118. If the numerator and denominator of a fraction be both multiplied, or both divided by the same quantity, its value will not be altered. For, if we multiply by m the two members of the equality (1), we will have these equivalent results, ma=mbXv.. (3); ...... dividing both by mb, we shall have m being any whole or fractional number whatever. 119. If the fraction is to be multiplied by m, it is the same whether the numerator be multiplied by it, or the denominator divided by it. For, if we divide by b, the two members of the equality (3), we obtain the following, The equality (1) may also be put under the form whence we derive, dividing each side by -b, a m =m Xv 1 b (7). m 120. If a fraction is to be divided by m, it is the same whether the numerator be divided by m, or the denominator multiplied by it. For, from the equality (1), we deduce these dividing the first by b and the second by mb, in order to have denominator b, and that we employ the greater line for separating the numerator from the denominator. 121. If two fractions have a common denominator, their sum will be equal to the sum of their numerators divided by the common denominator. For, let now the two equalities be corresponding to the fractions .... . (13), which have the same denominator; adding the two equalities (12) and (13), we shall have a+a=bv+bv'=b(v+v'); and dividing both members by b, in order to have the sum sought v+v', it becomes Note. In adding the above equalities, the corresponding members are added; that is, the two members on the lefthand side of the sign, are added together, and likewise those on the right. The same thing is to be understood when two equalities are subtracted, multiplied, &c. 122. If two fractions have a common denominator, their difference is equal to the difference of their numerators divided by the common denominator. For, if we subtract the equality (13) from (12), we shall have a—a—bv-bv'—b(v—v′) ; dividing each side by b, and we will obtain 123. Let us suppose that the fractions have different denominators, or that we have the equalities a=b.v, a'=b'.v' ; we will multiply the two members of the first by b', and those of the second by b, an operation which will give ab' bb'v, a'b bb'v'; then adding and subtracting, we have ab'a'b bb'(vv′), the double sign which we read plus or minus, indicating at the same time both addition and subtraction; dividing each side by bb', in order to find the sum and difference sought v±v', we will have from whence we might readily derive the rule for the addition and subtraction of fractions not reduced to the same denominator. 124. It would be without doubt more simple to have recourse to property (4) in order to reduce to the same denominator the fractions but our object is to show, that the principle of equality is sufficient to establish all the doctrine of fractions. 125. We have given the rule for multiplying a fraction by a whole number, which will also answer for the multiplication of a whole number by a fraction. Now, let us suppose that two fractions are to be multiplied by one another. multiplying one by the other, the two products will be equal; thus, aa' bb'. vv', = and dividing each side by bb', in order to have the product sought vv', we will obtain Therefore the product of two fractions, is a fraction having for its numerator the product of the numerators, and for its denominator that of the denominators. 126. It now remains to show how a whole number is to be divided by a fraction; and also, how one fraction is to be divided by another. Let, in the first case, the two equalities be m=m; a=b. v;. if we divide one by the other, the two quotients will be equal, that is, and multiplying both sides by b, in order to have the expres Therefore, to divide a whole number by a fraction, we must multiply the whole number by the reciprocal of the fraction, or which is the same, by the fraction inverted. Let, in the second case, the two equalities be a=b. v, a'=b' . v' ; if the first equality be divided by the second, we shall have multiplying each side by b' and dividing by b, for the purpose Therefore, to divide one fraction by another, we must multiply the fractional dividend by the reciprocal of the fractional divisor, or which is the same, by the fractional divisor inverted. 127. These properties and rules should still take place in case that a and b would represent any polynomials whatever. 1 ad According to the transformation a-d. demonstrated (Art. 86), we can change a quantity from a fractional form to that of an integral one, and reciprocally. So that, we have tity may In like manner any quan 62 d2 a2b2d2 be transferred from the numerator to the denominator, and reciprocally, by changing the sign of its index : bc-2 c-2 128. If the signs of both the numerator and denominator of a frac tion be changed, its value will not be altered. Which appears evident from the Division of algebraic quantities having like or unlike signs. Also, if a fraction_have the negative sign before it, the value of the fraction will not be altered by making the numerator only negative, or by changing the signs of all its terms. c+d c+d c+d And, in like manner, the value of a fraction having a negative sign before it, will not be altered by making the denominator only negative: Thus, a-b -ba-b 129. Note. It may be observed, that if the numerator be equal to the denominator, the fraction is equal to unity; thus, if a=b, then a a b a 1: Also, if a is >b, the fraction is great er than unity; and in each of these two cases it is called an improper fraction: But if a is <b, then the fraction is less than unity, and in this case, it is called a proper fraction. § II. Method of finding the Greatest Common Divisor of two or more Quantities. 130. The greatest common divisor of two or more quantities, is the greatest quantity which divides each of them exactly. Thus, the greatest common divisor of the quantities 16ab2, 12a2bc and 4abc2, is 4ab. 131. If one quantity measure two others, it will also measure their sum or difference. Let c measure a by the units in m, and b by the units in n, then a=mc, and b=nc; therefore, a+b=mc+nc=(m+n)c; and a-b-mc-nc=(m-n)c; or a+b=(m±n)c; consequently e measures ab (their sum) by the units in m+n, and a-b (their difference) by the units in m- n. 132. Let a and b be any two numbers or quantities, whereof a is the greater; and let p quotient of a divided by b, and cremainder; 9= quotient of b divided by c, and d= remainder; r quotient of c divided by d, and the remainder =0; thus, pb c) b (p. qc d) c (r Then, since in each case the divisor multiplied by the quotient plus the remainder is equal to the dividend; we have c=rd, hence gc=qrd (Art. 50); 0 |