MULTIPLICATION.

53. MULTIPLICATION is the operation of taking one number as many times as there are units in another.

54. THE MULTIPLICAND is the number to be taken.

55. THE MULTIPLIER is the number denoting how many times the multiplicand is to be taken.

56. THE PRODUCT is the result of the operation.

57. A COMPOSITE NUMBER is one produced by multiplying two or more numbers together. Thus, 60 is a composite number, because 3 x 4 x 5 = 60.

58. A FACTOR is any one of the numbers which, multiplied together, produce a composite number. Thus, 3, 4, and 5 are factors of the composite number 60.

NOTE.-1. The product, after multiplication, is a composite number, and the multiplicand and multiplier are factors or producers of the product.

2. Multiplication is a short method of addition. For, if the multiplicand be written as many times as there are units in the multiplier, and the numbers added, the sum will be equal to the multiplicand taken as many times as there are units in the multiplier.

Change now the multiplier into the multiplicand: that is, multiply 4 by 6.

Write, in a vertical line, as many 1's as there are units in the new multiplicand (4), and as many vertical lines as there are units in the new multiplier (6), when it is again evident that all the 1's will represent the number of units in the product. Hence,

The product of two factors is the same, whichever factor is used as the multiplier.

Thus, 3 x7 = 7×3 = 21:

also, 6×3 = 3×6=18

9×5=5×9= 45:

also, 8x6 = 6×8=48

60. Product of Several Factors.

Multiply the number 7 by the composite number 6 = 2 × 3.

ANALYSIS.-Write 6 horizontal lines with 7 units in each, and it is evident that the product of 7 × 6=42, will express the number of units in all the lines.

Let us first connect the lines in sets of two each, as at the right; the number of units in each set will then be expressed by 7 × 2=14. But there are three sets; hence, the number of units in all the sets is 14 × 3 = 42.

Again, if we divide the lines into sets of 3 each, as at the left, the number of units in each set will be equal to 7 x 3 = 21; and since there are two sets, the whole number of units will be expressed by 21 X 242.

53. What is Multiplication?-54. What is the number to be taken called?-55. What is the multiplier?-56. What is the product?— 57. What is a composite number?-58. What is a factor? Is the product, after multiplication, a composite number? What are its factors? Why is multiplication a short method of addition ?

59. In how many ways may 6 and 4 be multiplied together? How do the two products compare with each other? What principle does this prove?-60. If several factors be multiplied together, is the prod uct changed by changing their order?

Since the product of either two of the three factors, 7, 3, and 2, will be the same, whichever be taken for the multiplier (Art. 59), and since the same principle will apply to that product and to the other factor, as well as to any additional factor, if introduced, it follows that,

The product of any number of factors will be the same in whatever order they are multiplied.

61. When the multiplier is a composite number.

1. Multiply 215 by 36

Now,

3×3×2×2.

215 × 36

215 ×3×3×2×2:

hence, from the last principle,

I. Separate the composite number into its factors:

II. Multiply the multiplicand and the partial products by the factors, in succession, and the last product will be the entire product sought.

=

NOTE.-Any number whatever, as 440, ending with 0, is a composite number of which 10 is a factor: for, 440 44 X 10. If there are two O's on the right of the significant figures, then 100 is a factor; and so on for a greater number of ciphers. Hence, when there are ciphers on the right of significant figures, either in the multiplicand or multiplier, or both

Multiply the significant figures together, and then annex the ciphers to the product.

62. General Case and Rule.

1. Multiply the number 627 by 214. ANALYSIS.-The multiplicand 627 is to be taken 214 times; that is, 4 units times, 1 ten times, and 2 hundred times. Taking it 4 units times, gives 2508; taking it 1 ten times, gives 627, of which the lowest unit is 1 ten; hence, 7 is written in the tens place; taking it 2 hundred times, gives 1254, the lowest unit of which is 1 hundred. Adding, we have 134178 for the product.

OPERATION.

627

214

2508

627

1254

134178

NOTE.-When the multiplier contains more than one figure, the product obtained by multiplying the multiplicand by a single figure,

is called a partial product. In the example, there are three partial products, 2508, 627, and 1254. The sum of the partial products is equal to the product sought.

Principles from the Analysis.

1. If units be multiplied by units, the unit of the product will be 1.

2. If tens be multiplied by units, the unit of the product will be 1 ten.

3. If hundreds be multiplied by units, the unit of the product will be 1 hundred; and so on.

4. If units of the first order be multiplied by units of a higher order, the units of the product will be the same as that of the higher order.

5. If units of any order be multiplied by units of any other order, the unit of the product will be of an order one less than the sum of the units denoting the two orders.

2. Multiply the compound number £3 8s. 6d. 3far. by 6.

OPERATION.

£ 8. d. far. 3 8 6 3 6

ANALYSIS. Multiplying 3 farthings by 6, we have 18 farthings, equal to 4d. and 2far.; set down the 2far.: then, 6 times 6d. are 36d., and 4 pence to carry, are 40d., equal to 3 shillings and 4d.: then, 6 times 8s. are 48s., and 3s. to carry, are 51 shillings, equal to £2 and 11 shillings; then, 6 times £3 are £18, and £2 to carry, are £20, which set down.

20 11 4 2

NOTE.-The unit of each product will be the same as the unit of the multiplicand. Hence, for the multiplication of all numbers, we have the following

Rule.

Multiply each order of units in the multiplicand, in succession, beginning with the lowest, by each figure in the multiplier, and divide each product by so many units as make one unit of the next higher denomination: write down each remainder under the units of its own order, and carry the quotient to the next product.

NOTE.-In multiplying United States money, care must be taken to point off as many places for cents and mills as there are in the multiplicand.

63. Principles governing Multiplication.

The principles governing the operations of multiplication, are mainly the following:

1. There are three parts in every operation of Multiplication: First, the multiplicand; second, the multiplier; third, the product.

2. The multiplier is always an abstract number, and shows how many times the multiplicand is to be taken.

3. The unit of the product is always the same as the unit of the multiplicand.

4. The product is equal to the sum of the partial products which arise from multiplying the multiplicand, in succession, by each figure of the multiplier.

5. If the multiplier is 1, the product will be equal to the multiplicand.

6. If the multiplier is greater than 1, the product will be as many times greater than the multiplicand as the multiplier is greater than 1.

7. If the multiplier is less than 1, the product will be such a part of the multiplicand as the multiplier is of 1.

61. How do you multiply when the multiplier is a composite number? What is one factor of a number ending in 0? What is one factor when the number ends in two O's? In three O's? &c. How do you multiply such numbers together?

What is a

62. Explain the operation of multiplying 627 by 214. partial product? Explain the five principles which come from this analysis. Give the general rule for multiplication.

63. What is the first principle governing multiplication? What the second? What the third? What the fourth? What the fifth? What the sixth? What the seventh?

64. How many methods are there of proving Multiplication? What are they?