proper sign to the numerator; place the denominator underneath, and the equivalent improper fraction will be exhibited. For let a be the integral quantity to which any fraction is joined, furnishing the mixed quantity a+2: if a be multiplied by q, and the product, aq, connected to the numerator, we shall have aq+p; and if the denominator, q, be placed under this, there results the fraction aq+P; and that this is equiva q lent to the mixed quantity proposed is obvious, since, by actually dividing aq+p by q, the quotient is a+ CONVERSELY: P. Hence, a + 2. To reduce an improper fraction to a mixed quantity: Divide the numerator by the denominator: if there be no remainder, the so-called improper fraction will be an integral quantity in the form of a fraction; it will be exhibited in the quotient: if there be a remainder, the fraction whose numerator is this remainder, and whose denominator is the divisor, must be joined to the quotient, and we shall then have the mixed quantity required. It thus appears that what is called the reduction of an improper fraction to a mixed quantity is nothing more than the actual division of the numerator by the denominator; and examples of this operation are given in sufficient number at pp. 49, 54. The following reductions, however, are to be worked out by the learner, and the results here given shown to be true: in the first example or two, the required change of form may be written down at once, without any formal operation. (13) a—b+x—a(x—b)—b2 __ (a+x)2+(b......-x)2 a+2x a+2x NOTE. The term improper fraction has not exactly the same meaning in algebra as in arithmetic. In the latter science, an improper fraction is exclusively that of which the numerator is not less than the denominator; but an algebraic fraction is said to be an improper fraction whenever the division of the numerator by the denominator can be at least partially executed: we cannot, in general, say whether the numerator of an algebraic fraction is less than the denominator or not; because, so long as the symbols remain uninterpreted, we can seldom infer anything as to arithmetical value. 41. To reduce fractions to a common denominator: RULE.-Multiply each numerator by the product of all the denominators except its own, and the numerators of the changed fractions will be obtained. And the product of all the denominators will be the denominator common to all the changed fractions. For it is plain that, by proceeding in this way, the numerator and denominator of each fraction will be multiplied by the same thing, namely, the product of the denominators of all the other fractions. And since the multiplication of both terms of a fraction, by the same thing, merely changes the form, without disturbing the value of a fraction (page 79), it follows that the new form may be substituted for the old. NOTE. This rule is general, and may be applied at once to any series of fractions; but, like most general rules, it may be dispensed with in particular cases, and be replaced by shorter methods. This may always be done whenever the row of denominators is such that a common multiple of all of them may be found, which is a simpler quantity than the common multiple furnished by their product. A common multiple of a set of quantities is anything which admits of division by each: the product of all is, of course, such a common multiple. And whatever common multiple of the denominators is employed, in the reduction of fractions to a common denominator, we must always multiply the numerator of each fraction by what results from dividing the common multiple by the denominator: the numerators of all the changed fractions will thus be obtained, and the multiple employed will be their common denominator. That the fractions are not changed in value, by these operations, will appear from observing that the numerator and denominator of each one of the fractions are both multiplied by the same thing, namely, by what results from dividing the multiple used by the denominator of the fraction. When no two of the denominators of the proposed fractions have a common factor, then the product of the denominators is the least common multiple. But when common factors enter two or more of the denominators, the repeti tions should be struck out, or the factors which recur suppressed: the product of the denominators, when thus deprived of common factors, will be their least common multiple. [For the proof of these operations, see page 94]. It is necessary, therefore, in order to avoid useless multiplications and undue complexity in our results, that the denominators of the proposed fractions should be examined a little before applying the general rule: if we cannot detect the entrance of the same factor in two or more of them, we then proceed by the rule; but if we do detect the same factor in two or more, we expunge it in all but one, and then take the product of what is left for a common denominator. At the close of the chapter will be found formal directions for the discovery of the least common multiple of a set of quantities; but in the reduction of fractions here treated of, it is seldom thought necessary to follow these directions; if the least common multiple be not discoverable from inspection, the nearest to it we can get is made to serve, or the general rule implicitly followed: for there is frequently more work expended on the determination of the least common multiple, by the directions adverted to, than the advantage to be gained is worth. xyz, (2) Reduce to equivalent ones with a the com.denom. ayz baz, cxy, the equiv. fractions. xyz xyz'xyz' а ax a ху у х to equiv. fractions with a com. denom. Here the factors x, y, in the denominator of the first fraction, are repeated in those of the others: hence, suppressing the latter factors, the least common multiple of the denominators is xy; therefore, xy is to be the common denominator of the equivalent fractions, and the several numerators are found by dividing xy by each denominator, and multiplying the corresponding numerator by the result; so that the multiplier for the first numerator is 1, that for the second, x, and that for the a ax2 ay third, y: hence, the equivalent fractions are xy xy xy to equivalent fractions with a common Here the factor, x, occurs in two denominators, and the factor 2, also occurs in two denominators; therefore, suppressing one x and one 2, the least common multiple is 12aba, the denominator common to each of the sought forms: hence, the multiplier for the first numerator is 4b, that for the second 6a, and that for the third 3abx; so that the equivalent fractions 8b 6a2 3abcx 12abx' 12abx' 12abx are E common denominator. Here, since a-b2=(a+b)(a—b), the factor, a+b, enters two denominators: hence, the least common multiple of the three denominators a+b, a2—b2, 1, is a2-b2; so that the multiplier for the first numerator is a-b, that for the second 1, and that for the third a2-b2; and, consequently, the equiva(a—b)x 4(a2-b2) lent fractions are (5) Reduce y a2b22 a2-12 a2-b2 α b C d, to equivalent fractions with a x+y x- -y x common denominator. Here no two denominators have a common factor: hence, proceeding by the general rule, imagining the denominator 1 under the d, we have ax(x—y), bx(x+y), c(x2—y2), dx(x2—y2), for the numerators; and x(x2—y2) for the common denominator; hence the equivalent fractions are ax(x-y) bx(x+y) c(x2—y3) dx(x2—y3) 'x(x2—y2)'x(x2—y3)’x(x2—y3)' x(x2—y3) EXAMPLES FOR EXERCISE. Reduce the following fractions to equivalent ones having a common denominator. (9) 2(a+x)'3(a—x)'a2+2ax+x22 6(a2—x2)* 87 ADDITION AND SUBTRACTION OF FRACTIONS. 42. Addition.-Reduce the fractions to equivalent ones with a common denominator, and put this common denominator under the sum of the changed numerators. Subtraction.--Reduce the fractions to equivalent ones with a common denominator, and put this common denominator under the difference of the changed numerators. merators of these, the sum of the fractions is 2b(6+a2)+3 3ab2 (2) x+y+x—Y_(x+y)2 + (x—y)' _ 2(x2+y3). + .2 NOTE.-In performing such operations as these, it is not necessary to write the common denominator under each of the changed numerators; it is sufficient to connect these numerators together by the proper signs, to draw a line under the whole, and to write the denominator underneath. Thus the work of the last example may be indicated as follows: x+y xy= x-y 4xy === and this is the plan to be adopted in the examples below. And it is further to be observed, in reference to these examples, that the results of the proposed operations are to be freed from those factors common to numerator and denominator, which may be seen to enter them. These reduced results are given at the end of the book. |