reduced to a single whole number. Consequently, if unity is considered to be divided into as many equal parts as the numerical value of the denominator contains simple units, and the numerator to express as many of these parts as its numerical value contains simple units, it is evident that the elementary operations which it may be necessary to perform with literal fractions may be performed by the rules for arithmetical fractions, attention being given to the conventions established in Articles 3 and 4. However, as a letter may express an arithmetical fraction, the numerical values of the terms of an algebraic fraction may, either or both, be arithmetical fractions. Since the rules for performing the elementary operations of arithmetic with fractional expressions are demonstrated for those fractions only which have whole numbers for their terms, it seems necessary to investigate the rules anew, regard being had to this condition. It has been proved (Art. 25) that if the terms of any fraction are both multiplied or both divided by the same number, the value of the fraction is not changed; and this principle has been applied to the reduction of fractions to the lowest terms (Art. 26). It supplies the means also of reducing fractions which have different denominators to equivalent fractions having a common denominator; for it is evident that the product of all the denominators of any given fractions is a multiple of all the denominators of these fractions; That this multiple can be divided by each denominator; That if it is divided by the denominator of the first fraction, and the terms of this fraction are multiplied by the quotient, the result is an equivalent fraction having the product of all the denominators for its denominator; That if the multiple of all the denominators is divided by the denominator of the second fraction, and the terms of this fraction are multiplied by the quotient, the result is a fraction equivalent to the second fraction, and having the product of all the denominators for its denominator; And that in this manner any number of fractional expressions may be reduced to the same denominator. Whence, to reduce fractions which have different denominators to equivalent fractions having the same denominator, Rule. Multiply the terms of each fraction by the denominators of all the other fractions; the results are equivalent fractions reduced to the same denominator. Examples of the reduction of fractions having different denominators to equivalent fractions having a common denominator : a. If the factors (monomial and polynomial) which are common to two or more of the denominators, and all the factors which are found in one only of the denominators, are each taken once only and multiplied together, the product is a multiple of all the denominators, and ..Ans. bm2nx2-2bx2. .....Ans. less (unless all the denominators are prime to each other) than the product of all the denominators. The monomial factors are easily found by inspection of the quantities; the polynomial can be obtained by seeking the greatest common measure of all the denominators, taken two and two. But this is a very laborious process if the number of fractions is considerable. ADDITION OF FRACTIONAL EXPRESSIONS. 31. It is a self-evident principle that if two quantities are equal, and both are operated on in the same manner, the results are equal. Let, therefore, a=bv, a'=bv', (a, b,v, a', b', v', being any quantities, whole or fractional, monomial or polynomial,) and let both members of these equations be divided by b, which has no factor common to it and a or d. a ď b' b' =v. Whence v, v represent the values of the fractions or the quotients resulting from the division of a, a' by b. Whence, to obtain the sum of two fractions which have the same denominator, Rule. Form the algebraic sum of the numerators, and divide it by the common denominator. Examples of the addition of algebraic fractions having the same Suppose that the fractions have different denominators, and that a a or that a=bv, d'=b'v'. a bb' =v+o. a But =v, z=v; wherefore +==+d, and consequently += ab+a'b bb' Whence, to find the sum of two fractions which have different denominators, Rule. Reduce the fractions to a common denominator; form the algebraic sum of the numerators, and divide it by the common denominator. Examples of the addition of algebraic fractions which have different denominators: a с d a+cbr+db2r ?...... ......Ans. bn 10th. En+bn-r+bn-2r SUBTRACTION OF FRACTIONAL EXPRESSIONS. a α' 32. The difference of the expressions =v and v′ is indicated by the Whence, to obtain the difference of two fractions which have a common denominator, Rule. Subtract the numerator of the subtrahend from the numerator of the minuend, and divide the remainder by the common denominator. Examples of the subtraction of algebraic fractions which have the same denominator : a Let the fractions whose difference is required be =v; v', which have different denominators. Consequently, to find the difference of two fractions which have not the same denominator, Rule. Having reduced the fractions to the same denominator, subtract the numerator of the subtrahend from the numerator of the minuend, and divide the remainder by the common denominator. Examples of the subtraction of algebraic fractions which have different denominators : 33. The sum or difference of an integer and a fraction can be obtained by means of the preceding rules. Example 1st. Required the sum of a b and ac-bc2 ? a2b=a2b_a2b_ad__a3bd = ac-bc2__a3bd ac—bc2_a3bd+(ac—be2) ___a3bd+ac—be2 ...a2b+ = ad + = = ad ad ad ad This process is analogous to that in arithmetic (Part I. Art. 159) for the reduction of a mixed number to an improper fraction. and ar2. bx+cd-ab ac-bc2 Regarding ab+ ad ac3 as expressions which it is required to reduce to the form of improper fractions, it is evident that the reduction is made by multiplying the integer by the denominator of the fraction; adding the numerator of the fraction to the product, if the integer and fraction are connected by the sign+; subtracting the numerator of the fraction from the product, if the integer and fraction are connected by the sign; and giving this sum or difference the denominator of the fraction for its denominator. A fraction expresses a quotient; therefore a fraction which has the negative sign may be considered as the quotient of a dividend and divisor which are of contrary signs. In the processes of reducing to a common denominator, and adding or subtracting, the negative sign is considered to affect the numerator only. MULTIPLICATION OF FRACTIONAL EXPRESSIONS. a 34. Employing still the notation of Article 31, letv, v. Consequently, the product of two fractions is a fraction whose numerator is the product of the numerators, and its denominator the product of the denominators, of the given fractions. |