other forms, that is to say, to expressions whose denominators are equal. Of this we shall treat in the following chapter. 93. We conclude the present by remarking, that all integers may also be represented by fractions. For example, 6 is the same as, because 6 divided by 1 makes 6; and we may, in the same manner, express the number 6 by the fractions 12, 18, 24, 36, and an infinite number of others, which have the same value. CHAPTER IX. Of the Addition and Subtraction of Fractions. 94. WHEN fractions have equal denominators, there is no difficulty in adding and subtracting them; for is equal to, and is equal to . In this case, either for addition or subtraction, we alter only the numerators, and place the common denominator under the line; thus, 100 15 10% +1% is equal to ; 4-3 100 9 100 +31 + is equal to 36, or 18; 10 2-1+1 is equal to 25 16 20 16, or ; also + is equal to 3, or 1, that is to say, an integer; is equal to, that is to say, nothing, or 0. and nalor. 95. But when fractions have not equal denominators, we can always change them into other fractions that have the same denomiFor example, when it is proposed to add together the fractions and, we must consider that is the same as 3, and that 14 is equivalent to; we have therefore, instead of the two fractions proposed, these +, the sum of which is §. If the two fractions 3 were united by the sign minus, as, we should have or . 14 5 Another example: let the fractions proposed be + §; since 2 is the same as, this value may be substituted for it, and we may say+makes 11, or 13. Suppose further, that the sum of that it is; formakes, and and were required. I say makes. 3 96. We may have a greater number of fractions to be reduced to a common denominator; for example, 1, 4, 4, 4, §; in this case the whole depends on finding a number which may be divisible by all the denominators of these fractions. In this instance 60 is the number which has that property, and which consequently becomes. the common denominator. We shall therefore have instead of ; instead of ; 45 instead of ; instead of ; and instead of. If now it be required to add together all these fractions 38, 48, 45, 48, and 9, we have only to add all the numerators, and under the sum place the common denominator 60; that is to say, we shall have, or three integers, and 33, or 311. 40 97. The whole of this operation consists, as we before stated, in changing two fractions, whose denominators are unequal, into two others, whose denominators are equal. In order therefore to perform it generally, let and be the fractions proposed. First, mul tiply the two terms of the first fraction by d, we shall have the fracad a tion equal to next multiply the two terms of the second fracb d b ; tion by b, and we shall have an equivalent value of it expressed by bc b d ; thus the two denominators become equal. Now if the sum of the two proposed fractions be required, we may immediately answer that it is ad-bc b d ad + b c bd ; and if their difference be asked, we say that it is If the fractions and, for example, were proposed, we should obtain in their stead 45 and 9; of which the sum is 101 , and the difference 1. 98. To this part of the subject belongs also the question, of two proposed fractions, which is the greater or the less; for, to resolve this, we have only to reduce the two fractions to the same denominator. Let us take, for example, the two fractions and when reduced to the same denominator, the first becomes, and the second, and it is evident that the second, or, is the greater, and exceeds the former by Again, let the two fractions and be proposed. We shall have to substitute for them 24 and 2; whence we may conclude that § exceeds, but only by 99. When it is required to subtract a fraction from an integer, it is sufficient to change one of the units of that integer into a fraction having the same denominator as the fraction to be subtracted; in the rest of the operation there is no difficulty. If it be required, ៖ for example, to subtract from 1, we write instead of 1, and say, that taken from leaves the remainder. So subtracted from 1, leaves. If it were required to subtract from 2, we should write 1 and instead of 2, and we should immediately see that after the subtraction there must remain 11. 100. It happens also sometimes, that having added two or more fractions together, we obtain more than an integer; that is to say, a numerator greater than the denominator: this is a case which has already occurred, and deserves attention. We found, for example, article 96, that the sum of the five fractions,,,, and 5, was 23, and we remarked that the value of this sum was 3 integers and 33, or 1. Likewise + 3, or 11⁄2 + 1o1⁄2 makes, or 1. We have only to perform the actual division of the numerator by the denominator, to see how many integers there are for the quotient, and to set down the remainder. Nearly the same must be done to add together numbers compounded of integers and fractions; we first add the fractions, and if their sum produces one or more integers, these are added to the other integers. Let it be proposed, for example, to add 34 and 2; we first take the sum of and, or of and . It is for 1; then the sum total is 61. CHAPTER X. Of the Multiplication and Division of Fractions. - 101. THE rule for the multiplication of a fraction by an integer, or whole number, is to multiply the numerator only by the given number, and not to change the denominator: thus, But, instead of this rule, we may use that of dividing the denominator by the given integer; and this is preferable, when it can be used, because it shortens the operation. Let it be required, for example, to multiply by 3; if we multiply the numerator by the given integer we obtain 2, which product we must reduce to. But if we do not change the numerator, and divide the denominator by the integer, we find immediately, or 23 for the given product. Likewise multiplied by 6 gives 3, or 31. 102. In general, therefore, the product of the multiplication of a fraction by c is a ac and it and it may be remarked, when the integer is exactly equal to the denominator, that the product must be equal to the numerator. a And in general, if we multiply the fraction by the number b, the a product must be a, as we have already shown; for since expresses the quotient resulting from the division of the dividend a by the divisor b, and since it has been demonstrated that the quotient multiplied by the divisor will give the dividend, it is evident that multiplied by b must produce a. 103. We have shown how a fraction is to be multiplied by an integer; let us now consider also how a fraction is to be divided by an integer; this inquiry is necessary before we proceed to the multiplication of fractions by fractions. It is evident, if I have to divide the fraction by 2, that the result must be ; and that the quotient of divided by 3 is 4. The rule therefore is, to divide the numerator by the integer without changing the denominator. Thus, 12 divided by 2 gives; 104. This rule may be easily practised, provided the numerator be divisible by the number proposed; but very often it is not it must therefore be observed that a fraction may be transformed into an infinite number of other expressions, and in that number there must be some by which the numerator might be divided by the given integer. If it were required, for example, to divide by 2, we should change the fraction into, and then dividing the numerator by 2, we should immediately have for the quotient sought. In general, if it be proposed to divide the fraction by c, we change ac a it into and then dividing the numerator a c by c, write for the b c' quotient sought. a bc 105. When therefore a fraction is to be divided by an integer c, we have only to multiply the denominator by that number, and leave the numerator as it is. Thus divided by 3 gives 24, and & divided by 5 gives 3. 9 This operation becomes easier when the numerator itself is divisible by the integer, as we have supposed in article 103. For example, divided by 3 would give, according to our last rule, ; but by the first rule, which is applicable here, we obtain, an expression equivalent to, but more simple. a 106. We shall now be able to understand how one fraction b may be multiplied by another fraction. We have only to con means that c is divided by d; and on this principle, we shall first multiply the fraction by c, which produces the result ī after which we shall divide by d, which gives ac b ď ac Ъ Hence the following rule for multiplying fractions; multiply separately the numerators and the denominators. Thus by gives the product, or ; 107. It remains to show how one fraction may be divided by another. We remark first, that if the two fractions have the same number for a denominator, the division takes place only with respect to the numerators; for it is evident, that is contained as many times in as 3 in 9, that is to say, thrice; and in the same manner, in order to divide by, we have only to divide 8 by 9, which gives. We shall also have% in 1, 3 times in A 7 |