Art. 111. IN multiplication, we have two factors given, and are required to find their product By multiplying the factors 4 and 6, we obtain the product 24. But it is frequently necessary to reverse this process. The number 24, and one of the factors may be given, to enable us to find the other. The operation by which this is effected, is called Division. We obtain the number 4, by dividing 24 by 6. The quantity to be divided is called the dividend ; the given factor, the divisor; and that which is required the quotient.
112. DIVISION si finding a quotient, which multiplied into the divisor will produce the divident.
In multiplication the multiplier is always a number. And the product is a quantity of the same kind, as the, multiplicand. The product of 3 rods into 4, is 12 rods. When we come to division, the product and either of the factors may be given, to find the other: that is,
The divisor may be a number, and then the quotient will be a quantity of the same kind as the dividend ; or,
The divisor may be a quantity of the same kind as the dividend; and then the quotient will be a number.
Thus 12 rocs/4 = 3 rods. But 12 rods/3rods = 4.
And 12 rods/24 =1/2 rod. And 12 rods/24rods =1/2.
In the first case, the divisor being a number, shows into how many parts the dividend is to be separated; and the quotient shows what these parts are.
If 12 rods be divided into 3 parts, each will be 4 rods long. And if 12 rods be divided into 24 parts, each will be half a rod long.In the other case, if the divisor is less than the dividend, the former shows into what parts the latter is to be divided; and the quotient shows how many of these parts are contained in the dividend. In other words, division m this case consists in finding how often one quantity is contained in another.
A line of 3 rods, is contained in one of 12 rods, four times.
But if the divisor is greater than the dividend, and yet a quantity of the same kind, the quotient shows what part of the divisor is equal to the dividend.
Thus one half of 24 rods is equal to 12 rods.
113. As the product of the divisor and quotient is equal to the dividend, the quotient may be found, by resolving the dividend into two such factors, that one of them shall be the divisor. The other will, of course, be the quotient.
Suppose abd is to be divided by a. The factor a and bd will produce the dividend. The first of these, being a divisor, may be set aside. The other is the quotient. Hence,
When the divisor is found as a factor in the dividend, the division is performed by cancelling this factor.
In each of these examples, the letters which are common to the divisor and dividend, are set aside, a&d the other letters form the quotient. It will be seen at once, that the product of the quotient and divisor is equal to the dividend.
114. If a letter is repeated in the dividend, care must be taken that the factor rejected be only equal to the divisor.
In such instances, it is obvious that we are not to reject every letter in the dividend which is the sairie with one in the divisor.
115. If the dividend consists of any factors whatever, expunging one of them is dividing by it.
|Divide||a(b + d)||a(b + d)||(b + x)(c + d)||(b + y)(d - h)x|
|By||a||b + d||b + x||d - h|
|Quot.||b + d||a||c + d||(b + y)x|
116. In performing multiplication, if the factors contain numeral figures, these are multiplied into each other. (Art. 94.) Thus 3a into 7b is 21ab. Now if this process is tobe reversed, it is evident that dividing the number in the product, by the number in one of the factors, will give the number in the other factor. The quotient of 21ab/3a is 7b. Hence,
In division, if there are numeral coefficients prefixed to the letters, the coefficient of the dividend must be divided, by the coefficient of the divisor.
117. When a simple factor is multiplied into a compound one, the former enters into every term of die latter. Thus a into b + d is ab + ad. Such a product is easily resolved again into its original factors.
Thus ab + ad = a.(b + d).
ab + ac + ah = a.(b + c + h).
Now if the whole quantity be divided by one t)f these factors, according to Art 115, the quotient will be the other factor.
If the divisor is contained in every term of a compound dividend, it must be cancelled in each.
|Divide||ab + ac||aah + ay|
|Quot.||b + c||ah + y|
And if there are cvefficients, these must be divided, in each term also.
|Divide||6ab + 12ac||12hx + 8|
|Quot.||2b + 4c||3hx + 2|
118. On the other hand, if a compound expression containing any factor in every term, be divided by the other quantities connected by their signs, the quotient will be that factor. See the first part of the preceding article.
119. In division, as well as in multiplication, the caution must be observed, not to confound terms with factors.
Thus (ab + ac)/a = b + c.
But (ab.ac)/a = aabc/a = abc.
And (ab + ac)/(b + c) = a.
But (ab.ac)/(b.c) = aabc/bc = aa.
120. In division, the same rule is to be observed respecting the signs, as in multiplication ; that is, if the divisor and dividend are both positive, or both negative, the quotient must be positive : if one is positive and the other negative, the quotient must be negative.
This is manifest from the consideration that the product of the divisor and quotient must be the same as the dividend.
+a.+b = +ab
-a.+b = -ab
+a.-b = -ab
-a.-ab = +ab
+ab/+a = +a
-ab/+b = -a
-ab/-b = +a
+ab/-b = -a
|Divide||abx||8a - 10ay||6am.dh|
|Quot.||-bx||-4 + 5y||-3m.dh = -3dhm|
121. If Tins letters of the divisor are not to be found in the dividend, the division is expressed by writing the divisor under the dividend, in the form of a vulgar fraction.
This is a method of denoting division, rather than an actual performing of the operation. But the purposes of division may frequently be answered, by these fractional expressions. As they are of the same nature with other vulgar fractions, they may be added, subtracted, multiplied, &c. See the next section.
122. When the dividend is a compound quantity, the divisor may either be placed under the whole dividend, as in the preceding instances, or it may be repeated under each term, taken separately. There are occasions when it will be convenient to exchange one of these forms of expression for the other.
Thus b + c divided by x, is either (b +c)/x, or b/x + c/x.
For it is evident that half the sum of two or more quantities, is equal to the sum of tlievr halves. And the same principle is applicable to a third, fourth, fifth, or any other portion of the dividend.
So also a - b divided by 2, is either (a - b)/2, or a/2 - b/2.
For half the difference of two quantities is equal to the difference of their halves.
123. If some of the letters in the divisor are in each term of the dividend, the fractional expression may be rendered more simple, by rejecting equal factors from the numerator and denominator.
124. If the divisor is in some of the terms of the dividend, bwt nottin all; those which contain the divisor may be divided as<in Art*' 116, and the others set down in the form of a fraction.
|Divide||dxy + rx - dh||bm + 3y|
|Quot.||dy + r - dh/x||-m - 3y/b|
125. The quotient of any quantity divided by itself or its equal, is obviously a unit.
Cor. If the dividend is greater than the divisor, the quotient must be greater than a unit: But if the dividend is less than the divisor, the quotient must be less than a unit.
1. Divide 18aby + 6abx - 18bbm + 24b, by 6b.
2. Divide (a - 2h).(3m + y).x, by (a - 2h).(3m + y)
3. Divide ax - ry + ad - 4my - 6 + a, by -a.
4. Divide ard - 6a + 2r - hd + 6, by 2ard.
126. From the nature of division it is evident, that the value of the quotient depends both on the divisor and the dividend. With a given divisor, the greater the dividend, the greater the quotient. And with a given dividend, the greater the divisor, the less the quotient. In several of the succeeding parts of algebra, particularly the subjects of fractions, ratios, and proportion, it will be important to be able to determine what change will be produced in the quotient, by increasing or diminishing either the divisor or the dividend.
If the given dividend be 24, and the divisor 6 ; the quotient will be 4. But this same dividend may be supposed to be multiplied or divided by some other number, before it is divided by 6. Or the divisor may be multiplied or divided by some other number, before it is used in dividing 24. In each of these cases, the quotient will be altered.
127. In the first place, if the given divisor is contained in the given dividend a certain number of times, it is obvious that the same divisor is contained,
In double that dividend, twice as many times;
In triple the dividend, thrice as many times, &c.
That is, if the divisor remains the same, multiplying the dividend by any quantity, is, in effect, multiplying the quotient by that quantity.
Thus, if the constant divisor is 6, then 24/6 = 4 the quotient.
Multiplvmg the dividend by 2, 2.24/6 = 2.4
Multiplying by any number n, n.24/6 = n.4.
128. Secondly, if the given divisor is contained in the given dividend a certain number of times, the same divisor is contained,
In half that dividend, half as many times;
In one third of the dividend, one third as many times, &c.
That is, if the divisor remains the same, dividing the dividend by any other quantity, is, in effect, dividing the quotient by that quantity.
Thus 24/6 = 4
Dividing the dividend by 2, (1/2)24/6 = (1/2).4
Dividing by n, (1/n)24/6 = (1/n).4
129. Thirdly, if the given divisor is contained in the given dividend a certain number of times, then, in the same dividend,
Twice that divisor is contained only half as many times;
Three times the divisor is contained one third as many times.
That is, if the dividend remains the same, multiplying the divisor by any quantity, is, ia effect, dividing the quotient by that quantity.
Thus 24/6 = 4
Multiplying the divisor by 2, 24/2.6 = 4/2
Multiplying by n, 24/n.6 = 4/n
130. Lastly, if the given divisor is contained in the given dividend a certain number of times, then, in the same dividend,
Half that divisor is contained twice as many times;
One third of the divisor is contained thrice as many times.
That is, if the dividend remains the same, dividing the divisor by any other quantity, is, in effect, multiplying the quotient by that quantity.
For the method of performing division, when the divisor and dividend are both compound quantities, see one of the following sections.