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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 y 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 seven rows can be made of thein, 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 1 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; reserving 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);
then 7 times 2 (hundreds) are 14 (hundreds) and 3 which were reserved are 17 (hundreds). The answer is 1778 cents; and since 100 cents make a dollar, we may say 17 dollars and 78 cents.
The process of multiplication, by a single figure, may be expressed thus : Multiply each figure of the multiplicand by the multiplier, beginning at the right hand, and carry as in addition.
What will 24 oren come to, at 47 dollars apiece ? It does not appear so easy to multiply by 24 as by a num ber consisting of only one figure ; but we may first find the price of 6 oxen, and then 4 times as much will be the price of 24 oxen.
282 dolls. price of 6 oxen.
U128 dolls. price of 24 oxen.
188 dolls. price of 4 oxen,
1128 dolls. price of 24 oxen. A number which is a product of two or more numbers is called a composite or compound number. The numbers, which, being multiplied together, produce the number, are called factors of that number. 4 is a composite number, its factors are 2 and 2, because 2 times 2 are 4. 6 is also a composite number, its factors are 2 and 3. The numbers 8, 9, 10, 12, 14, 15, &c. are composite numbers ; some of them have only two factors, and some have several. The sign X, a cross, in which neither of the marks is either horizontal or perpendicular, is used to express multiplication, Thus 3 X2 = 6, signifies 2 times 3 are equal to 6. 2 X 3 X 5 = 30, signifies 3 times 2 are 6, and 5 times 6 are 30.
Numbers which have several factors, may be divided into a number of factors, less than the whole number of factors, in several ways. 24, for example, has 4 factors, thus, 2 X 2 X 2 X3= 24. This may be divided into 2 factors and into 3 factors in several different ways.
Thus 4 x 6 = 24; 2 X 2 X 6 = 24; 3 x 8 = 24; 2 x 12 = 24; 2 X 6 X 2: = 24.
When several numbers are to be multiplied together, it will make no difference in what order they are multiplied, the result will always be the same.
What will be the price of 5 loads of cider, each load con taining 7 barrels, at 4 dollars a barrel ?
Now 5 loads each containing 7 barrels, are 35 barrels. 35 barrels at 4 dollars a barrel, amount to 140 dollars. Or we may say one load comes to 28 dollars, and 5 loads will come to 140 dollars. Or lastly, 1 barrel from each load will come to 20 dollars, and 7 times 20 are 140. Thus 7 Or 7 Or
What is the price of 23 loaas of hay, at 34 dolls. a loai ?
Multiply 328 by 112,
112 =4X7 X 4
1312 product by 4
9184 product by 28
36736 product by 112 It is easy to see that we may multiply by any other number in the same manner.
This operation may be expressed as follows. To multiply by a composite number. Find two or more numbers, which being multiplied together will produce the multiplier ; multiply the multiplicand by one of these numbers, and then that product by another, and so on, until you have multiplied by all the factors, into which you had divided the multiplier, and the last product will be the product required.
If the multiplier be not a composite number, or if it cannot be divided into convenient factors : Find a composite number as near as possible to the multiplier, but smaller, and multiply by it accor:ling to the above rule, and then add as many times the multiplicand, as this number falls short of the multiplier.
V. I have shown how to multiply any number by a single figure ; and when the muluplier consists of several figures, how to decompose it into such numbers as shall contain but one figure. İt remains to show how to multiply by any number of figures; for the above processes will not always be found convenient.
The most simple numbers consisting of more than one figure are 10, 100, 1000, &c. It will be very easy to multiply by these numbers, if we recollect that any figure written in the second place from the right signifies ten times as many as it does when it stands alone, and in the third place, one hundred times as many, and so on. If a zero be annexed at the right of a figure or any number of figures, it is evident that they will all be removed one place towards the left, and consequently become ten times as great; if įwo zeros be annexed they will be removed two places, and will be one hundred times as great, &c. Hence, to multiply by