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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 1 hundred-dollar bill with the other hundred-dollar bills, we find 22 of them; 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 one-dollar bills, and the amount 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, thai 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 some times 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 * froin each other.
be written 4000 + 600 + 70 + 3* 5764
5000 + 700 + 60 + 4 9287
9000 + 200 + 80 + 7
18000 + 1500 + 210 + 14
Placing the results under each other, we have
18,000 + 1,500 + 210 + 14
= 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.
4,673 Here as before the sum of the units + 5,764 is 14, but instead of writing 14 we + 9,287 write only the 4, and reserving the 1
ten, we say 1 (ten, which we reserved) 19,724 and 7 are 8, and 6 are 14, and S 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 7 hundreds ; writ
* 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.
ing down the 7 hundreds, and reserving the 1 thousand, we say, 1 (thousand, which we reserved) and 4 are 5, and 5 are ten, 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.
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 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 downward the first time, to add them upwards the second time, and vice versa.*
From what has been said it appears, that the operation of addition may be reduced to the following
Rule. Write down the numbers in the most convenient manner, which is generally so that the units may stand under units, tens under tens, &c. First add to-gether all the units, and if they do not exceed nine; write the result in the units' place ; but if they amount to ten or more than ten, reserve the ten or tens, and write down the excess above even tens, in the units' place. Then add the tens, and add with them the tens which were reserved from the column ; reserve the tens as before, and set down the excess, and so on, till all the columns are added.
Multiplication. III. 1. Questions often occur in addition in which a number is to be added to itself several times.
How much will 4 gallons of molasses come to at 34 cents a gallon?
34 cents This example may be performed 34 cents
very easily by the common method 34 cents of addition. But it is easy to see 34 cents
that if it were required to find the
price of 20, 30, or 100 gallons, the Ans. 136 cents operation would become laborious on
* 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 mucha better to omit the lower line 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 coincidence if a mistake of exactly the same number were made.
account of the number of times the number 34 must be written down. X
I find in adding the units that 4 taken 4 times amounts to 16, I write the 6 and reserve the ten; 3 taken 4 times amounts to 12, and 1 which I reserved makes 13, which I write down, and the whole number is 136 cents.
If I have learned that 4 times 4 are 16, and that 4 times 3 are 12, it is plain that I need not write the number 34 but once, and then I may say 4 times 4 are 16, reserving the ten and writing the 6 units as in addition. Then again, 4 times 3 (tens) are 12 (tens) and 1 (ten which I reserved) are 13 (tens.)
Addition performed in this manner is called Multiplication. In this example 34 is the number to be multiplied or repeated, and 4 is the number by which it is to be multiplied; that is, it expresses the number of times 34 is to be taken.
The number to be multiplied is called the multiplicand, and the number which shows how many times the multiplicand is to be taken is called the multiplier, The answer or result is called the product. They are usually written in the following manner :
136 product Having written them down, say 4 times 4 are 16, write the 6 and reserve the ten, ihen 4 times 3 are 12, and 1 (which was reserved) are 13.
In order to perform multiplication readily, it is necessary to retain in memory ihe sum of each of the nine digits repeated from one to nine times; that is, the products of each of the nine digits by themselves, and by each other. These are all that are absolutely necessary, but it is very convenient to remember the products of a much greater number. The annexed table, which is called the table of Pythagoras, contains the products of the first twenty numbers by the first ten.