FRACTIONS. DEFINITIONS AND NOTATION. a a 111. We have seen (12) that division may be indicated by writing the dividend and divisor on opposite sides of a horizontal line. The term Fraction, in Algebra, relates to this mode or form of indicating division. Hence, 112. A Fraction is a quotient expressed by writing the dividend above a horizontal line, and the divisor below. Thus is a frac b tion, and is read, a divided by b. 113. The Denominator of the fraction is the quantity below the line, or the divisor. 114. The Numerator is the quantity above the line, or the dividend. 115. Any fraction may be decomposed as follows: 1xa Ő Xa b Hence, 1.-The value of a fraction is equal to the reciprocal of the denominator multiplied by the numerator. 2.-In any fraction, the reciprocal of the denominator may be regarded as a fractional unit; and the numerator shows how many times this unit is taken in the fraction. Hence, 3.-A fraction is a fractional unit or a collection of fractional units, the value of each depending upon the denominator. 116. An Entire Quantity is an algebraic expression which has no fractional part; as x_3xy. 117. A Mixed Quantity is one which has both entire and fractional parts; as a + b GENERAL PRINCIPLES OF FRACTIONS. 118. Sinee a fraction is a form of expressing division, it is evident that all the operations in fractions must be based upon the general relations subsisting between the dividend, divisor, and quotient. These principles relate, first, to change of value; second, to change of sign. 1st. Change of value. 119. By modifying the language of (84), we may express the mutual relations of the numerator and denominator of a fraction, as follows: I. Multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction. II. Multiplying the denominator divides the fraction, and dividing the denominator multiplies the fraction. III. Multiplying or dividing both numerator and denominator by the same quantity, does not alter the value of the fraction. 2d. Change of sign. 120. The Apparent Sign of a fraction is the sign written before the dividing line, to indicate whether the fraction is to be added or subtracted. Thus, in aax" y+ 4a-2.c the apparent sign of the fraction is plus, and indicates that tho fraction is to be added. 121. The Real Sign of a fraction is the sign of its numerical value, .when reduced to a monomial, and shows whether the fraction is essentially a positive or a negative quantity. Thus, in the fraction just given, let a=2 and x=3. Then a’-ax? 4-18 -14 -7 40-2x 8-6 2 The real sign of this fraction therefore is minus, though its apparent sign is plus. 122. Each term in the numerator and denominator of a fraction has its own particular sign, distinct from the real or apparent sign of the fraction. Now the essential sign of any entire quantity is changed, by changing the signs of all its terms. Hence, I. Changing all the signs of either numerator or denominator, changes the real sign of the fraction; (85, I). II. Changing all the signs of both numerator and denominator, does not alter the real sign of the fraction; (85, II). III. Changing the apparent sign of the fraction, changes the real sign. REDUCTION. 123. The Reduction of a fraction is the operation of changing its form without altering its value. OASE I. 124. To reduce a fraction to its lowest terms. A fraction is in its lowest terms, when the numerator and denominator are prime to each other. And since it does not alter the value of a fraction to suppress the same factor in both numerator and denominator, (119, III), we have the following RULE. I. Resolve the numerator and denominator into their prime factors, and cancel all those factors which are common Or, II. Divide both numerator and denominator by their greatest common divisor. EXAMPLES FOR PRACTICE. a_1 1. Reduce to its lowest terms. Ans. 3a-2a-1 2. Reduce to its lowest terms. 4a-2a-3a+1 The greatest common divisor of the numerator and denominator, as found by (105), is a-1; hence, (3a-2a-1)=(a-1)=3a+1 (4a-2a-3a+1)+(a-1)=4a+2a-1 And we have for the reduced fraction, 3a+1 Ans. 4a’+2a-1' Reduce each of the following fractions to its lowest terms : 7x'yz X 3. Ans. 21xy®z 3y? 2-1 4. Ans. xyty y a'-ab' a'-ab 5. Ans. a’+2ab+7 att x2 62x2 co 6. Ans. x* +63 OASE II. 125. To reduce a fraction to an entire or mixed quantity. The division indicated by a fraction may be at least partially performed, when there is any term in the numerator whose literal part is exactly divisible by some term in the denominator. Hence, RULE I. Divide the numerator by the denominator as far as possible, and the quotient will be the entire quantity. II. Write the remainder over the denominator, annex the fraction thus formed to the entire part, with its proper sign, and the whole result will be the mixed quantity. EXAMPLES FOR PRACTICE. Reduce the following fractions to entire or mixed quantities : ab+x 1. b 2. a'+ba bx Ans. at ở Ans. at a a 3. 5ay+ab+y y 2–2 ab Ans. 5a+1+ y 4. Ans. 23*+xy+y'). 5. 6. +3 7. 3x*_12ax—9x+y y Ans. -—40–3+ 3х Ans. 3—28x 4.x 3.0—7x*+7+30 3x-10 Ans. 3x+5+ x*—4x+8 X4x+8 56x*+126x—140 2 Ans. 8x--6- 2yo XY 8. x+3 9. |