SECTION VI. FRACTIONS. ART. 96. Algebraic Fractions are similar to vulgar fractions in Arithmetic; they express a part, or parts, of a quantity or a unit. 97. They consist of two parts, the numerator and denominator, the former being written above the line, and the latter below it; and, when taken together, are the terms of the fraction. 98. The denominator shows into how many parts the quantity or unit is divided; and the numerator, how many of these parts are represented by the fraction. 99. A proper fraction is one whose numerator is less than its denominator; as, 100. An improper fraction is one whose numerator is equal to or greater than its denominator; as, 101. A mixed quantity is a whole number or quantity, with a fraction annexed, with the sign either plus or minus; as, 102. A compound fraction is a fraction of a fraction; as, 103. A complex fraction is a fraction having a fraction in its numerator or denominator, or in both: as, 104. The value of a fraction depends on the ratio which the numerator bears to the denominator. 105. The value of a fraction is not changed by multiplying or dividing both numerator and denominator by the same quantity. 106. The greatest common measure of two or more quantities is the largest quantity that will divide all of them without a remainder. 107. The least common multiple of two or more quantities is the least quantity that can be divided by them all without a remainder. 108. A fraction is in its lowest terms when no quantity, excepting a unit, will divide both of its terms. 109. Quantities are said to be prime to one another when their greatest common measure is a unit. 110. Prime factors of quantities are those factors which can be divided by no quantity but themselves or a unit; thus, the prime factors of 35 are 7 and 5. 111. A composite quantity is that produced by multiplying two or more quantities together. 112. A fraction is, in value, equal to the number of times the numerator contains the denominator. 113. A fraction is increased in value either by multiplying its numerator or dividing its denominator. 114. A fraction is diminished in value either by dividing its numerator or multiplying its denominator. CASE I. 115. To find the greatest common measure or divisor of the terms of a fraction. RULE. Arrange the two quantities according to the order of their powers, and divide that which is of the highest dimension by the other, having first cancelled any factor that may be con. tained in all the terms of the divisor, without being common to those of the dividend. Divide this divisor by the remainder, simplified as before, ano so on for each successive remainder, and its preceding divisor till nothing remains; and the last divisor will be the greatest common measure or divisor required. If any of the divisors, in the course of the operation, become negative, they may have their signs changed, or be taken affirmatively, without altering the truth of the result; and, if the first term of a divisor should not be exactly contained in the first term of the dividend, the several terms of the latter may be multiplied by any number or quantity that will render the division complete. EXAMPLES. 1. Find the greatest common measure or divisor of cx+x2)a2c+a2x cx+x2 a2c+a2x2 As x is found in both terms of the divisor, we divide those terms by x before the operation. The greatest common measure of both terms we perceive is c+x; that is, it will divide them both without a remainder. We cancel 2bx in both terms of the second divisor, as it is common to both. As x+b is the last divisor, it is the greatest factor or common measure of the quantities proposed. 3. Required the greatest common divisor of 3a2-2a-1, and 4a3-2a2-3a+1. 3a2-2a-1)4a3-2a2—3a+1(4a 3 12a-6a2-9a+3 12a-8a2-4a 2a2-5a+3)3a2-2a-1 6a2-4a-2(3 6a2-15a+9 11a-11 a-1)2a-5a+3(2a-3 2a2-2a -3a+3 -3a+3. As 11 is common to both terms of the third divisor, it is cancelled; therefore a-1 is the greatest common factor of both quantities. 4. What is the greatest common divisor of x3—a3, and x2-a2? Ans. x-a. 5. What is the greatest common factor of x2-1, and ax+a? Ans. x+1. 6. Required the greatest common factor of y1-x2, and y3— y2x—yx2+x3. Ans. y2-x2. 7. Required the greatest common measure of a3-a2x+ax2 —x3, and aa—x1. Ans. a3-a2x+ax2—x3. 8. Required the greatest common factor of a1-x1, and a3+ Ans. a2+x2. a3x2. CASE II. 116. To reduce fractions to their lowest terms. RULE. Divide the terms of the fraction by the prime factors common to both. Or, divide both terms of the fraction by their greatest common divisor. 117. That fractions after reduction have the same value as before, is evident from the fact that their numerators retain the same ratio to their denominators; for equi-multiples and sub-multiples of any two numbers have the same ratio to each other as the numbers themselves. Letters or numbers common to all the quantities in each term of the fraction may be cancelled. In this operation we find 2ab to be the largest factor in both terms; it, therefore, may be cancelled, and the answer is 2c 3ad 2. Reduce abxy admny to its lowest terms. abxy bx admny dmn Ans. |