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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, fc. First add together 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 preceding column ; reserve the tens as before, and set down the excess, and so on, till all the columns are added.
III. 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 very 34 cents easily by the common method of addition. 34 cents But it is easy to see that if it were required 34 cents to find the price of 20, 30, or 100 gal
lons, the operation would become laborious Ans. 136 cents on account of the number of times the number 34 must be written down.
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 that 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 Multiplica tion. 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, then 4 times 3 are 12, and 1 (which was reserved) are 13.
In order to perform multiplication readily, it is necessary to retain in n omory the 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.
TABLE OF PYTHAGORAS.
22 24 26 28
28 30 32 34 36
10 11 12 13 14
56 60 64 68 72 76 80
70 75 80 85 90 95 100 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90
1141 14 21 28 35
70 77 84 91 98 105 112 119 126 133 140 10 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160
99 | 108 | 117 | 126 135 144 153 162 171 180
40 | 50
90 100 110 120 130 140 150 160 170 180 190 200
To form this table, write the numbers 1, 2, 3, 4, &c. as far as you wish the table to extend, in a line horizontally. This is the first or upper row.
To form the second row, add these numbers to hemselves, and write them in a row directly under the first. Thus 1 and 1 are 2 ; 2 and 2 are 4; 3 and 3 are 6; 4 and 4 are 8; &c. To form the third row, add the second row to the first, thus 2 and 1 are 3; 4 and 2 are 6; 6 and 3 are 9; 8 and 4 are 12; &c. This will evidently contain the first row three times. To form the fourth row, add the third to the first, and so on, till you have formed as many rows as you wish the table to contain.
When the formation of this table is well understood, the mode of using it may be easily conceived. If for instance the product of 7 by 5, that is, 5 times 7 were required, look for 7 in the upper row, then directly under it in the fifth row, you find 35, which is 7 repeated 5 times. In the same manner any other product may be found. If
you seek in the table of Pythagoras for the product of 5 by 7, or 7 times 5, look for 5 in the first row, and directly under it in the seventh row you will find 35, as before. It appears therefore that 5 times 7 is the same as 7 times 5. In the same manner 4 times 8 are 32, and 8 times 4 are 32; 3 times 9 are 27, and 9 times 3 are 27. In fact this will be found to be true with respect to all the numbers in the table. From this we should be led to suppose, that, whatever be the two numbers which are to be multiplied together, the product will be the same, whichsoever of them be made the multiplier.
The few products contained in the table of Pythagoras are not sufficient to warrant this conclusion. For analogical reasoning is not allowed in mathematics, except to discover the probability of the existence of facts. But the facts are not to be admitted as truths until they are demonstrated. I shall therefore give a demonstration of the above fact; which, besides proving the fact, will be a good illustration of the manner in which the product of two numbers is formed.
There is an orchard, in which there are 4 rows of trees, and there are my trees in each row.
If one tree be taken from each row, a row may be made consisting of four trees; then one more taken
from each row will make another row of four trees ; and since there are seven trees in each
row, it is evident that in this way seven rows, of four trees each, may be made of them. But the number of trees remains the same, which way soever they are counted.
Now whatever be the number of trees in each row, if they are all alike, it is plain that as many rows, of four each, can be made, as there are trees in a row. Or whatever be the number of rows of seven each, it is evident that severi rows can be made of them, each row consisting of a number equal to the number of rows. In fine, whatever be the number of rows, and whatever be the number in each row, it is plain that by taking one from each row a new row may be made, containing a number of trees equal to the number of rows, and that there will be as many rows of the latter kind, as there were trees in a row of the former kind.
The same thing may be demonstrated abstractly as follows : 6 times 5 means 6 times each of the units in 5; but 6 times 1 is 6, and 6 times 5 will be 5 times as much, that is, 5 times 6.
Generally, to multiply one number by another, is to repeat the first number as many times as there are units in the second number. To do this, each unit in the first must be repeated as many times as there are units in the second. But each unit of the first repeated so many times, makes a number equal to the second ; therefore the second number will be repeated as many times as there are units in the first. Hence the product of two numbers will always be the same, whichsoever be made multiplier.
What will 254 pounds of meat cost, at 7 cents per pound?
This question will show the use of the above proposition ; for 254 pounds will cost 254 times as much as 1 pound; but I pound costs 7 cents, therefore it will cost 254 times 7. But since we know that 254 times 7 is the same as 7 times 254, it will be much more convenient to multiply 254 by 7. It is easy to show here that the result must be the same ; for 254 pounds at 1 cent a pound would come to 254 cents; at, 7 cents a pound therefore it must come to 7 times as much
Here say 7 times 4 are 28; reserv7
ing the 2 (tens) write the 8 (units);
then 7 times 5 (tens) are 35 (tens) and Ans. 1778 cents. 2 (tens) which were - reserved are 37 (tens); write the 7 (tens) and reserve the 3 (hundreds) ;