ed to twenty. But twelve is the same as ten and two, there. fore we may say twenty and ten are thirty and two are thirtytwo. A man bought a cow for 27 dolls. and a horse for 68 dolls. How much did he give for both ? Operation. Cow 27 dolls. In this example it is proposed to add Horse 68 dolls. together 27 and 68. Now 27 is 2 tens and 7 units; and 68 is 6 tens and 8 Both 95 dolls. units. 6 tens and 2 tens are 8 tens ; and 8 units and 7 units are 15, which is 1 ten and 5 units ; this joined to 8 tens makes 9 tens and 5 units, or 95. A man bought ten barrels of cider for 35 dolls., and 7 bar. rels of flour for 42 dolls., a hogshead of molasses for 36 dolls., a chest of tea for 87 dolls., and 3 hundred weight of sugar for 24 dolls. What did the whole amount to ? Operation. Cider 35 dolls. In this example there are five numFlour 42 dolls. bers to be added together. We obMolasses 36 dolls. serve that each of these numbers conTea 87 dolls. sists of two figures. It will be most Sugar 24 dolls. convenient to add together either all the units, or all the tens first, and then Amount 224 dolls. the other. Let us begin with the 3 tens and 4 tens are seven tens, and 3 are 10 tens, end 8 are 18 tons, and 2 are 20 tens, or 200. Then adding the units, 5 and 2 are 7, and 6 are 13, and 7 are 20, and 4 are 24, that is, 2 tens and 4 units; this joined to 200 makes 224. It would be still more convenient to begin with the units, in the following manner ; 5 and 2 are 7, and 6 are 13, and 7 are 20, and 4 are 24, that is 2 tens and 4 units; we may now set down the 4 units, and reserving the 2 tens add them with the other tens, thus : 2 tens (which we reserved) and 3 tens are 5 tens, and 4 are 9 tens, and 3 are 12 tens, and 8 are 20 tens, and 2 are 22 tens, which written with the 4 units make 224 as before. A general has three regiments under his command; in the first there are 478 men; in the second 564; and in the third 593. How many men are there in the whole ? tens. Operation. First reg. 478 men In this example, each of the Second reg. 564 men numbers is divided into three Third reg. 593 men parts, hundreds, tens, and units. To add these together it is most In all 1,635 men convenient to begin with the units as follows ; 8 and 4 are 12, and 3 are 15; that is, 1 ten and 5 units. We write down the 5 units, and reserving the 1 ten, add it with the tens. 1 ten (which we reserved) and 7 tens are 8 tens, and 6 are 14 tens, and 9 are 23 tens, that is, 2 hundreds and 3 tens. We write down the 3 tens, and reserving the 2 hundreds add them with the hundreds. 2 hundreds (which we reserved) and 4 hundreds are 6 hundreds, and 5 are 11 hundreds, and 5 are 16 hundreds, that is, 1 thousand and 6 hundreds. We write down the 6 hundreds in the hundreds' place, and the 1 thousand in the thousands' place. The reserving of the tens, hundreds, &c. and adding them with the other tens, hundreds, &c. is called carrying. The principle of carrying is more fully illustrated in the following example. A merchant had all his money in bills of the following description, one-dollar bills, ten-dollar bills, hundred-dollar bills, thousand-dollar bills, fc. each kind he kept in a separate box. Another merchant presented three notes for payment, one 2,673 dollars, another 849 dollars, and another 756 dollars. How much was the amount of all the notes ; and how many bills of each sort did he pay, supposing he paid it with the least possible number of bills ? Operation. The first note would require 2 of the thousand-dollar bills; 6 of 8 9 of the hundred-dollar bills ; 7 ten7 5 6 dollar bills; and 3 one-dollar bills. 4 2 7 8 The second note would require 8 of the hundred-dollar bills; 4 ten-dollar bills; and 9 onedollar bills. The third note would require 7 of the hundreddollar bills; 5 ten-dollar bills; and 6 one-dollar bills. Count ing the one-dollar bills, we find 18 of them. This may be paid with 1 ten-dollar bill and 8 one-dollar bills; putting this i ten-dollar bill with the other ten-dollar bills, we find 17 of them. This may be paid with 1 hundred-dollar bill, and 7 ten-dollar bills; putting this one-hundred dollar bill with the other hundred-dollar bills, we find 22 of thern ; this may be paid with 2 of the thousand-dollar bills, and 2 of the hundred-dollar bills ; putting the 2 thousand-dollar bills with the other thousand-dollar bills, we find 4 of them. Hence the three notes may be paid with 4 of the thousand-dollar bills, 2 of the hundred-dollar bills, 7 ten-dollar bills, and 8 onedollar bills, and the annount of the whole is 4,278 dollars. Besides the figures, there are other signs used in arithmetic, which stand for words or sentences that frequently occur. These signs will be explained when there is occasion to use them. A cross + one mark being perpendicular, the other horizontal, is used to express, that one number is to be added to another. Two parallel horizontal lines = are used to express equality between two numbers. This sign is generally read is or are equal to. Example 5 + 3 = 8, is read 5 and 3 are 8; or 3 added to 5 is equal to 8; or 5 more 3 is equal to 8; or ' more frequently 5 plus 3 is equal to 8; plus being the Latin word for more. These four expressions signify precisely the same thing. Any number consisting of several figures may sometimes be conveniently expressed in parts by the above method. Example, 2358 = 2000 + 300 + 50. + 8 = 1000 + 1200 + 140 + 18. A man owns three farms, the first is worth 4,673 dollars ; the second, 5,764 dollars ; and the third, 9,287 dollars. How many dollars are they all worth ? Perhaps the principle of carrying may be illustrated more plainly by separating the different orders of units from each Other. Operation. 4673 may be written 4000 + 600 + 70+ 3* 5764 5000 + 700 + 60 + 4 9287 9000 + 200 + 80 + 7 14 21. 15.. 18... Placing the results under each other, we have 18,000 19,724 = 19,724 In this example the sum of the units is 14, the sum of the tens is 21 tens or 210, the sum of the hundreds is 15 hundreds or 1,500, the sum of the thousands is 18 thousands or 18,000; these numbers being put together make 19,724. If we take this example and perform it by carrying the tens, the same result will be obtained, and it will be perceived that the only difference in the two methods is, that in this, we add the tens in their proper places as we proceed, and in the other, we put it off until we have added each column, and then add them in precisely the same places. Operation. 4,673 Here as before the sum of the units is 14, +-5,764 but instead of writing 14 we write only the 4, 19,287 and reserving the 1 ten, we say 1 (ten, which we reserved) and 7 are 8, and 6 are 14, and =19,724 8 are 22 (tens) or 2 hundreds and 2 tens ; setting down the 2 tens and reserving the hundreds, we say, 2 (hundreds, which we reserved) and 6 are 8, and 7 are 15, and 2 are 17 (hundreds) or 1 thousand and a hundreds ; writing down the 7 hundreds, and reserving the 1 thousand, we say, 1 (thousand, which we reserved) and 4 are 5, and 5 are 10, and 9 are 19 (thousands) or 1 ten-thousand and 9 thousands ; we write the 9 in its proper place, and since there is nothing more to add to the 1 (ten thousand) we write that down also, in its proper place. The answer is 19,724 dollars. * It will be well for the learner to separate, in this way, several of the examples in Addition, because this method is frequently used for illustration in other parts of the book. We may now observe another advantage peculiar to this method of notation. It is, that all large numbers are divided into parts, in order to express them by the different orders of units, and then we add each different order separately, and without regard to its name, observing only that ten, in an inferior order, is equal to one in the next superior order. By this means we add thousands, millions, or any of the higher orders as easily as we add units. If on the contrary we had as many names and characters, as there are numbers which we have occasion to use, the addition of large numbers would become extremely laborious. The other operations are as much facilitated as Addition, by this method of notation. In the above examples the numbers to be added have been written under each other. This is not absolutely necessary; we may add them standing in any other manner, if we are careful to add units to units, tens to tens, &c., but it is generally most convenient to write them under each other, and we shall be less liable to make mistakes. In the above examples we commenced adding the numbers at the top of each line, but it is easy to see that it will make no difference whether we begin at the top or bottom, since the result will be the same in either case. Proof. The only method of proving addition, which can properly be called a proof, is by subtraction. This will be explained in its proper place. The best way to ascertain whether the operation has been correctly performed, is to do it over again. But if we add the numbers the second time in the same order as at first, if a mistake has been made, we are very liable to make the same mistake again. To prevent this, it is better to add them in a reversed order, that is, if they were added downwards the first time, to add them upwards the second time, and vice versa.* * The method of omitting the upper line the second time, and then adding it to the sum of the rest is liable to the same objection, as that of adding the numbers twice in the same order, for it is in fact the same thing. If this method were to be used, it would be much bet. ter to omit the lower lino instead of the upper one when they are added upward ; and the upper line when added downward. This would change the order in which the numbers are put together. The danger of making the same mistake is this : if in adding up a row of figures we should somewhere happen to say 26 and 7 are 35, if we add it over again in the same way, we are very liable to say so again. But in adding it in another order it would be a very singular eoincidence if a mistake of exactly the same number were made. |