Art. 87. In addition, one quantity is connected with another. It is frequently the case, that the quantities brought together are equal; that is, a quantity is added to itself.
As a + a = 2a;
a + a + a = 3a;
a + a + a+ a = 4a;
a + a + a + a + a = 5a;
This repeated addition of a quantity to itself, is what wag originally called multiplication. But the term, as it is now used, has a more extensive signification. We have frequent occasion to repeat, not only the whole of a quantity, but a certain portion of it. If the stock of an incorporated com* pany is divided into shares, one man may own ten of them, another five, and another a part only of a share, say two-fifths. When a dividend is made, of a certain sum on a share, the first is entitled to ten times this sum, the second to five times, and the third to only two-fifths of it. As the apt-portioning of the dividend, in each of these instances, is upon the same principle, it is called multiplication in the last, as well as in the two first.
Again, suppose a man is obligated to pay an annuity of 100 dollars a year. As this is to be subtracted from his estate, it may be represented by -a. As it is to be subtracted year after year, it will become, in four years,-a - a - a - a = -4a. This repeated subtraction is also called multiplication. According to the view of the subject;
88. Multiplying by a whole number is taking the multiplicand as many times, as there are units in the multiplier.
Multiplying by 1, is taking the multiplicand once, as a.
Multiplying by 2, is taking the multiplicand twice, as a + a.
Multiplying by 3, is taking the multiplicand three times, as a + a + a.
Multiplying by a FRACTION is taking a certain PORTION of the multiplicand as many times, as there are like portions of a unit in the multiplier.
Multiplying by 1/3, is taking 1/3 of the multiplicand, once, as (1/3)a. Multiplying by 2/3, is taking 1/3 of the multiplicand, twice, as
(1/3)a + (1/3)a.
Multiplying by 3/3, is taking 1/3 of the multiplicand, three times.
Hence, if the multiplier is a unit, the product is equal to the multiplicand: If the multiplier is greater than a unit, the product is greater than the multiplicand: And if the multiplier is less than a unit, the product is less than the multiplicand.
Multiplication by a NEGATIVE quantity, has the same relation to multiplication by the positive quantity, which SUBTRACTION has to addition. In the one, the sum of the repetitions of the multiplicand is to be added, to the other quantities with which this multiplier is connected. In the other, the sum of these repetitions is to be subtracted from the other quantities. This subtraction is performed at the time of multiplying, by changing the sign of the product.
89. Every multiplier is to be considered a number. We sometimes speak of multiplying by a given weight or measure, a sum of money. But this is abbreviated language. If construed literally, it is absurd. Multiplying is taking either the whole or a part of a quantity, a certain number of times. To say that one quantity is repeated as many times, as another is heavy, is nonsense. But if a part of the weight of a body be fixed upon as a unit, a quantity may be multiplied by a number equal to the number of these parts contained in the body. If a diamond is sold by weight, a particular price may be agreed upon for each grain. A grain is here the unit; and it is evident that the value of the diamond, is equal to the given price repeated as many times, as there are grains in the whole weight. We say concisely, that the price is multiplied by the weight: meaning that it is multiplied by a number equal to the number of grains in the weight. In a similar manner, any quantity whatever may be supposed to be made up of parts, each being considered a unit, and any number of these may become a multiplier.
90. As multiplying is taking the whole or a part of a quantity a certain number of times, it is evident that the product, must be of the same nature as the multiplicand.
If the multiplicand is an abstract number; the product will be a number.
If the multiplicand is weight, the product will be weight. If the multiplicand is a line, the product will be a line. Repeating a quantity does not alter its nature. It is frequently said, that the product of two liqes is a surface, and that the product of three lines is a solid. But these are abbreviated expressions, which if interpreted literally are not correct.
91. The multiplication of fractions will be the subject of a future section. We have first to attend to multiplication by positive whole numbers. This, according to the define tibn (Art. 88.) is taking the multiplicand as many times, as there are units in the multiplier. Suppose a is to be multiplied by b, and that b stands for 3. There are then, three units in thee multiplier b. The multiplicand must therefore be taken three times; thus, a + a + a = 3a, or ba.
So that, multiplying two letters together is nothing more, than uniting them one after the other, either with, or without the sign of multiplication between them. Thus b multiplied into c is b.c, or bc. And x into y, is x.y or xy.
92. If more than two letters are to be multiplied, thev must be connected in the same manner. Thus a into b and c, is cba. For by the last article, a into b, is ba. This product is now to be multiplied into c. If c stands for 5, then ba is to be taken five times thus,
ba + ba + ba + ba + ba = 5ba or cba.
The same explanation may be applied to any number of letters. Thus, am into xy, is amxy. And bh into mrx, is bhmrx.
93. It is immaterial in what order the letters are arranged The product ba is the same as ab. Three times five is equal to five times three. Let the number 5 be represented by air many points, in a horizontal line; and the number 3, by off many points in a perpendicular line.
• • • • •
• • • • •
• • • • •
Here it is evident that the whole number of points is equal, either to the number in the horizontal row three times repeated, or to the number in the perpendicular row five times repeated; that is, to 5.3, or 3.5. This explanation may be extended to a series of factors consisting of any numbers whatever. For the product of two of the factors may be considered as one number. This may be placed before or after a third factor: the product of three, before or after a fourth.
Thus 24 = 4.6 or 6.4=4.3.2 or 4.2.3 or2.3.4.
The product of a, b, c, and d, is abcd, or acdb, or dcba, or badc. It will generally be convenient, however, to place the letters in alphabetical order.
94. WHEN THE LETTERS HAVE NUMERICAL COEFFICIENTS, THESE MUST BE MULTIPLIED TOGETHER, AND PREFIXED TO THE PRODUCT OF THE LETTERS.
Thus, 3a into 2b, is 6ab. For if a into b is ab, then 3 times a into b, is evidently 3ab: and if, instead of multiplying by b, we multiply by twice b, the product must be twice as great; that is 2.3ab or 6ab.
95. If either of the factors consists of figures only, these must be multiplied into the coefficients and letters of the other factors.
Thus 3ab into 4, is 12ab. And 36 into 2x, is 72x. And 24 into hy, is 24hy.
96. If the multiplicand is a compound quantity, each of its terms must be multiplied into the multiplier. Thus b + c + d into a is ab + ac + ad. For the whole of the multiplicand ie to be taken as many times, as there are units in the multiplier. If then a, stands for 3, the repetitions of the multiplicand are,
b + c + d
b + c + d
b + c + d
And their sum is 3b + 3c + 3d, that is, ab + ac + ad.
97. The preceding instances must not be confounded with those in which several factors are connected by the sign . . In the latter case, the multiplier is to be written before the other factors without being repeated. The product of b.d into a, is ab.d, and not ab.ad. For b.d is bd, and this into a, is abd. (Art. 92.) The expression b.d is not to be considered, like b + d, a compound quantity consisting of two terms. Different terms are always separated by+ or -. The product of b.h.m.y into a, is a.b.h.m.y or abhmy. But b + h + m + y into a, is ab + ah + am + ay.
98. If both the factors are compound quantities, each term in the multiplier must be multiplied into each in the multiplicand.
Thus a + b into c + d is ac + ad + bc + bd.
For the units in the multiplier a + b are equal to the units in a added to the units in b. Therefore the product produced by a, must be added to the product produced by b.
99. When several terms in the product are alike, it will be expedient to set one under the other, and then to unite them, by the rules for the reduction in addition.
100. It will be easy to see that when the multiplier and multiplicand consist of any quantity repeated as a factor, this factor will be repeated in the product, as many times as in the multiplier and multiplicand together.Mult. a.a.a Here a is repeated three times as a factor.
Into a.a Here it is repeated twice.
Prod. a.a.a.a.a Here it is repeated five times.
The product of bbbb into bbb, is bbbbbbb.
The product of 2x.3x.4x into 5x.6x, is 2x.3x.4x.5x.6x.
101. But the numeral coefficients of several fellow-factors may be brought together by multiplication.
Thus 2a.3b into 4a.5b is 2a.3b.4a.5b, or 120aabb.
For the coefficients are factors, and it is immaterial in what order these are arranged. (Art. 93.)
The product of 3a.4bh. into 5m.6y, is 360abhmy.
102. The examples in multiplication thus far have been confined to positive quantities. It will now be necessary to consider in what manner the result will be affected, by multiplying positive and negative quantities together. We shall find,
That + into + produces +,
- into + produces -,
+ into - produces -,
- into - produces +
All these may be comprised in one general rule, which it will be important to have always familiar. If the signs of the factors are ALIKE, the sign of the product will be affirmative; but if the signs of the factors are UNLIKE, the sign of the product will be negative.
103. The first case, that of + into +, needs no farther illustration. The second is - into +, that is, the multiplicand is negative, and the multiplier positive. Here -a into +4 is - 4a.
104. In the two preceding cases, the affirmative sign prefixed to the multiplier shows, that the repetitions of the multiplicand are to be added to the other quantities with which the multiplier is connected. But in the two remaining cases, the negative sign prefixed to the multiplier, indicates that the sum of the repetitions of the multiplicand are to be subtracted from the other quantities. (Art. 88.) And this subtraction is performed, at the time of multiplying, by making the sign of the product opposite to that of the multiplicand. Thus +a into -4 is -4a. For the repetitions of the multiplicand are,
+a + a+ a + a = +4a.
But this sum is to be subtracted, from the other quantities with which the multiplier is connected. It will then become -4a.
Thus in the expression b - (4.a) it is manifest that 4.a is to be subtracted from b. Now 4.a is 4a, that is +4a. But to subtract this, from b, the sign + must be changed into -. So that b - (4.a) is b - 4a. And a.(-4) is therefore - 4a.
Again, suppose the multiplicand is a, and the multiplier (6 - 4). As (6 - 4) is equal to 2, the product will be equal to 2a. This is less than the product of 6 into a. To obtain then the product of the compound multiplier (6 - 4) into a, we must subtract the product of the negative part, from that of the positive part.
Multiplying a in 6 - 4, is the same as multiplying a into 2. And the product 6a - 4a, is the same as the product 2a. Therefore a into -4, is -4a.
But if the multiplier had been (6 + 4), the two products must have been added. Multiplying a into (6 + 4), is the same as multiplying a into 10. And the prod. 6a + 4a is the same as the product 10a.
This shows at once the difference between multiplying by a positive factor, and multiplying by a negative one. In the former case, the sum of the repetitions of the multiplicand is to be added to, in the latter, subtracted from the other quantities, with which the multiplier is connected. For every negative quantity must be supposed to have a reference to some other which is positive; though the two may not always si and in connection, when the multiplication is to be performed.
105. If two negatives be multiplied together, the product will be affirmative : (-4).(-a) = +4a. In this case, as in the preceding, the repetitions of the multiplicand are to be subtracted, because the multiplier has the negative sign. These repetitions, if the multiplicand is -a, and the multiplier -4, are -a - a - a - a = -4a. But this is to be subtracted by changing the sign. It then becomes +4a.
Suppose -a is multiplied into (6-4). As 6 - 4 = 2, the product is, evidently, twice the multiplicand, that is, -2a. But if we multiply -a into 6 and 4 separately; -a into 6 is - 6a, and - a into 4 is - 4a. As in the multiplier, 4 is to be subtracted from 6; so, in the product, -4a must be subtracted from -6a. Now -4a becomes by subtraction +4a. The whole product then is - 6a+4a which is equal to - 2a. Or thus,
Multiplying -a into 6 - 4 is the same as multiplying -a into 2. And the prod. -6a + 4a, is equal to the product -2a.
It is often considered a great mystery, that the product of two negatives should be affirmative. x But it amount** to nothing more than this, that the subtraction of a negative quantity, is equivalent to the addition of an affirmative one; and, therefore, that the repeated subtraction of a negative quantity, is equivalent to a repeated addition of nn affirmative one. Taking off from a man's hands a debt of ten dollars every mouth, is adding ten dollars a month to the value of his property.
106. As a negative multiplier changes the sign of the quantity which it multiplies; if there are several negative factors to be multiplied together,
The two first will make the product positive;
The third will make it negative;
The fourth will make it positive;
Thus-a × -b = +ab
+ab × -c = -abc
-abc × -d = +abcd
+abcd × -e = -abcde
That is, the product of any even number of negative factors is positive; but the product of any odd number of negative factors is negative.
The product of several factors which are all positive, is invariably positive.
107. Positive and negative terms may frequently balance each other, so as to disappear in the product. A star is sometimes put in the place of a deficient term.
108. For many purposes, it is sufficient merely to indicate the multiplication of compound quantities, without actually multiplying the several terms. Thus the product of a + b + c into h + m + y, is (a + b + c).(h + m + y). The product of a + m into h + x and d + y, is (a + m).(h + x).(d + y).
By this method of representing multiplication, an important advantage is often gained, in preserving the factors distinct from each other.
When the several terms are multiplied in form, the expres sion is said to be expanded. Thus,
(a + b).(c + d) becomes when expanded ac + ad + bc + bd
109. With a given multiplicand, the less the multiplier, the less will be the product. If then the multiplier be reduced to nothing, the product will be nothing. Thus a.0 = 0. And if 0 be one of any number of fellow-factors, the product of the whole will be nothing.
Thus, ab.c.3d.0 = 3abcd.0 = 0. And (a + b).(c + d).(h - m).0 = 0.
110. Although, for the safee of illustrating the different points in multiplication, the subject has been drawn out into a considerable number of particulars; yet it will scarcely be necessary for the learner, after he has become familiar with the examples, to burden his memory with any thing more than the following general rule.
Multiply the letters and coefficients of each term in the multiplicand, into the letters andcoefficients of each term in the multiplier; and prefix to each term of the product, the sign required by the principle, that like signs produce +, and different signs -.
1. Mult. a + 3b - 2 into 4a - 6b - 4.
2. Mult. (7ah - y).4 into 4x.3.5.d.
3. Mult. 3ay + y - 4 + h into (d + x).(h + y).
4. Mult. 7ay -1 + h.(d - x) into -(r + 3 - 4m).