Thus, the chance of error, on account of the figures recurring in the same order, is very small. For, if we made a mistake in adding 4 to 13, it is very improbable that we shall make the same mistake in adding it to 8; and so, in the other cases.
Another method is, to make a second Addition, arranging the numbers differently; placing, for example, the number which was uppermost, in the middle, and that which was in the middle, at the top or bottom; so as to avoid adding them twice in the same order of arrangement. This is a very good method, but often long and tedious.
Another method is, to add each column, and set down its whole amount sepa. rately, and then to add the amounts as they stand, thus,
TOTAL AMOUNT. But, after all, it is better to avoid errors by care and attention, than to detect them after they are made. If, however, any mode of proof be employed, there is one, which, for ingenuity, certainty, and expedition, is superior to any of those which have been mentioned. But, as it is somewhat complex in principle, it was thought proper to defer its insertion to this place. It is called, the proof by casting out of nines, or, more simply, the proof by nines. Before proceeding to it, however, we will attend to a few simple properties of numbers. ART. I.
1. Two men start together, and travel the same way, one at the rate of 10, the other of 9 miles, an hour. At the end of one hour, the former will be 10, the latter, 9 miles from the starting point. Of course, they will be 1 mile apart. If they were, then, both to proceed at the rate of 9 miles an hour, they would always continue 1 mile apart. But, during the second hour, the forward man goes 10 miles, instead of 9, and, consequently, makes the dis. tance between them a mile greater. So that in 2 hours, they are 2 miles apart. For the same reason, in 3 hours, they are 3 miles apart; in 4 hours they are 4 miles apart; and so on. Here we see, that each hour makes one mile more of distance between them. Therefore, the number of miles, between them, will always be equal to the nnmber of hours.
2. I wish to measure a stick of timber. I use a pole 10 feet long, and make a mark at the end of every ten feet I measure.
Another man follows me, using a pole 9 feet long. He, likewise, makes a mark at the end of every nine feet. As his pole is 1 foot shorter than mine, his fist mark will be one foot behind mine. If he